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This research was supported in part by the National Institutes of
Health, Institute of General Medical Sciences, Grant No. 12868-04,
and partially by the National Science Foundation, Grant No. GP-5790.
1-)
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ASYMPTOTICALLY OPTIMAL NONPARAMETRIC
TESTS FOR MISCELLANEOUS PROBLEMS
OF LINEAR REGRESSION
by
Malay Ghosh
Department of Statistics
University of North Carolina at Chapel Hill
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:-1.
•-
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Institute of Statistics Mimeo Series No. 634
JULY 1969
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TABLE OF CONTENTS
CHAPTER
Page
ACKNOWLEDGEMENTS
ABBREVIATIONS AND NOTATIONS
ABSTRACT
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II
iv
vi
viii
INTRODUCTION AND SUMMARY
1
1.1.
1.2.
1.3.
1
6
A Review of the Literature
A Summary of the Results in Chapters II-VI
Some Definitions and well-known Results
10
RANK ORDER TESTS FOR REGRESSION WITH PARTIALLY
INFORMED STOCHASTIC PREDICTORS
14
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2.1.
2.2.
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2.3.
2.4.
2.5.
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III
Introduction, Notations and Assumptions
A class of conditionally distribution-free
rank-order tests
Asymptotic permutation distribution of M
Asymptotic non-null distribution of MV v
Asymptotic optimality of tests and A~R.E.
NONPARAMETRIC TESTS FOR GROUPED DATA IN THE UNIVARIATE
LINEAR REGRESSION MODEL
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
IV
18
22
31
46
Introduction, Notations and Assumptions
Asymptotically Optimal Parametric Test
Asymptotically Optimal Nonparametric Test
Extension to the case of a countable set of
class intervals
Asymptotic Relative Efficiency (ARE)
Applications
Concluding Remarks
54
54
57
73
86
89
95
98
ON TESTING THE IDENTITY OF MULTIPLE REGRESSION
SURFACES FOR GROUPED DATA
100
4.1.
4.2.
100
103
107
115
4.3.
4.4.
Notations and Assumptions
Asymptotically Optimal Parametric Test
A Class of Nonparametric Tests
An Important Application
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CHAPTER
v
Page
NONPARAMETRIC TESTS FOR GROUPED DATA IN THE
MULTIVARIATE LINEAR REGRESSION MODEL
123
5.1.
5.2.
5.3.
123
127
139
5.4.
5.5.
VI
Introduction, Notations and Assumptions
A Class of Parametric Tests
A Class of Nonparametric Tests
A Multivariate Generalization of tests in
Chapter IV
Concluding Remarks and Scope for further
Research
148
162
A NUMERICAL EXAMPLE
162
6.1.
6.2.
165
Introduction and the Data
Calculation of actual test statistics
and analysis of the data
REFERENCES
169
174
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ACKNOWLEDGEMENTS
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It is a pleasure and privilege to acknowledge my sincerest
gratitude to Professor P. K. Sen, my adviser, for proposing the
problems in this dissertation and making numerous suggestions and
comments during its progress.
I wish also to thank Professor H. A. David, Professor Wassily
Hoeffding and Professor N. L. Johnson for going through the manuscript
and offering many helpful suggestions.
Thanks are also due to
Professor G. E. Simmons for some useful discussions and Professor
W. H. Graves, the other member of my doctoral committee.
I wish to acknowledge the Department of Statistics for
financial assistance during the first two years of my graduate
program iind the Department of Biostatistics and the National Institutes
of Health for financial assis tance in the third and final year of my
graduate studies.
I wish also to thank Professor James E. Grizzle of the
Department of Biostatistics for his never-failing help during my
graduate studies and Professor H. K. Nandi of the University of
Calcutta for stimulating my interest in Mathematical Statistics.
I also owe a deep sense of gratitude to my parents and grandparents
for their active co-operation and enthusiasm, but for which I would
not have reached this stage of my academic career.
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Finally, thankful appreciation is due to Mrs. Bobbie Wallick
and Mrs. Mary Flanagan for their careful and patient typing of the
manuscript.
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ABBREVIATIONS AND NOTATIONS
Abbr~vi,ations
r.v.
random variab le
d.f.
distribution function
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i.i.d.
independent and identically distributed
i,.i.d.r.v.
independent and identically distributed random variables
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p.d.
positive definite
a.e.
almost everywhere
U.M.P.
uniformly most powerful
C.L.T.
Central Limit Theorem
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Notations
N (1I,L:)
p variate normal with mean vector Mand dispersion
P ..::. ""
matrix k,
Central chi-square with p degrees of freedom
Noncentral chi-square.withp degrees of freedom
and non-centrality parameter n
a'"" b (b
\)
\)
;t
\)
0)
a /b
\)
\)
-+
1 as
\)-+<>0
identity matrix
of order p
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p
(unless otherwise mentioned)
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Notations (cant.)
all ••• a Iq
............
a
pI
••• a
pq
==
A= «a ij ))
pxq
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ABSTRACT
The following four problems are considered in this dissertation.
The first problem concerns testing the hypothesis of no
regression in a multiple (linear\) regression model with stochastic
predictors.
A class of permutationally(conditionally) distribution-
free tests for the above problem has been proposed and its asymptotic
properties together with asymptotic relative efficiency results have
been studied.
The second problem deals with testing the hypothesis of no
regression in a multiple (linear) regression model with non-stochastic
predictors, but with grouped data.
tests, their asymptotic properties and some specific applications of
such tests have been studied.
The third problem deals with the same model, but we are
interested in testing that the intercept is zero along with the
hypothesis of no regression.
Tests analogous to 'signed-rank tests'
have been used here with an interesting application to the problem
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of 'paired comparison.'
I--_
multivariate set up.
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Permutationally distribution-free
Finally, the above findings have been generalized to a
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CHAPTER I
INTRODUCTION AND SUMMARY
A Review of the Literature.
1.1.
Hajek (1962) considered the following problem:
Having observed
X. = a + Bc. + O€., we test the hypothesis
111
HO : B =
(1.1.1)
° against the
alternatives B > 0,
where a and 0(>0) are nuisance parameters, B the parameter under test,
c i are known constants and
€I' €2""
are i.i.d.r.v.
A class of
genuinely distribution-free rank-statistics has been proposed by
Hajek for the abo"le testing problem.
The asymptotic approach of Hajek
consists in regarding the given testing problem as a member of a
sequence {H o ' H ; v ; : : l} of similar testing problems. In this
v
v
.
. h
sequence, the "th
v
test1ng
probl em concerns..N 0 b servat i ons W1t
v
N
+
V
H
ov
00
as v
+
= Ho(Nv )'
As a rule, H
depends on v through N only, i.e.,
Ov
v
00.
(e.g. we might assume HOv
= Ho '
Ho being applied to
N = N observations), while H depends on some parameters c .
v
V
V1
(i = 1, 2, ••. , N )in addition.
v
More specifically, he has
considered a sequence of random vectors
the X . 's are independent and
V1
(1.1.2)
i
Pr(X .
V1
= 1, 2, ••. , Nv ,
~
x)
v ; : : 1.
=
F .(x)
V1
~v = (XVI"'"
XVN )', where
v
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2
In the above, F is a known distribution function,
c . (i
V1
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= 1,
2, ••• , N ) are known constants, a and 0(> 0) are nuisance
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parameters and B is the parameter under test.
(1.1.1) has been considered.
The asymptotic theory has been studied
by using the "contiguity" lemmas of LeCam (see e.g. [19] or [25];
see section 1. 3 for definition of contiguity).
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It
has been shown
that the test one would have obtained by using the Fundamental
Lemma of Neyman and Pearson (see e. g. [26], p. 65) is asymptotically
power-equivalent to Hajek's rank order tests under certain assumptions
on the c\\i' sand F.
Assume that
the density f(x)
(1.1. 3)
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The testing problem
= F'(x) exists and
Let the constants c . (i = 1, 2, ••• , N ) satisfy a condition (known
V1
v
in the literature as Noether condition, and, henceforth,
referred to as such)
~::[,~~~"
where
cv
-1
== N
v
v
L c .•
i=l V1
(c . - C
V1
V
)2 !~v
(c . _ C
• 1
1=
V1
)2]
=
0,
Further the cvi's satisfy the boundedness
condition
N
(1.1.5)
V
sup LV (c . - C )2
V1
v
v i=l
<
00
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-1
Let F
denote the inverse of F and let
ep(u)
(1.1.6)
f
=
I
(F- 1 (u))
O<u<1.
f(F-1Cu))
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.
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oep2 eu) du
Then,
I:00[f'(X)/f(X)]2f(X)dX < 00.
=
We call ep a "score" function.
Now, let R . be the rank of X . in the ordered sample
V1
VV1
~
•••
~
V1
VVN ' i.e.
v
(1.1. 7)
V
vR.
with probability 1, 1
~
i
~
N , v
V
V1
Further, let av(i), 1
~
i
~
~
1.
N be some arbitrary scores
v
satisfying
lim Jl ravel + [uN ]) - ep(u)]2 du
v
OO 0
(1.1.8)
=
0,
V-+
[x] being the largest integer less than or equal to x.
Let the rank-
order statistics Sv be expressed as
(1.1.9)
Let e:
v
=
N
L: v
i=l
(c . -
v1
c )a
v
V
(R .) ,
V1
V
~ 1.
and Q «(3, 0') be the size and power of test based on critical
v
region
(1.1.10)
S
v
> K d , v
e: v
~
1,
where Ke: is the upper 100e:% point of a standard normal variable and
(1.1.11)
~v
i=l
(c . - C )2 IOO_oo[f ' (x)/f(x)]2 f (x)dx.
V1
v
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The main theorem of Hajek is as follows:
THEOREM 1.1.
Under the model (1.1.2) and under the assumptions
(1.1.3) - (1.1.8), the critical region (1.1.10) provides an asymptotically UMP test of limiting size
power equals
1 - q, (KE -~,
d.)
0'
V
-
and K and d
--E--V
'
for (1.1.1), and the asymptotic
E
where , <p(x)
=
r;;- .
(V27T)
-Ix
t2
! -ooexp. (- T)dt,
are defined in (1.1.10) and (1.1.11) respectively.
If however, the true d.f. is G instead of F (where G satisfies
similar regularity conditions as F), Hajek's test is no longer
asymptotically UMP, but the A.R.E. of the proposed test relative to
the UMP test has been obtained.
It turns out to be the square of the
correlation coefficient between "true score" and the "assumed score"
i.e.
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where \}J(u)
g(x)
=
- g' (G- I (u))
g(G-I(u))
o<
u < 1,
G'(x), J:00[g'(X)/g(X)]2 g(x)dx
='
<
00.
The notation X for 'dependent' variable (i.e. 'predictand')
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has been used all through except in Chapter II, where the c . are
v~
stochastic.
We have used the notation Y for 'dependent' variable in
that chapter and
XIS
for 'independent' variables, (i.e. 'predictors').
A situation likely to arise in practice is when X is not
"'\)
observable, but we have a finite (or a countable) set of c1assintervals
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(1.1.12)
a
J
< x ~ a
j
j
+1 ,
j = ••• , -2,1, 0, 1, 2, .••
[where -00 <.•• <a_ 1 <a <a 1 < ••• < 00 is any (finite or countable) set of
O
ordered points on the (extended) real line
stochastic vector is X*
00]], and the observable
h
* ••• , X*(XvI'
vN- .)' ,were
=
~v
[~oo,
v
(1.1.13)
00
X*.
L: I. Z ..
j =-00 J
v J.J
VJ.
and
(1.1.14)
(i
Z
ifX.sI.
v J.
J
••
v J.J
1,2, ... , N,
v
v::O:
1;
otherwise ,
j
... , -2, -1,0,1,2, ... ) .
The model, the testing problem and the assumptions are the
same as before.
In practice, we may very often come across situations when
data are collected in several groups or ordered categories, thus,
rendering any analysis suitable for ungrouped data, impossible.
Even in studying the height distribution of a large number of individuals, it
is very often convenient to record the observations
classified into several groups.
Another example may be, when we want to compare the efficacies
of several contraceptive devices.
A number of couples are studied
for a certain length of time, and the distributions of the conceptions
are compared.
studied.
By nature, the exact time of conception cannot be
Rather, the cycle in which the conception tak~s place, can
be recorded, though the actual distribution of the conceptions can
be taken to be absolutely continuous.
Hence, the data can be
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recorded only in terms of intervals of approximate 4-weekly cyclelengths.
In the case of grouped data, Sen (1967) has obtained a class
of permutationally (conditionally) distribution-free tests for the
testing problem in (1.1.1).
He has also obtained a class of asymptoti-
cally UMP parametric tests under the same conditions as of Hajek
(1962) and has established the asymptotic power equivalence of the two
kinds of tests under "contiguous" alternatives.
Sen has also obtained
an A.R.E. expression similar to that of Hajek when the true d.f. is G
instead of F.
He has also shown that the loss of efficiency due to
grouping can be made arbitrarily small provided the probabilities of
observations belonging to class intervals {I.} can be made arbitrarily
J
small.
1.2.
Throughout our investigation, we have taken N = v, v
v
~
1.
A Summary of the Results in Chapters II-VI.
The results of Hajek and Sen, described in the earlier section,
have been extended in several directions.
In Chapter II, we have
extended the findings of Hajek to the multiple (linear) regression
model with stochastic predictors including the situations where the
predictors carry only partial information, and, may be referred to as
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partially informed stochastic _predictors.
the general linear model Y
vi
=
I
For example, we may have
(30 + (3 I Xl V1. +
...
+ (3 X 1. + os 1.,
P pv
v
i
=
1, 2, .•. , v, v
~
1, where the observations (Y., Xlvi"'"
V1
X .),
pv 1
i
=
1, 2, •• " , v , v
~
1 may not be available, but only the ranks on
these are available, ranking being done independently within each
Y or X '
kv
v
We want to test
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(1. 2 .1)
(Sl"'"
against the alternatives
~ ~
Q.
S)'
p
=
Q
A class of permutationally (condi-
tionally) distribution-free tests has been obtained for the above
problem which is unconditionally distribution-free for the particular
case p = 1.
This class of tests is asymptotically optimal in the
sense of Wald [34].
The A.R.E. results are given and in the particular
case p = 1, the expression foruthe
A~R.E.
turns out to be the product
of the squared correlation.coefficients between the "true score" and
"assumed score" for each of Y and X.
In Chapter III, the results of Sen have been generalized to
the multiple linear regression model.
have worked with grouped data.
From this chapter onwards, we
Sen's model is s simple linear
regression model where the hypothesis of no regression has been tested
against two-sided alternatives.
Unlike his case, since we have here
two-sided alternatives, we have not been able to get a liMP test, but
a test which is an "asymptotically most stringent test"; (for definition, see Wald [34]).
This has been achieved by showing the
"asymptotic equivalence" (definition given in Section 1.3) of the
proposed test statistic to the likelihood ratio test statistic,
(see [34] or [35]).
Consequently, other optimal properties of the likelihood
ratio test as proved by Wald
[34]~
hold for our test as well.
Regarding the mathematical tools used in this chqter, there
is a basic difference from that of Sen.
In proving many of the
results in this chapter, we have avoided the "contiguity" argument.
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The reason behind this is to extend the results to the multivariate
case, where "contiguity" of the. sequence of probability measures
corresponding to the alternatives to the corresponding sequence for
the null hypothesis, at least, does not seem apparent.
A corollary
to the lemma I of LeCam (see e.g._[l9J, p. 204) gives a sufficient
condition under which "contiguity"_holds.
general in the multivariate case.
This does not hold in
We are not aware of any condition
ensuring "contiguity" in the multivariate situation.
In Sen's case,
however, "contiguity" has been used as a major tool.
Further, Sen
has considered a countable.set.of.class,-intervals, whereas, we have
for all essential purposes,considered a finite set of classintervals, and ultimately have extended the results to the case of a
countable set of class-intervals. by using the "contiguity" argument.
We have obtained the same efficiency expressions as in the case of
Sen and consequently, his conclusions regarding the same hold in our
case as well.
Our results also include the
K(~2)-sample
problem as
a particular case.
In Chapter IV a similar problem has been considered, the only
difference being that here we are interested in testing for the
intercept along with the regression coefficients.
More specifically,
if we have a linear model,
=
So
+
Slc1V-t
.L
+...+ SP c pVJ.. +
OE:
VJ.• (i
1, •. "
v; v
~ 1)
,
the c . 's being known constants, we have tested
VJ.
(0, .•• , 0) in Chapter III against all possible
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alternatives, whereas we have tested
Ho : «(30 ' S1 ' ••• , (3p)
tives in Chapter IV.
(0, 0, ••• , 0) against all possible alterna-
The parametric test procedures for the two
problems are exactly the same, but the permutation test procedures
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vary in the two cases, since in Chapter III, statistics analogous to
rank statistics have been considered, whereas in ChapterIV statistics
analogous to signed-rank statistics have been considered.
A detailed
explanation of these has been given in Chapter IV.
As an interesting application of the test procedure considered
in this chapter, we have been able to find a test for the pairedcomparison problem as considered by Mehra and Puri [28].
It has been
shown that the two tests are asymptotically power-equivalent when
each pair of treatments is compared the same number of times.
But
if that is not so (and, obviously, the number of treatments compared
is
~
3), the two tests may not be asymptotically power-equivalent.
A counterexample has been obtained to this effect, and, in fact, in
the particular example considered, our test is asymptotically more
powerful than that of Mehra and Puri.
This shows that the test
considered by Mehra and Puri (which seems rather arbitrary apart
from the fact that the test statistic is asymptotically a chi-square)
need not be asymptotically optimal in the sense of Wald [34], whereas
our statistic can claim that property.
Moreover, it seems that our
statistic is computationally simpler than its rival.
The findings of Chapter III and IV have been extended to the
multivariate case in Chapter V.
Our attempts seem to be the first
in this respect, since very little seems to have been said or done
regarding the general linear regression problem for multivariate
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grouped data even under the normal theory set-up.
We may, however,
remark that unlike the univariate case, the asymptotic optimality in
the sense of Wald cannot be claimed in the multivariate case in a
quite general sense.
In Chapter VI, a specific example has been considered.
This,
however, illustrates the test procedure developed in Chapter III,
although, as we shall see later on in Chapter V, the multivariate
extension of such tests is quite simple, even in practice.
A general manner of presentation which has been followed in
Chapters III-V is this:
In section 1, We have introduced the
problem and have given the basic notations and assumptions.
In
section 2, we have derived a class of parametric tests and have
established the asymptotic optimality (if any) of such tests.
In
section 3, we have developed a class of permutationally distributionfree non-parametric tests.
The remaining few sections deal with
miscellaneous remarks, derivation of the A.R.E.
results and
applications to specific cases.
1.3.
Some Definitions and well-known Results.
In this section, we give some definitions and well-known
results which have been used repeatedly in the subsequent chapters.
i)
Continguity (Definition:
Hajek [17]).
Let {p } and {Q } be two sequences of probability measures
v
v
defined on a sequence of measurable spaces {)tv,)lv}' v ~ 1.
If for
any sequence of events A €~, P (A ) ~ 0 entails Q (A ) ~ 0, we
v
v
v
v
v
v
say that the probability measures Q are contiguous to the probability
v
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measures P.
v
Contiguity implies that any sequence of r.v. 's Y convergv
ing to Y in P -probability converges to Y in Q -probability as well.
v
v
ii)
Uniformly Continuous Integrals (Definition:
Consider the probability space (Q,;(_, p).
Loeve [27]).
Let {X , v ;::: I},
v
be a sequence of r.v. 's (measurable functions) on this space.
Then
the integrals' of the r.v.'s X are said to be uniformly continuous,
v
°
if, for any A E~, JIX IdP +
uniformly in v as P(A) + 0; in other
A v
words, for every E > 0, there exists a 0 independent of v such that
E
Jlx IdP
A v
iii)
< E
for any set A with P(A)
<
0 .
Uniformly Integrable (Definition:
Let B = [Ix I;::: a],
v
v
X
v
E
Loeve [27]).
's being r.v.'s as in (ii).
the r.v.'s Ix I are uniformly integrable, if J Ix IdP
v
B
v
v
v, as a + 00.
NOTE:
+
We say
° uniformly in
The r.v. 's X are uniformly integrable if, and only if, their
v
integrals are uniformly continuous and uniformly bounded.
iv)
Kronecker or Direct Product (Definition).
Let A = «a .. )) be a p x q matrix and S =
1J
matrix.
Then, the Kronecker or Direct Product of A and B denoted by
A@B, is given by
A@B
=
a
pq
B
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12
° (f(v»
v)
p
(Definition).
Consider a sequence of r.v. 's {Xv , v
(X is of probability order
v
° (f(v»),
> 0, such that Pr·{IX 1 ~ A f(v)} ~
E V E
o (f(v»
p
1 -
0
P
E,
E >
0, there exists
for all values of v.
(f(v», if for every
1}.
We write
0, lim Pr·{[IX I/lf(v)
>
E
~
V-700
V
(i.e. X If(v) converges in probability to zero as v
v
vii)
We write Xv =0
(f(v»_
..-...p
(Definition).
Consider a sequence of r.v. 's {X , v
v
X =
v
1l.
if, for each
an A
vi)
~
I)
~
Asymptotically Equivalent (Definition).
cally equivalent" i f X v
v
Y
0 in probability as v
+
V
+
00, and {X } or
asymptotically N (M,
~),
p
for every vector
-
=
~
.1.
~'M)/[~'~~]2
~
where
X )' of r.v. 's is
pv
is a p.d. matrix if, and only if,
(a l , •.. , a )'
p
~
Q, with finite and real elements,
is asymptotically Nl(O, 1).
Bonferroni Inequality (see e.g. Feller [12]).
Let (~,A, p) be a measure space, and let Ai E~
(i
= 1,
2, ••• , r).
Let 8 1
A. ),
J.
8
m
=
V
Asymptotic Normality (Definition).
We say a sequence X = (Xl ,""
-v
v
ix)
"asymptoti~
has a non-degenerate limiting distribution.
viii)
(~'~v
= 1,
00).
+
We say two sequences {X } and {y } of r.v. 's are
v
v
{y }
E}
E ... E P
1~il<i2<···<im~r
2
=
r
E P(A.),
i=l
J.
... ,
A.
J.
2
n... n A. ),
J.
m
I
I
.-
13
Then the first set of Bonferroni Inequalities says that the
probability Pm that m or more of the events AI' A2 , •.• , A occur
r
simultaneously, satisfies the inequality
I
x)
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~
S - S
m
m+l
~
P
m
(m = 1, 2, ••• , r - 1).
S
m
A Convergence Theorem (see Cramer [11], p. 254) •
{Z } are sequences of r.v.' s
If {X }, {y } and
v
v
a,b
v
are numbers such that X converges in law to X as v
v
+
00,
converge in probabilities to a and b respectively as v
zvXv
+ Y converges in law to bX + a.
v
in law to X as v
+
00,
X' - X
V
V
converges in law to X as v
xi)
+
+
a
+
Y
v
00
,
;t
a
and Z
v
then
In particular, if Xv converges
in probability as v
+
00,
X'
v
00.
Slutsky's Theorem (see Cramer [11], p. 255).
If sl ' s2 , ••• ,
v
v
Spv are sequences of r.v. 's converging in
probability to the constants cl, c2, ••• , c
p
respectively, any rational
function R(sl , ••• , S ) converges in probability to the constant
v
pv
R(cI' ••• ' c ), provided that the latter is finite.
p
any power Rk(Sl , ••• , S
v
k
R (Cl, ••• , c).
P
pv
) with k
>
a
It follows that
converges in probability to
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CHAPTER II
RANK ORDER TESTS FOR REGRESSION WITH
PARTIALLY INFORMED STOCHASTIC PREDICTORS
2.1.
Introduction, Notations and Assumptions.
Consider a (double) sequence of stochastic matrices
z
"'V
= (z"'V 1' ••• '
'
-
Z
"'Vi
-
(Y
Z V ), 1
"'\)
X')
vi' "'vi
~
v
<
00, where
- (Y
XCI)
X(p))
vi' vi'···' vi '
(p ~ 1) are i.1." .d. random
vectors having an absolutely continuous d.f.
p-dimensiona1 real space.
d.f.
F01(~)
= Fv(OO,
density function
of Y • given X .
v 1.
'''V 1.
~)
Fv(y, ~), ~ s IRP , the
We assume that the p-variate marginal
does not depend on v and possesses a continuous
fOl(~).
We denote the conditional density function
~ by fv(yle).
We also assume that
1
So - v- 2 ~'~)/a], where So and
parameters and
~
0) are nuisance
0(>
= (Sl,eee, Sp )' is the parameter (vector) under test.
Thus the density function corresponding to the d.f.
Fv (y,
~)
is
given by
(2.1.1)
We want to test the null hypothesis of no regression i.e.
(2.1.2)
~
=Q
against the alternatives
~ ~
Q.
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15
It may be noted that under HO, fv(y,,e) = f(y,,e) = fOl(,e)f20(cr
for all v
v
(y - So»,
1 and thus Y . and X . are stochastically independent for
v1
"'v 1
~
all i = 1, 2, •.. , v and all v
=
-1
~
1.
We shall use the notations
v
-1
(k = 1, ••• , p) and Y
L:
v
v
i=1
v
-1
L:
Y ••
V1
i=1
The
following assumptions are made:
(2.1.3) f20(y)
E(X(k»
(2.1.4)
vI
=
= 0 (assumed without any loss of generality) and
(k) 14+0
E!XvI
v
~
<
(d/dy)f20(y) exists and I(f 2 0)
c
<
<
00,
for some 8
>
0 and for all k = 1, ... , p,
1;
(2.1.5)
Let
~(u) = -
(2.1.6)
f20(F26(u»/f20(F26(u»,
-1
where F20(U) = inf{x : F20(x)
~
u
<
<
1,
u} ;
~*(u) = - g'(G- 1 (u»/g(G- 1 (u»,
(2.1. 7)
0
0
<
u
<
1,
where G is a d.f. with density function g satisfying (2.1.3); thus
l
(2.1.8)
I(g) =
0
f
~
*2 (u)du
We introduce a class of score functions a (i), i
v
satisfying
(2.1.9)
limfl {av(l+ [uv]) \)"+00
0
~*(u)}2
du= 0,
1,2, ••. , v,"
,
00·
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= f(y,;e) = fOI(,e)f20(cr
It may be noted that under HO, fv(y,;e)
-I
(y - SO)),
for all v 2 1 and thus Y . and X , are stochastically independent for
v 1.
"'v 1.
1,2, •.. , v and all v 2 1.
all i
v
=
We shall use the notations
v
-I
(k
L:
i=I
1, ... , p) and Y
v
=v
-1
v
r
i=l
YV1."
The
following assumptions are made:
(2.1.3) f20(y)
=
(d/dy)f20(y) exists and I(f 2 0)
E(X(k)) = 0 (assumed without any loss of generality) and
(2.1.4)
vI
(k)
_ 14+6
ElxVI
< c <
00, for some 0
>
0 and for all k = 1, ... , p,
v 2 1;
(2.1.5)
L: =
Var(X
"'VI
) is p.d.
Let
(2.1.6)
-1
where F20(u) = inf{x : F20(x) 2 u} ;
~*(u)
(2.1. 7)
= -
I
g'(G- 1 (u))/g(G- (u)),
0 < u < 1,
where G is a d.f. with density function g satisfying (2.1.3); thus
(2. L 8)
We introduce a class of score functions a (i), i = 1, 2, ••. , v,
v
satisfying
(2.1.9)
lim
v-+ oo
II
0
fa (1 + [uv)) v
~*(u)} 2
du = 0,
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where [x] denotes the highest integer less than or equal to x.
In
particular, we can take the following types of score functions as
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suggested by Hajek [19]:
(b)
a (i) = <p*(i/(v + 1)), i = 1, .•. , v, v
(c)
a (i) =
v
i
v
J
~
l',
V
/
<p*(u)du, i
v (i-l)/v
= 1, .. .
denote ordered statistics in a sample U
VI
,'V ,
,.~.,
~
V
1, where
of size v from a
U
vv
uniform distribution on (0, 1), U . 's being distributed independently
v~
of X(k.) 's (."...
v~
= 1 ,"', ".
v,. k -- 1 , •.. ,
p) •
Let Fk(x) be the marginal
distribution of x(k) and F k'(x, y) the joint bivariate marginal
k,
vI
(k')
distribution of x(k) and X
VI
VI
.
{~v},
Consider another sequence of random vectors
where,
Ev = (W~I) , ••• , w~p)),
is assumed to be distributed independently of
y
Also, assume that the conditions (2.1. 4.) and
~v
= (y
VI , •• " y vv )'.
(2.1.5) are satisfied with ~v replacing ~1'
of
Fk
Let
denote the d.f.
w~~) and ~~(u) = F~-I(u), k = 1, 2, ••• , p. Consider another class
of "score" functions b
kv
(i), (i = 1, 2, ••• , v, v
~
1; k = 1, 2, ••• , p)
satisfying
1
(2.1.10)
lim
V-?<JO
J {bkv(l +
0
[uv]) -
w.~(u)}2
du
= 0.
In section 2.2, a class of permutationa11y (conditionally)
invariant distribution-free tests has been proposed and studied.
In
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17
section 2.3 the asymptotic permutation distribution of the proposed
class of test statistics (under Ha) has been developed.
The asymptotic
non-null distributions have been obtained in section 2.4 by using the
contiguity lemmas of LeCarn (see e.g. [19] or [25]).
In section 2.5
the optimality properties of the proposed tests have been developed
under the conditions of Wald and the ARE results of the allied tests
(in Pitman's sense) have been derived.
In the remainder of this section, we prove a lemma which
will be used repeatedly in the subsequent sections.
Let {X ., i = 1, ••. , V; v
LEMMA 2.1.1.
--
Vl.
~
I} be a double sequence
-
of r.v.'s such that for each v, Xv 1' .•• ' Xvv are independently
2 0
distributed with d.f. F ,(x) for which Elx .1 + < c < 00 for some
Vl.
o>
Vl.
0 and all i = 1, •.. , v.
E (X )
Then, Xv
-+
v
0 a. e. as v
-+
00.
The proof of the above lemma is based on the following result:
LEMMA 2.1.2.
Let
- Xv l ' Xv 2' ••• ' Xvv be samples from distributions with
d.f. F lex), F 2(x), ••• , F (x) respectively.
-- v
v
vv
n
2) such that Efx .I n
c
<
-
Vv I
and some positive K, where X
v
=v
(>
then given any €
Proof.
<
Vl.
>
0, p{IX
v
for all i
00,
>
-1.
If there exists
= 1,
2, .•. , v, v ~ 1,
€} < K!(€n v n!2) for all v
v
~
i=l
X.,
Vl.
~
v
= E(X
v
>
)
The proof of the lemma 2.1.2 follows along the lines of
Brillinger [8] (who however considered the case of i.i.d.r.v.
Proof of LEMMA 2.1.1.
"IXv -
E (X )
v
I
> €."
Given any
E >
0, let A denote the event
v
Let C be the event that "infinitely many A
v
va
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18
happen."
Using lemma 1.1 we get,
P{
Ixv
- E ex) I
>
0
o
d ~ K/ (8 2 + '.11+"2""), for all v
>
vO,
i2
<
00.
so that
00
v
l:
peA ) ~
v
v=l
-1-
Using now the well-known Borel-Cantelli lemma (see e.g. [27]) we
get P(C)
=
°
Le. A can happen only "finitely many times."
v
xv - E (Xv )
2.2.
-+
°
a. e. as v
-+
Hence
00.
A class of conditonally distribution-free rank-order tests.
We rank the observations within each row of the stochastic
matrix Z , ranking being done independently between rows. Let R(0)
vi
"'v
(k)
be the rank of Y . among y , ••. , Y and R. the rank of X(~) among
vI
V1
X(k )
vl
, ... ,
X(k)
(k
vv
= 1, .•• , p;
VV
v
V1
V1
~ 1); we define the collection
(-rank) matrix by
R(0)
vI
R(l)
vI
(2.2.1)
R(1)
R(1)
vv
"'V
=
vI
"'v
"'V
• e •••••••••••••
R(P)
(k)
where R
vv
R
"'v
r R(O)
R(0)
R(P)
vv
R(P)
"'v
(k)
(k)
= .(RVI
, ••• , R ), k = 0, 1, ••• , p. Each row of the above
vv
stochastic matrix is a random permutation of the integers 1,2, ••• , v.
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19
Let
(2.2.2)
Skv = v
where b
= v -1
kv
-1/2
v
E [b (R(k )) - b , Ha
kv
kv v i
i=l
V
v
E b (i), (k
i=l k v
= 1, ••• ,
(R(~)) -
p) and
V1
~
v
=
v
a ], k = 1, ... ,
V
-1
v
E
i=l
a (i).
v
The test statistic to be proposed is a quadratic form in the Sk
\)
IS.
The nonparametric methods developed in this paper are useful
X(~),
s are partially informed, i.e., they are not observable,
V1
when the
but only the ranks on them are available.
Our techniques are equally
applicable when we use mixed rank statistics,. i.e., using the observations X(~)'s as such instead of rank~scores on them.
V1
The test criterion is formulated by using-the rank-permutation
principle of Chatterjee and Sen (1964), (also discussed in Puri and
Sen (1966)).
Since each row of R is a random permutation of the numbers
"'v
1,2, ••• , v, Bvis a random matrix having (-v!)p+1 possible realiza-
tions •
Now, as has been noted after (2.1.2), tmderthe null
hypothesis the Yv 's are distributed independently of the -KV
~ 's
(k
= 1, ••• ,
p).
By assumption, the Y 's are identically distributed.
v
I
I
Hence, under Hd ,the v! possible permutations of Y 's are independent
I
'~v
I.
I
I
v
of the permutations of
~v' s
and are equiprobableamong themselves,
each permutation having the probability (v!)-l.
So, i f we delete
R(O) in R, we are left with a rank matrix involving ranks of the
~v's
"'v
which we denote by Bv(o).
Two collection matrices, say,
Bv(o) and B~(o) are said tobe permutationallyequivalent when it is
p,
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20
possible to arrive at one from the other by only a number of interchanges of its columns.
50 if
B~(O) is a matrix having the same
column vectors as Bv(o)' but the columns are so arranged that the
first row of B~(o) consists of the numbers 1, 2, •.. , v in the natural
order, Le.
2
1
v
R* (0)
(2.2.3)
~v
R*(P)
vv
then E~(o) is permutationally equivalent to Bv(o).
to each B~(o)' there will be a set L(~(O)) of v!
Thus corresponding
realizations of
Bv(o) such that any member of the set is permutationally equivalent
to B~(o).
The probability distribution of Bv(o) over the (v!)p possible
realizations will depend on
FOl(~)
even under HO in (2.1.2),
P
(unless F 0 1 (~)
=
IT Fk(x )); thus rank-statistics
k
k=l
are, in general, not distribution-free under HO.
~v
=
(5 1v , .•• , 5
pv
However, given a
particular set L(B~(o)) (of v! realizations), the conditonal distribution of Bv(o) over the v! permutations of the columns of E~(o)
would be uniform under HO, i.e.
irrespective of
FOl(~).
tete? denote the permutational (conditional)
v
probability measure generated by the conditional law in (2.2.4).
Then,
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after some algebraic computations, we get the following results:
(2.2.5)
(2.2.6)
=
(v/(v-1»){v-
I
I
1 v
(k)
L: (b
(R.) -b. )(b k,
k
kV
. 1
v Vl.
V
{v
-1 v
L: (a (i) -
i=l
v
a )2 }
v'
(k I)
(R.
l.=
Vl.
) -bk,V )}'
(k, k I = 1, ••. , p).
We propose the following test statistic:
(2.2.7)
M
v
=
S I V* S
"'v "'v "'v
where:Y~ is a generalized inverse of :Y
v
=
«Vkk IV»'
We shall show
in the next section that under the assumptions (2.1.3), (2.1.4),
(2.1.9) and (2.1.10),
yv
converges in probability to the matrix
fa I(g), where
(2.2.8)
with
O'kk
O'kk '
J:
*2
1}!k (u)du,
J: J:
k = 1, ••• , p,
1}!~(u)1}!~,(u')dHkk'(u,u')'
Hk,k'(u,U ' ) being the bivariate joint d.f. of
1 :::; k
;;e
k ' :::; p.
critical function
I.
Cov (Skv ' SkI v I~)
v
and
1
Fk(X~~»
:::;
k
and
;;e
k'
:::;
p,
Fk'(X~~'»,
The following test procedure is proposed based on the
1;;l(~v):
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if M > M
1
(2.2.9)
1';1
(k)
8
=
V,E
V
if M = M
IVE
V,E
V
if M < M
0
V,E
V
where MV,E and 8 1VE are so chosen that E~v(1';l (kv))
that E(1';l (k) IH a ) =
E,
=
E.
This implies
E(·IHa) standing for expectation under Ha,Le.
1;;1(Z ) is a similar size E test.
However, this procedure requires
"'v
evaluation of v! possible realizations of Mv '
The task becomes
prohibitively laborious for large. v and this leads us to .the study
of the asymptotic distribution of the permutation test statistic M .
v
2.3.
Asymptotic permutation distribution ofM •
v
We shall now prove a theorem which will be used in deriving
the asymptotic permutation distribution of M.
v
This theorem
generalizes Theorem 4.2 of PuriandSen (1966) in the sense that
unlike Puri and Sen we do not have to assume the existence of the
~
derivatives of the score functions
boundedness conditions thereon.
* (u)
or 1)Jk(u)'s
and the
*
In proving the result all we need to
assume is the square integrability of ~* and the uniform boundedness
of the (2 + 8)
for some 8
*
!1)Jk(u)
I
>
th
(k)
order moments ofW . 's (i
V1
=l~ •.•
, v,
k = 1, ..• , p)
0; the latter implies the conditions
~ [u(l - u)]
-1/2+8'
for some appropriate 8' > 0 as assumed
in [29].
THEOREM 2.3.1.
Proof.
v
-1
Under (2.1.8)
v
l:
i=l
[a (i) -
v
Yv
av ]2 =
I(g)k
+
v
-1
a
v
l:
i=l
in pro.bability as v +
-2
- a
v
Now,
00.
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if M > M
1
(2.2.9)
Z;;1
(k)
=
8
if M = M
IVE:
V,E:
and 8
IVE:
V,E:
V
ifM < M
v
V,E:
0
where M·
V,E:
V
are so chosen that
E~ (Z;;
If
V
(2 )) = E:.
1 ""'v
This implies
that E (Z;; 1 (k) IHo) = E:, E (,1 Ho) s tanding for expectation under HO , i . e.
Z;;1
(k)
is a similar size
E:
test.
However, this procedure requires
evaluation of v! possible realizations of Mv '
The task becomes
prohibitively laborious for large v and this leads us to the study
of the asymptotic distribution of the permutation test statistic M .
v
2.3.
Asymptotic permutation distribution of M .
v
We shall now prove a theorem which will be used in deriving
the asymptotic permutation distribution of M.
v
This theorem
generalizes Theorem 4.2 of Puri and Sen (1966) in the sense that
unlike Puri and Sen we do not have to assume the existence of the
derivatives of the score functions ~*(u) or ~~(u)'s and the
boundedness conditions thereon.
In proving the result all we need to
assume is the square integrability of ~* and the uniform boundedness
of the (2 + 8)
th
(k)
order moments of W . 's (i =1, ••
V1
o,V,
k = 1, ••• , p)
for some 8 > 0; the latter implies the conditions
I~~(u) I
:s; [u(l - u)]
-1/ 2 +8'
for some appropriate 8' > 0 as assumed
in [29].
THEOREM 2.3.1.
Proof.
v
-1
Under (2.1.8)
~v +
I(g)k
v
~
i=1
[a (i) - a ] 2
v
v
v
-1
o
v
L:
i=l
in probability as v
_2
- a
v
Now,
+
00.
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
23
1
a
1
f0
f
=
v
0
1
~*(u)du = 0).
get, a
v
~
= fo
a (1 + [uv])du
v
~
2
a (i) =
v
i= 1
(since
Using now (2.1.9) and the Schwarz inequality, we
0 as v
V-I ~
~*(u)]du,
[a (1 + [uv]) v
Further,
00.
fI a2 (1
v
0
+ [uv])du =
fI [a
0
fo~*(u)
(1
v
1
+ 2
~*(u)]du
[a (1 + [uv]) v
~*(u)]2du
+ [uv]) -
1
+ (
;0
~*2(u)du
Using now, (2.1.9) and the fact that
1
Jo I~*(u){a
with I(g)
v
<
~*(u)}ldu ~
+ [uv]) -
(1
00,
we get, v
-1 v
-1 v
V
E
2
E a (i)
i=I
(~.3.1)
1
1
v
[a (i) V
i=I
aV ]2
1
IT(g){J [a (1 + [uv]) 0
~
v
I(g) as v
~
~ I(g) as v ~
00.
~*(u)]2du}T
Thus,
00.
Proceeding exactly in the same way and using (2.1.10), we can show
that
as v
~
00,
(k
= 1, ••• ,
p).
Using (2.2.6), (2.3.1) and (2.3.2) we get,
(2.3.3)
For k
~
o
Vkk,v ~ 0kk I(g)
k', we get from (2.2.6),
as v ~
00.
I
I
24
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I·
I
~
I
I
{V
-1
- ..,..2}
a
•
v
For proving the theorem, all we have to show now is that
v
(2.3.4)
-1
We introduce the following notations:
(2.3.5)
(2.3.6)
(2.3.7) Ckv(i!(v+l))
=
bkv(i);
(i
=
1, ••• , v; 1 s; k;t k' s; p).
Hence,
(2.3.8)
'v
-1
v
E
i=l
1 1
=
faf
0 C (v H (u)! (v +1) ) Ck 'v
kv
kv
1 1
=
faIo
[C
kv
(v~v (u)! (v +1)) -1jI~ (vHkv (u)!(v+ l ))] Ck 'v (vHk 'v (v)!(v+l))dHkk,v (u, v)
1 1
+
(v~ 'v (v)! (v+ l )) dHkk 'v (u, v)
fofo [1jI~(VHkv
(u)!(v+ l ))
-1jI~(u)]1jI~,( Hk,v (V)!(v+l))dHkk,v (u,v)
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
25
1 1
+
foJO[~~'(VHk'V(V)/(V+l»
-
~~,(V)]~~(U)dHkk'V(U,V)
(I) + (II) + (III) +, (IV) + (V), (say).
=
{v
-1 v
L:
i=l
Now,
v- 1 :v [bkv(i) - ~~(i/(V+l»]2 = IIo[bkv(l+[uv]) - ~~«1+[uv])/(v+l)~2du
i
1
<
2 f:[b kV (l+[UV])
-
~~(u)]2dU + 2 f:[~~(l+[UV]/(V+l)) - ~~(U)J2dU
Since ~~(u) is a monotonically non-decreasing and square
integrable function of u',using lermna a (p.164) of [19], we get,
1
*
* 2 du
- ~k(u)]
f o[~k«l+[uV])/(V+l»
which implies
(I) + 0 as v +
2
v-l.~
b (i)
~=1 k v
00.
+
+
0 as v
f\k*2(U)dU
<
+
00,
00.
we get from (2.3.9),
0
Similarly (II)
+
0 as v
+
Us.ing now (2.1.10)
00.
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
26
(2.3.10) :
1
1(111)
1
1
1~[JO{W~(VHkV(U)~V+1)-W~(U)}2dHkV(U~2[JOW~~(VHk'V(V)~V
+1)dHkV (V)]1/2
But,
as v
-+
00.
Also,
(2.3.12)
= E[v- 1
~
i=l
({Wk**(R(k ) /(v+1)) - Wk**(U .)} + {Wk**(U .) - W*(U .)})2
vi
.. V1
V1
k V1
where W~*(A) is a function defined on 0 < A ~ 1 by
(2.3.13)
W~*(A) = W~(i/(v+1)) for (i-1)/v
<
A ~ i/v, (i = 1, ... , v)
so that Wk** (i/(v+1)) = Wk*(i/(v+1)) for all i = 1, ... , v.
Then using
the elementary inequality (a + b)2 ~ 2(a 2 + b 2), we get from (2.3.12),
1
(2.3.14)
E[Io
[W:(vHkv (u)/(v+1)) -
+ 2 E[v
-1
W~(u)]2dHkv(u)]
v
**
*
2
L {Wk (U .) - Wk(U .)} ]
i=l
V1
V1
I
I
27
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
= 2 E[~**(R(k) /(v+1»
k
- ~**(U )]2
k vI
vI
1
+ 2
I
**
*
2
[~k (u) - ~k(U)] du
0
f>~~*<U) - ~~(U)]2dU = f>~~(l + [UV]/(v+1» _~~(U)]2du.
Now,
*
~k(u)
Since
is a monotonically non-decreasing and square integrable
1
Io[~~*(U)
function of u, using lemma a, p. 164 of [19],
as v
~
00.
-
~~(u)]2du ~ a
Again using lemma 2.1 of [16], we get,
(2.3.15)
~
I~k** (i/(v+1»
r;:; (v -1/2 max
2 v2
l~i~v
-1. ~t... (**('/(
- =**1)(
1J!k
v
~k 1 v+ 1»
1=1
v
where ....:Je*
1J!k = v-1 E ~k** (i/(v+1»
i=1
as v ~
=
v-I
Also, v-I E ~**2(i/(v+1»
i=1
v -1/2 max
1
**.(1/(v+1» .,
~k
= v
~ cr~k
1/2
l~i~v
~
00
since cr
o
kk
<
00.
<
max'
~ max {
as v
'f'
=
0
l~i~v
l~i~v
~k*)2)1/2,
~ ~~(i/(V+1» ~ II ~~(u)du
i=1
v
00.
-
00
as v ~
(i-D/v
J(i-l)/V
and
*
Ii/V
i /V
00,
~k(u)dul
~~2(U)du}1 2 ~ a
/
Now, from (2.3.15),
Thus, from (2.3.14),
1
(2.3.16)
E[Io[~~(VHkV(U)/(V+1»
-
~~(u)]2
dHkv(u)]
~a
as v
~
00.
a
I
I
.e
I
I
I
28
Using (2.3.11) and (2.3.16) with the fact that convergence in mean
implies convergence in probability, we get from (2.3.10), (III)
in probability as v
v
-+
00.
-+
00.
Similarly (IV)
-+
-+
° in probability as
°
Again,
-1 v
*
*
(k)
.
(k')
~k,(Fk'(~\l.' ».
= v i=1
I ~ (F (X
»
k k vi
(V)
y
Since,
I;
I
I
Ie
I
I
I
I
I
I
I
.e
I
<
00,
using Markov's weak law of large numbers, we get,
in probability as v
asymptoticallyN
Let
~
00.
Hence (2.3.4) is proved.
Under the permutational model
-
THEOREM 2.3.2.
Proof.
-+
p
<Q,
(?v and
-
(2.1. 8) S
is
~v-
I(g)·kO) in probability.
= (e 1 , ••• , e )'
p
~
Q be
a vector of finite real constants.
Then
E(e'S
(2.3.17)
~
~v
Il?)
v =
0, E[(e'S )2/6>]
v = (?,'Y,,?,)I(g).
.- ' - y ' ~
~v
We have to show that for every vector e as above, e'S
~
asymptotically normal (0,
(2.3.18)
e'S
~
~v
=
v
I
i=1
P
{I
(~'kO~)I(g»
(k)
~
~v
in probability.
(0)
ek(bk(R"l.') - b k )}{a (R , )
k=1
v
v Vl.
y
is under6?v
We have,
-aV }
I
I
.-
29
{~
k=l
Under p ,
v
ek(b
kv
(R(~))
V1
- D. ), i = 1, ••• , v} is equiprobable over
kv
the v! permutations of the columns of R~(O) while {av(i), i = 1, ..• , v}
I
I
I
I
I
I
Noether condition (see e.g. (16)) in probability and the sequence
Ie
which implies the Noether condition, and then apply Theorem 4.2 of
I
I
I
I
I
I
I
.I
is equiprobable over the v! permutations of the integers 1, 2, ••• , v
independently of the permutations of{
ek(b (R(~))) i = 1, ... , vL
kv V1
. k=l
t
Since Y,v
-+
10 in probability as v
-+
00,
to prove the theorem it is
(k)
p
sufficient to show that the sequence { E ekbk(R . )} satisfies the
k=l
V1
{a (i)} satisfies the condition Q of Hajek (16), namely,
v
<. -<max<,
1-1
••• -1
1
kv
'" [ a (.)
1
~
-
a ]2
kv a=l v a
v
O=>lim - - - - - - - ' - - - - - - - - - v-+oo
~ [a (i) 2
(2.3.19) lim
. kv Iv =
v-+ oo
i=l
L et a 2 (l) -< 8 2 (2)
V
V
[16], p. 514.
<
-V
:::::;
...
=
a ]
v
v
2
::; av(v)
be the ordeml
2
a (i)'s and let us define a step function n (u) as
v
v
2
(2.3.20) n (u) = a (i) for (i-I)/v < u ~ i/v, i
v
v
Then v
k
-1
max
=
E
/' <
1 ':>1
- ••• <'
_1
1
k
a= 1
v
f
lo
Hence~
f
l-k
a
v
-+
v
0 as v
Iv
-+
n (u)
v
=
n (u)du
v
-+
00.
Hence,
v
0
f
l-k
)v
v
1, ... , v.
n (u)du.
But,
v
Iv
v
2
E a (i) -+ I(g) <
-1
i=l
as
v -+
00
as v
-+
00.
v
00
o,
since then k
v
Iv
-+
O.
Also
I
I
••
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I.
I
I
30
(2.3.21):
k\!
_ 2
-1
k
max
E [a\! (i a ) - a \! ]
2\!
max
E
1~i1< •.• <ik ~\! a=l
l~il< ••• <ik ~\! a=l
\!
\!
-------'----------~-------~---------+O
\!
-1 \!
_ ]2
-2
\!
E a \!2(.)
1
- a
• b [a)i) ..; a \!
\!
1=1
i=l
as \! -+
00.
Putting in particular k\! = 1, it follows that the sequence
{a (i)} satisfies the Noether condition i.e.
\!
(2.3.22)
1 ~
max
[a (i) i ~\!
\!
~
i=l
[a (i) -
\!
a
\!
a
)2
\!
-+ 0
as \! -+
00
•
]2
Also,
(2.3.23)
\!
~
\!
-+
00
{ p
E
k=l
~
as
-1
\!
-1
ek
[b k «k»
R.
\!
\!1
_ -bk\! ]}2
p
max
p E
1 ~ i ~ \!
k=l
(p
(the proof being similar to that in the case of a \! (i)'s).
Further,
(2.3.24)
in probability (using lemma 2.3.1 and the fact that
Hence,
fa is p.d.).
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
II
I
31
_
-b
(2.3.25)
Hence the theorem.
-THEOREM 2.3.3.
2.4.
2
kv
]}
~ 0 in prob~bility.
An immediate consequence of the
~bove
theorem is:
2
Under (?, M is asymptotically distributed as X •
v
v
p
Asymptotic non-null distribution of Mv '
We first prove the following elementary lemma:
LEMMA 2. 4• 1.
(i)
.
v -1/2 max
I$;i$;v
IX.
(k) I
v~
~
0
a.e. as v
for all
~
k=l, ... ,p.
(ii)
~.e.
for all k
Proof.
We first prove (ii).
= 1, ... ,
p.
We can write,
(2.4.1)
Elx(k)1 4+O <
VI
c < ""
'
using lemma 2.1.1,
(2.4.2)
a.e.
as v
~
00.
as v
~
"",
I
I
.e
I
I
I
I
I
I
32
[Note that the
p~variate
2+0
,
.'".- (Elx~\kl)
14+ 0 )4+ 0
Again, E Xv(k),2+0
1
v
X(k)
(2.4.3)
E(x(k))
vI
+
v
~Vl
marginal d.f. of
0
=
<
2+0
c4tO
does not depend on v].
<\Xl
a.e. as v
+
So,
00
•
(ii) follows now from (2.4.1) - (2.4.3).
To prove (i), we first recall two equivalent foms of Noether
condition given by Hoeffding (1951).
We state these in a slightly
different form convenient for our purpose:
(2.4.4)
Ie
I
I
I
I
I
I
I
.e
I
a.e. for some r
>
2.
The right hand expression in (2.4.4) can be -rewritten as
(2.4.5)
l- r
v
I
(k)
a.e. for some r
>
2.
lim v
v+oo
2"
{v -
1
I
X.
i=l V1
We shall prove first (2.4.5).
(k) r
1 v
(k)
(k) 2 /2
,} / { v I (X. - X
)} r
v
i=l V1
v
- X
Taking r
=2
and using an elementary inequality, we get,
(2.4.6)
Taking now, 0"
=
o-
20'
2 + 0' , we have,
+ 0' where (0
=0
< 0' < 0/2)
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
33
(Elx(~)12+0')2+011 ~ Elx(~)1 (2+0')(2+0") =
Vl
Vl
Hence, by lemma 2.1.1,
(2.4.7)
Also X(k)
v
-+ 0
a . e. as v
we get (2.4.5).
_1
(v 2" max
1~i~v
Ix .1)
-+
Using now (ii), (2.4.6) and (2.4.7),
00 •
(i) follows immediately from (2.4.7) since
2+~'
u
/;' v
~ v- 1 -2 ~
i=1
Vl
Ix 12 +
~,
u
vi
Consider a sequence of measure spaces 0v'
,l v'
lJ)' v ;::: 1,
a typical element of 1
. v being z , where
~v
(p)
(1)
(1)
.tv = (yv l ' Xv 1 , ... , Xv 1 , ... , yvv ' x vv , ••• ,
Now take a particular alternative, say So = So* '
~
=
~* = (S~o, ... , S;o)', cr
hypothesis So
= S~, ~ = Q, cr
cr~
=
and associate with it the particular
= cr~.
Denote the distribution of ~v
under the alternative and the hypothesis by Q and P
v
where dQ
v
= q dlJ ,dP
v v
v
= pv d~.
v
v
respectively,
Let the respective densities for
Z . under the alternative and the hypothesis be qVl.(x) and pVl.(x),
~Vl
~
=
~
(Xl' ... , x)', i
p
= 1, •.. , v.
Let
i f p . (x)
Vl '"
(2.4.8)
otherwise
Consider the following two statistics:
v
(2.4.9)
W = 2
v
~
~
i=1
( r 1/2 _ 1)
vi
0
I
I
I·
I
I
I
I
I
I
Ie
I
I
I
I
I
I
34
(2.4.10)
L
v
II
r
i=l
,
Vl.
We introduce the notations:
(2.4.11)
. _1
=v
2 rI"
v'X
, (i = 1, ... , v).
~Vl.
The asymptotic distribution of M will be obtained by using
v
the contiguity lemmas of LeCam (see [19] or [25]).
We need show
that the distributions Q are contiguous to the distributions P •
v
v
We adopt a procedure analogous to that of Hajek [17].
The following
lemmas will be needed:
LEMMA 2.4.2.
Under the given model and (2.1.3)-(2.1.5),
vi P) ~ -i+ I (f 2 a)(x 'kx)
E (W
Proof.
as ·v ~
1/2
= f 20
(x).
Write S20(X)
00.
Using the notations in (2.4.11),
we get from (2.4.9),
(2.4.12)
=
W
v
v
2
I:
[ {s 2 a (y', - h ,) / s 2 a (y' .)} - 1].
Vl.
Vl.
Vl.
1=1
Use the Bonferroni inequality, the fact that p is fixed (independent
of v) and lemma 2.4.1 to get
(2.4.13)
max
<'<
1 -l.-V
Ih Vl.,I
~ 0
a. e. as v ~
00.
Now, as in [17], we can write after some algebraic simplifications,
E (Wi P , hI'...·' h
I.
I
I
v
=
v
v
v
vv )
=-
I
I
I·
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I.
I
I
35
note that since max jh
l~i~v
.1
a.e. as v-+-
-+- 0
for fixedh .(w)'s,
00,
V1
V1
w's being points in measurable sets in the basic probability space,
proceeding as in [17], go (h vi )
-+-
i- 1(f20)
set of points w of zero measure).
a.e.·. (Le. except for a
Also,
v
L h2 . =
a. e. ,
i=l V1
since
i, ••• , p, and,
so that lemma 2.1.1 can be applied.
Thus to prove the lemma, all
v 2
we have to show is that the random variables L h . go(h .) have (i)
i=l V1
V1
'uniformly continuous' integrals and (ii) the integrals are 'uniformly
bounded' (see chapter I), and then apply the L -convergence theorem
r
given in Loeve [27, p. 163] with r = 1.
Since it can be shown that
it
(see e.g. [19]), Ig o (h ) j ~
1(f 2 0), uniformly in i = 1, ... ,v,v 2': 1
vi
v
v
2
and E( L h .) = x.,'J:;X, Ej ( L h .go (h 2
I ~ 1(f20)x.,''f,x.,· (i) follows,
. 1 V1
1=
.
1=
1 V1
t
.»
V1
since for any measurable set A, with 1
Abeing
if
defined as :
WEA
otherwise ,
v 2
E( L h .go(h .)1 )
.
V1
V1
A
1=1
I
I
36
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
..I
I
:s;; c 4/ ( 4+ 0) p 6 ( max Y2 ) 2 •
k
l:S;;k:S;;p
uniformly in v as E(I ) = P(A)
A
+
O.
Q.E.D.
Let T = (T 1 , ..• , T )', where
""V
v
pv
=-
_1
2 v T
v
()
L: X
i=1
~
v~
[S20(Y' .)/S20(Y' .)],
v~
v~
k = 1,2, ••• , p.
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
•e
I
37
Proceeding along Hajek [17] and using the technique of conditional expectation as in the preceeding lemma, we get,
LEMMA 2.4.3.
Under the given model and the assumptions (2.1.3)-
(2.1.5) Var(W \) - v'T!P
~ ~v
v)
+
as v
0
+
00.
Using lemmas 2.4.2 and 2.4.3, it now follows that
which implies
LEMMA 2.4.4.
UnderP, the given model (2.1.1) and (2.1.3)-(2.1,5),
v
is asymptotically multinormal (Q, I(f 2 0)f) a.e.
Proof.
We have to show that for every vector
of finite real constants,
a.e.
~'!v
~
= (e 1 , ••• , e )'
p
is asymptotically normal (0,
~
I(f20)~'k~)
We can write,
(2.4.15)
e'T
~
v
~v
v
-1/2
P
(k)
( 2: ekX i
2:
i=l
v
k=l
)H F 20(Y' .».
V1
(2.4.16)
Also,
P
(k)
max I 2: ekX .
l:S;i:S;v k=l
\)1
2:\){ ~ e x(~)}2
i=l k=l k V1
+
Iv
~'~
1/2
I
+
0 as V
a.e. as v
+
00
+
00
from lemma 204.1 (i) and
from lemma 2.4.1 (ii).
a.e. as v
+
00 •
i.e.
0
So,
I
I
I·
I
I
I
I
I
I
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I
I
I
I
I
I
I.
I
I
38
(2.4.17)
the sequence {
~ e x(~)} satisfies the Noethercondition a.e.
k=1 k Vl.
Under P , Y' 1' ••• , Y' ,
v
independent •
v
vv
~(F20(Y'
VI
», .•. ,
~ e x(k ) , ... , ~ ekx(k) are mutually
k=1 k v 1
k=1
vv
vv » (all identically distributed
~(F20(Y'
under the hypothesis) are Borel-measurable functions of Y' " •. , Y'
VI
respectively.
vv
Now, using (2.4.16) and (2.4.17) ,conditional: on the
x(~)'s
fixed, e'T is asymptotically
Vl.
~
~V
Since,
a.e. as v
~
e'T
00,
~
~V
is unconditionally
Using the above lemma, we get from (2.4.14),
(2.4.18)
We next prove the following lemma:
LEMMA 2.4.5.
Under (2.1.1) and (2.1.3)-(2.1.5),
lim max P (I{q .(2 i)/P ,(2 ,)} - 11
<'<
V
Vl. ~v
Vl. ~Vl.
V ~ l_l._V
~
£-1
E[I{qvi(~v±)/PVi(~Vi)} -
= £-1 E[lh
,I
Vl.
JOO
-00
11
>
£) = O.
IP v ]
1{f2o(y-h .) - f20(y)}/h ,Idy]
Vl.
Vl.
But, it can be shown after some elementary algebra that
J
OO
_00
l{f2o(y - h ,) - f20(y)}/h ,Idy
Vl.
Vl.
~
1 2
r /
(f20), (i = 1,2, ••• , v).
I
I
I·
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I.
I
I
39
Hence, P v (I{q v ~.(Z
i)/Pv ~,(Z
,)} ~v
~v ~
I
::; e -I 1 1/2 (f 2 0)E h ,
I
v~
•
11
> e)
80,
(2.4.19)
::;
e
::;
e
-I 1 1 / 2
(f20)
_I
1
1/2
max E( Ih ,I)
v~
l::;i:5:v
.
(f20) E( maxi h v~,I)
l::;i::;v
We know,
max
1<'<
-~-v
(where
(~,
Ih ,I
V~
-+
0
-+
00, and
k' P Ok ) is the measure space,
1, if w e
1~ (w)
a. e. as v
~
e
k and
~
=
·
X(k)
8 ~nce
v I ,""
" ~ . d • r. v. , using a well-known result (see
X(k) are ~.
vv
e.g. Hartley and David [21]),
I
I
I·
I
I
I
I
I
I
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I
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I
I
I
I
I.
I
I
40
Hence,
uniformly in v as
max
l$i$v
x(~)
Vl.
POk(~) ~
O.
Hence, integrals of r.v.
= 1, •.• , p and
are uniformly continuous for each k
using (2.4.20) and the L -convergence theorem of Loeve [27], p. 163),
r
with r
= 1,
we get,
(2.4.21)
E( _max
1 <'<
-l.-V
Ih . I)
~ 0
as v ~
00
•
Vl.
The lemma now follows from (2.4.19).
We now state LeCam's second lemma (see [19], p. 205) which
is an immediate consequence of (2.4.18) and lemma 2.4.5.
LEMMA 2.4.6.
(ii)
The statistic log Lv
~s~at~i~s~f~i~e=s
log Lv is asymptotically normal (-; I(f 2 0) (X'tX), I(f20)(X'tX))
a.e.
(iii)
the distributions Q are contiguous to the distributions P •
v
v
I
I
I-
I
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I
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I
I
I
41
We are now in a position to derive the asymptotic non-null
distribution of
by using LeCam's third lemma (see [19], p. 208).
We first define the following statistics:
(2.4.22)
1, ... , p;
*
*(
) *( F20 (Y' ). ) ,k=l, .•. ,p.
T ,,=v -1/2 v2:ljJkU,ep
(2.4.23)
kv
•
1=
1
V1
V1
Then,
But I(g) <
00,
b
kv
~
~
0 as v
00
and proceeding as in Theorem 1.5. a
(p. 157) and 1.6. a (p. 163) of [19]
as v
~
00,
(2.4.25)
for all i = 1, ..• , v.
E[(S~v - T~v)2IPv] ~ 0
Hence, S~v - T~v
Qv-probabi1ity because of
~
E[bkv(R~~))
-
ljJ~(UVi)]2 ~
0
Hence, from (2.4.24),
as v +
00,
for all k = 1, ••. , p.
0 inP -probabi1ity (and hence in
v
contigu~ty)
for all k = 1, ••• , p.
Using Bonferroni inequality, we get,
(2.4.26)
II
I
~v
S* - T* ~
~v
~v
h
S*(S*
were
=
1v
~v
, ••• ,
0 in Pv -probability (and hence"in"Q v-probability),
~
S* )'
and T* = (T*"I""'"
pv . . ~v
v
T* )'.
pv
Further,
I
I
42
,e
I
I
I
I
I
I
k=1, •.. ,p.
Hence,
(2.4.27)
E[(Skv - Sk*v )21p v ]
(av(R~~»
-
~*(U~i»(av(R~~~)
-
~*(U~i'») 1Pv]'
(writing
for F20 (Y' .. )) .
Vl.
Ie
I
I
I
I
I
I
I
,e
I
U~i
But,
v
_IV
_
2
0
! [b (i) - b ] + 0kk <
kv
i=1 k V
for all i = 1, ... , v.
(2.4.28)
(0)
00
*
and Era (R . ) - ~ (U*.)]2
V
Vl.
Hence, from (2.4.27),
V1
+
0
I
I
,e
43
s
(2.4.29)
~v
- s*
~v
+
0 in Pv (and hence in Qv )-probability.
~
(2.4.26) and (2.4.29) give,
I
I
I
I
I
I
s
(2.4.30)
~
- ~
T*
+
0 in P
~
v (and hence in Qv )-probability.
From (2.4.14) and lemma 2.4.6 (i),
in P -probability (also used in proving lemma 2.4.6 (ii)).
v
for any vector .?
~
Hence,
.-P. of finite real constants, (log Lv' ~e'S)
~v
has asymptotically the same distribution as of
Using (2.4.30), the latter has the same asymptotic distribution as' of
Ie
I
I
I
I
I
I
I
,-
I
Let
(2.4.31)
(k, k' = 1, ... , p).
We prove the following lemma:
LEMMA 2.4.7.
Under P , the model (2.1.1) and the assumptions
v
(2.1,3)-(2.1.5), (2.1.8)-(2.1.10), (X'1: v - ~ I(f20)X'll, ~':!~) is
asymp tot;ically N2 (-} I (f2 0) (X 'll), 0, I (f2 0) (X' ll), (~' k,O~) I (g) ,
(~'Ioo;~) J; cj>(u)CP*(u)du) ~.
Proof.
We have to show that for every (a,b)
~
(0,0), where a, b
are finite and real constants, av'T
+ be'T* is asymptotically
'"lo ~v
~
+ 2
~
ab(~'Ioox) J:Cp(U)CP*(U)dU)
a.e.
I
44
I
••I
I
I
I
I
I
Ie
I
I
I
I
I
I
I.
I
I
Now, av'T
+ be'T
*
,... "'V
'" "'V
=v
_1 V
P
2L: [L: aY 1}Jk(U .)cp(F2 0(Y'.))
k
i=l k=l
p
+
V1
V1
L:
bek1}J~(U .)ep*(F2 0(Y' .))]
-1.
V
k=l
.
V1
V1
*
= V 2 L: (Z 1 . + Z* .), say,
i=l
V1
2V1
where,
_1 P
a Yk 1}Jk(U .)¢(F20(Y'.)),
= v 2L:
k=l
Z2*V1.
Let
*
,)
(i = 1,2, ... , v).
be 1}Jk(U.)ep (F 2 0(Y .),
k
V1
V1
k=l
+ Z*
1,2, .•.
i
2vi'
Ivi
V1
*
_1 P
v T L:
z*. = Z*
Vl
V1
,v.
We may also note that E[av'T
+ be'T,.....,v* IH o ] = 0 and
I-l. ,....",'V
~
Var(aX'lv + b~'l~IHo)
=
a2(X'~X)I(f20) + b2(~'~o~)I(g)
1
+ 2
ab(~'kOOx)foCP(U)CP*(U)dU.
v
show now is that
o
>
L:
i=l
E[Z*~
V1
As in [19] (pp. 217-218) all we need
XZ*.(o)
Vl
IP v ]
+
as v
0,
if
XZ*.(o)
x
Z* . (0)
Vl
:::; X
Z* . (0/ 2 )
IVl
Iz*.1
Vl
otherwise.
Vl
But
0
+ X
Z2* Vl. (0/ 2)
> 0
+
00, where for any
I
I
,I
I
I
I
I
I
Ie
I
I
45
Hence,
as v
~
00
= 1, 2 (these conditions being equivalent to the
for j
Lindeberg condition of the classical Central Limit Theorem for
v
E z~
i=l
.
and
v
E z~
i=l
V1
.).
The proof is exactly the same as in [19,
V1
p. 218], and, hence, is omitted.
Using new LeCam's third lemma (see [19], P. 208) and lemma
2.4.6, we get, under Qv'
* is
~'Iv
asymptotically
1
Nl«~'kOOX)Jo~(U)~*(~)dU, (~'kO~)I(g))
for every ~ ~
Q of
a.e.
finite real constants i.e. l~ (and hence £v)
is under Q asymptotically
\J
I
I
I
I
I
.-
I
«EooV)fl~(U)~*(U)dU'
Np~rl.o
Also, from Theorem 2.3.1, ""'v
V
EoI(g)) a.e.
~
~
EOI(g) in probability.
Recalling
f'V
definition of M in (2.2.8) and using Slutsky's theorem, we get that
v
under Q ,M is distributed asymptotically as
v
v
1
X~«X'k~okolkoox)(fo~(U)~*(u)du)2/I(g))
a.e.
I
I
46
.-
2.5
I
(2.5.1)
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I
f
I
Aymptotic optimality of tests and A.R.E.
Let
(x v = (Y1
~v
,"',
Ypv )
being the maximum likelihood estimator (m.l.e.)
of X), be the likelihood ratio test criterion for the testing problem
considered here.
Let
(2.5.2)
We make the assumptions on the likelihood function L(~vlx)
as in Wald [34], (given in details in chapter III, and, hence, are
not restated here for the sake of brevity).
Under these assumptions,
and also under the assumptions (2.1.3)-(2.1.5), (2.1.8)-(2.1.10),
we get the following result:
THEOREM 2.5.1.
Proof.
M* + 2 log A
e v
v
+
0 in P -probability.
----
v
with~v'
Same as in Theorem 3.2.3 of chapter III
k' I(f 2 0)
and S respectively.
md~v replacing x*, A , A2 (F, {I.})
J
v
~v
~v
The theorem implies that M.
+ 2 log e Av
v*
(because of contiguity).
+
0 in Q -probability
v
So, if the assumptions (2.1.3)-(2.1.5) and
(2.1.8) along with the regularity assumptions of Wald are satisfied,
a test procedure, similar to the one we have considered, based on
M* possesses the optimal properties of the likelihood ratfutest as
v
described in Theorem 3.2.2 of chapter III.
We might also remark that using LeCam's third lemma, it can
be shown that, under the alternatives, M* is asymptotically
v
I
I
.I
I
I
I
I
I
Ie
I
I
I,
I
I
I
I
.-
I
47
X~«X'.tX)I(f20)) a.e.
If, under the same sequence of alternatives,
two test statistics are asymptotically non-central chi-squares with
the same degrees of freedom, it has been shown by Hannan [20] and
Andrews [2] that the A.R.E. of one test procedure with respect to
the other is given by the ratio of theirnon'"-centrality parameters.
Using this, we get the A.RoE. of the test based on-M with respect
v
to the one based onM* is given by
v
(2.5.3.)
e =
where p 2 =
l
[(X'f~oI~lk.OOX)/(X'.u) ]p~
!j>(u)<P* (u) dul [(II ¢2 (u) du)
Jo
1/
2
(
(\*2 (u) du) 1 / 2]
We may also observe that in the particular case, ¢*
all k
= 1, ... ,
p, e = 1
•
J0
0
=
¢, W~
= Wk'
for
i.e. the test procedure based on the statistic
M\) as we have proposed is asymptotically optimal in Wald's sense.
We may note that the general efficiency expression-depends on
X
= a-1
!,
However, using a well-known theorem of Courant (see Bodewig
[1956]), we get
(2.5.4)
. 1e1genva
·
1 ue
max e = maX1ma
0
f ~
"-1,,,
"-1,,~OO
kOOkO
Y.,
min e
= minimal eigenvalue of
~
-1
-1
k.~Oko ~OO.
Y.,
It is interesting to observe that instead of considering the
rank statistics Skv(k =1'00"
p) as defined in (2.202) i f we consider
the mixed statistics
SO
k'J
(k)
(which would be more logical to consider when the X.
V1
k
=
(i
= 1, .•. , v,
1, ... , p) are observable), the appropriate statistic will be
I
I
.I
I
I
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I
Ie
I
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I
I
I
I
I
.-
I
48
O
the corresponding quadratic fonn in Skv 's and the effi.ciency
expression
.
will be
e'
(2.5.5)
=
2
P2 •
Thus in the particular case when ep *
=
ep, the resulting test is
optimal according to criteria described earlier.
However, as already
mentioned, the mixed statistics cannot be used when the
not observable.
based on
x(~) are
v~
We may also remark that unlikeM-, the statistic
v
X(~);s is not unconditionally distribution.... free even when
v~
p = 1.
We next investigate some particular cases where the efficiency.
expression can be simplified considerably leading to some interesting
results.
Case I
(2.5.6)
2
P = 1.
PI
2
Here, e = PI P 2 ' where,
=
I ~1 (u)~f(u)du/[(
I
0
J10 ~l(u)du)
2
1
(fo ~l*2 (u)du) 1/2 J.
1/2·
We may note that this efficiency expression is the same as we would
have obtained by an extension of Hajek's ideas to simple linear
regression with stochastic predictors.
The test statistic which we
would consider in that case will be simply
Slv~
This will be uncon-
ditionally distribution-free since in this situation under the
hypothesis of no regression (or independence according-to our model),
the pennutation distributionofS1Vbecomesidentical. with its
unconditional distributiono
For two-sided alternatives, the test
procedure is optimal in Wald'ssense.
For one-sided alternatives,
this will be asymptotically uniformly most powerful (UMP) as in
I
I
.e
I
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.e
I
49
* = cp
Hajek (J,962) in case cp.
*
and 1JJ 1 = 1JJ
1
It might also be remarked
•
that the test considered here is an alternative approach to the
study of independence of two random variab J,e~ - as has been considered
by BhuchongkuJ, [5].
Bhuchongkul followed the Chernoff-Savage [lOJ
approach in proving the asymptotic normality of her test statistic
and as such had to put rather stringent conditions
r~gardingthe
exis tence of the del:'ivat! ves of the score function and on itsPol,lnds.
But unlike our case the asymptotic normality of her statistic
follows even when regression is non,-linear.
that the normal
scor~s
Howevel;', her observation
t~sf;:
test is the locally most powerful r,ank
under the alternative that the variables under consideration have a
bivariate normal distribution is equally
v~lid
in our case.
In fact,
in case of linear regression our result is slightly more genefiill in
the sense that whatever be the parent distribution, if1JJr
and cp*
= cp,
optimal.
the resulting test as considered by us is asymptotically
We give expl:'essions for e in some particular cases:
TABLE 2.5.1.
Showing some efficiency expressions.
= 00-1 ~ -1 (u),
CP(u)
cp *
1JJ*
1
e
= WI
1JJ 1 (u)
2 u - 1
1
= 01~ -1 (u),
2 u - 1
2
02 iJ?
(3hr) 2
3/n
u-~
-1
(u)
~(x)
=
(2n)
-l/Z
-1 ~-l (u)
°00
u
1
-2"
3/n
J-
x-00 exp.(-t!2)dt
2
-1 ~-1 (u)
°00
02 if?
-1
1
(u)
I
I
••I
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50
Case II.
p
*
~k(u)
(i)
2
Here P2
2.
~
- 21 (k
=u
=
1, 2), ~k(u)
crk~
-1
(u), (k
= 1,2)
= 3/TI and we get after some simplifications,
1
. (1./2TI)Sin- (P 12 / Z)
l/12
~o
=
=
G(1/2TI)sin
-1
(P12 I 2)
1/12
]
using these, we get after some lengthy algebra,
where,
and L = Lo
Then,
e
M
= max
Q.
e
=
(3/TI) 2 (1 + P 12 )/(1 + (6/TI)sin -1 (P 12 I 2»,0
$
P
12
<
I.
I
I
<
1
1
I
I
••I
I
I
I
I
I
Ie
I
I
I
I
I
I
I.
I
I
51
We can draw the following conclusions as in Bickel [6].
(1)
.912:0;; eM :0; .917.
As a function of p,e is symmetric about
M
° :0; P
P = 0, concave for
<
1 and -1
P :0; 0, attains its minimum
<
of (3/TI)2 at P = 0, approaches it as Ipi
of .917 at
(2)
+
1 and ~ttains its maximum
.78.
° and
e m considered as a function of p is symmetric-about
decreases from a maximum of (3/TI)2 at p =
° to an infimum of
.827 as p + 1.
(ii)
1/J *(u)
k
~
u -
*(u) = 0 -1
00
1
2 '
~
-1
2
(k = 1, 2), 1/Jk(u) = ok
(u)~ ~(u)
2
Here P2 = 1 and so e '" PI •
= 0 -1
0
~
-1
~
_1
(u), (k
~
1, 2)
(u) .
Hence for eM' em we
g~t
exactly the s.ame
expressions as in Bickel (1965) and hence,allhis conclusions hold
in this case.
~*(u)
2
Here, P 2
= 2u
- 1, ~(u)
2
3/TI and PI = 1.
Hence ,e :;: 3/TI = .955 .
(iv)
~
* (u) = 0 -1
00
~
-1
(u),
~(u)
'"
0
-1
0
~
-1
(u).
Then, e '" 1.
-1
It can be shown in the general situation that if 1/J~(u) = Ok* ep(u),
1/Jk(u) = Ok
~
-1
2
. (u), (k= 1, .. ., p), then PI
=1
so that e
=
2
P2.
Also,
I
I
I·
I
I
I
I
I
52
* = u - 21
when, ~k(u)
I.
I
I
~k(u)
= Ok
~
-1
(u), (k = 1, ... , p), it can be
found as in the case p = 2,
(2.5.7)
where ~ = (OI""'op)' = (y l a l
£0
r
= «Pkk'»' 2
-1
~o
-1
- V
~
show that l'£ol
But
=
n
,···,
1
for all l
P
'J"J"[2
L
j,j'=1
p
L
j,j'=1
YkOk) , '£ =
-1
£02 £0 '
where~
(~ Pkk ,»)' We shall next show th~t
definit~.
~ ~'2l
~'2l ... X'Lol =
where c
«2 sin'"
,=:
is positive
,I
Ie
I
I
I
I
I
I
'
To prove this, it is sufficient to
Q (see e.g. Fraser [13], p. 55).
~
sin~l(pjJ.,/2) - PJ. j
,]
, J.' J"
are positive constants and are the same for all j, j'
~l,.~.,op.
Thus
l'2l - l'Lol
(2.5.8)
p
2n-l
L c
L ,.,., Pjj ,
n=2 n j,j'=1 J J
00
=
Using now a result of 1. Schur (see [4], p. 96) we get that if
A = «a
c
=
jj ,»
and B
==
«b .. ,»
JJ
are positive definite matrices,
«c jj ,»
«P jj ,»
= «a .. ,b. ,» is positive definite. Hence, since
JJ J j
is positive definite, «p~j'» is positive definite for all
r = 1,2, .•...
Thus from (2.5,8),
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-1
Since £0
i.e.
o'r
I"'oJ
-
0 >
~Ot"J
-1
2
is positive definite,
o'ro.
rv
ro.JI"U
Hence, from (2.5.7), e
2
< P2
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CHAPTER III
NONPARAMETRIC TESTS FOR GROUPED DATA
IN THE UNIVARIATE LINEAR REGRESSION MODEL
Introduction, Notations and Assumptions.
3.1.
Consider a (double) sequence of random vectors
x
~
(X
vI
,.,., X )', v
vv
~
1, consisting of v independent r.v.
IS,
where Xvi has a continuous d.f. Fvi(X) given by
FV1.(x)
(3.1.1.)
where~'
=
(Sl"'"
= F(a -1 [x -
So - ~Q'c
.]), i
~V1
= 1,
2" •. , v, v
~
1,
Sp) is the parameter (vector) of interest, So
and a(>O) are real (nuisance) parameters and c .
~V1
=
(c
., ..• , c . ) ' ,
1V1
PV1
i = 1, 2, ••• , v, are vectors of known constants satisfying (without
any
I~ss
(3.1.2)
of generality)
v
E c kvi = 0,
i=l
k = 1, 2, •• "
p.
Define
v
(3.1.3)
E
i=l
c~.,
k=I,2, ... ,p.
V1
We assume
(3.1. 4)
max Ic~ .1
l:5:isv
V1
= 0(1),
for all k
= 1, •.. ,
p.
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Let
v
-v = «
(3.1.5)
it
We assume that
~v
L
i=1
and
.»
c k*V.1
c k*VI 1
~
= (00
A2 (F)
J
AV
« Akk
I ) )
•
~v
7
~
as v
~
00.
= F'(x) exists and
[f ' (x)/f(x)]2f(x)dx
<
00.
-00
Clv') v'
We consider the measure spaces
of the space~
=
are positive definite, and
Further, we assume that f(x)
(3.1. 6)
and ~
it
being denoted by x
~
=
Il
v )' a typical point
x ), v ~ 1.
v
(xl"'"
Our
problem is to test the null hypothesis
(3.1. 7)
HO:,g =
Q (Le. no regression) against ,g
;:Ii:
Q.
HO :: HO\l depends on v only through the number of observations, while
the sequence of alternatives (as will be considered) will depend on
the parameters c
. (i
kV1
= 1,
2, ... , v; k
= 1, •.• , p) in addition.
More specifically, we consider the sequence of alternatives
1, ... , p,
(3.1.8)
where T
.~
=
(11"", Tp )1 is a vector with fixed real elements.
Let
{P } and {Q } be two sequences of probability measures (associated
v
v
with the sequences of null and alternative hypotheses respectively)
defined on the sequences of measurable spaces
dP
v
= P dll , dQ
v
v
v = qv dll v , v
~
1.
{* v , IIA V },
where
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..
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Introduce the likelihood ratio LOv
LO\/l})
=
as follows:
i f Pv (~) >
°
1
i f pv(~)
qv(~) = 0
'"
i f pv(,t;Sv) =
o<
qv (,e) I P,V<.~)
(3.1.9)
,
= LOv (X)
~v
qv (~v)
We are concerned with the situation when X is not observable.
~v
We have a finite (or countable) set of class-intervals
1. :a, < x .:::; a,
(3.1.10)
J
J
[where, _00 ': :; ••• < a_I
<
aO
a1
<
_00, ••. ,
j =
J+1
-1, 0, 1, ... , 00,
00 is any (finite or countable)
<o •• ~
set of ordered points on the (extended) real line [_00,00]].
observable stochastic vector is~*
v
=
(3.1.11)
=
(X
*
VI
,.
eo
03
The
X* )', where
vv
,
00
1,2, ... , v,
1.2 .. , i
l:
j=-oo J vJ.J
and
1,
(3.1.12)
2 Vij
= .
[ 0,
if X ,
VJ.
E:
I,
J
otherwise,
for all i = 1, 2, ... , v, v
~
1, j =
_OO,e:_lIIe,
-1, 0, 1, ..
0,00.
We shall first consider the case when we have a finite
set of class intervals I
-s,
~
s2
so that X*. =
VJ.
l:
j=-s 1
1,2 ..
J VJ.J
,
I
_
-s1+ 1
,
••
G
,
I
s2
,
(a
(i = 1, 2,eGe, v·, v
possible by amalgamating classes on either end.
~
-s1
1) •
_00 , a S +1 = 00)
2
This is always
Also, if s1 + s2
since there are only a finite number of class-intervals, we can
relabel the suffixes of the Ij's by replacing j by s1 + j
(j
= -s1"'"
s2)'
Thus the relabelled class-intervals are
Q, ,
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I
..
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Q,
00), so that X*.
-00 ,
V1
(i
= 1, 2, ••• , v, v
~
1).
=
L
j=O
LZ ..
J
V1J
Later on, however, the results are extended
to the case of a countable set of class-intervals.
In the present chapter, we have considered some permutationa11y distribution-free tests for the above problem.
It has been
shown that under a set of mild regularity conditions, these are
asymptotically optimal in a certain sense.
The assumption that rank
~v
=
p is made without any loss of
generality, since, otherwise, a reparametrization in (301.1) will
lead to a lower order A which will be of full rank.
""v
3.2.
Asymptotically Optimal Parametric Test.
Define
-
0, 1, •.. ,
(3.2.1)
F.
F«a.
(3.2.2)
P.
F. _
(3.2.3)
Hu)
= -f'(F- (u»/f(F-1(u»,
(3.2.4)
fj, •
J
J
J +1
J
(Fj +1
j
F
+ 1;
Q,
Fj' j = 0, I, ... , £;
1.
=
J
-
(30)/a), j
-
U
<.
l',
du, when P.
~
°
°
<:
J
0
J
= 0, otherwise.
We can rewrite
(3.2.5)
when P.
J
(3.2.6)
fj,.
J
as
0,1, ... , Q"
/::,.
J
;f
O.
Let
I
J = {j j
E:
[0, 1, ••• , Q,], Po> O}
J
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58
A2 (F,{L}) =
(3.2.7)
J
-i
L:
tJ.
.Q,
2
P. =
0
j=O
J
2:
[r
2
}";
tJ. •
jeJ J
J
P.
J
Hence,
o
(3.2.8)
<
A2 (F,{I,})
J
F.
J+1
jeJ
<p(u) duJ 2/
F.
J
(j+l
J
du ~ A2 (F)
<
00.
F.
J
Assume,
P,
max
(3.2.9)
O$j$)(.
J
<
1.
Let,
x=
(3.2.10)
(y"oo., y )' = (Tl/o, ••• , T la)' = J./a
!
p
P
=
h .
V~
1,2, ... , v).
(i
Then,
Pr(X.
e I,IH
)
v~
J v
(3.2.11)
= F[{(a '+'l
J
-
S0 )/a} - hv~.J
- F[{(a, - So)!a} - h .J .
JV~
Using Taylor expansion, (3.2.1), (3.2.2) and (3.2.4), we get now,
i = I, 2" •. , v,
for all j
~
(3.2.12)
Pr(X ,
where 0
(3.2.13)
J,
vl
<
e.
< 1
J
C
I.IH )
v
J
g, (h .) = p.
V~
P.[l + h .tJ.. + h .e,g.(h .)]
J
v~
J
V~
J J
V~
=
P .. (say),
V~J
and,
-1
J
=
J
fhov~. [f'({(a
j +1
- So)!a} - e .y)
0(1), uniformly in j e J,
J
i
1,2, ... , v.
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59
Fix first a and
.e
I
but not Yk'S (k = 1, 2, ••• , p).
Then the likelihood
function of X* under H is given by
·-v
\)
0, 1, ... ,
where a:
J
Q,).
Let
(3.2.15)
\)
Q,
= [(d/dYkH E log[ E Z .•
i=l
j=o \)lJ
(F(a~
J+1
- h .) \)1
F(a~
J
- h .)]}J
\)1
x=Q
Q,
E
\)
=
Ie
I
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I
I
I
I
0,
E
c
i=l
'1(
j=O
Q;
kvi
E
j=O
v
E
i=l
c*
kvi
(noting that Z .. Z .. f =
\)lJ
i
V1J
Z . •LLP.
V1J J J
Z .. P.
V1J J
fI.,
L:
j=O
a
k
Z •• b. •
V1J J
for all j
~
1, 2, ..• , p,
j', 0
~
j, j' ~ £,
= 1, 2, ... , \), v ;::: 1).
We fix first a and
0.
For each fixed a and
0,
it can be
shown that a quadratic form involving the statistics Tkv's is asymptotically equivalent to the likelihood ratio test criterion (to be
defined later).
Also, we shall see in section 2.3 that if we use a
quadratic form involving permutation test statistics, it is asymptotically equivalent to the likelihood ratio test criterion for each
fixed a and
0.
Since the permutation test-statistics and hence,
the quadratic form in them is asymptotically distribution-free, the
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60
test has the correct limiting size for all possible particular
hypothesis; so the test statistic is indeed "asymptotically equivalent"
to the likelihood ratio test criterion.
As we have already mentioned, the test statistic we are going
to consider will be an appropriate quadratic form in the statistics
Tkv's (k = 1,2, .•• , p).
Apart from the intuitive appeal of these
statistics being co-ordinatewise extensions of Sen's [33) statistic,
the rationale behind using this quadratic form (to be proposed later)
is that it can be proved to be "asymptotically equivalent" to the
'likelihood ratio test criterion', and, hence, (under regularity
assumptions of Wald [34) enjoys the asymptotic optimal properties
of the latter.
To formulate the optimal parametric test procedure, we need
the following lemmas:
LEMMA 302.1.
Under HO' the model (3.1.1) and the assumptions (3.1.2)2
Tpv)' is asymptotically
~
- Np (0,
~
-i\A (F,{I.})).
J
(3.1.6), T
= (T'I"'.'"
~
v
Proof.
It is sufficient to show that for any vector
~
with fixed and real elements, efT
is asymptotically
f"O.I\)
.."V
We can write,
v
efT
(3.2.16)
~
~
k=l
ekc~vi '
W =
vi
since, E(Zi,lp) = P. , 0 ::; j
'J
J
mvi Wvi
~
i=l
J1,
p
where, mvi =
~v
\!
J
l:
fj"
j=O
:5 fI, ,
i
Z •• , i = 1, 2, ••• ,
J V1J
1, 2, ••• , v,
\!
\I.
Then,
;::: 1, it is
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61
1
immediate to check that E(W
.Ip ) = J ~(u)du = 0,
VJ.
Var(W
.Ipv ) = A2 (F,{I,}),
i = 1, ... , v, v ~ 1. Hence,
J
Ip v ) = 0, Var(e'T Ipv ) = (e'A e)A2 (F,{I.})
- (e'Ae)A2 (F,{I.}).
J
J
VJ.
E(e'T
~
~v
~
Under HO, WV1 "'"
= 1, ••• , v.
i
°
V
~v
~
~v~
~
W are i.i.d.r.v. with 0
vv
<
~~
Var(W )
vi
<
00,
Using now a theorem given in p. 153 of [19], to
prove the lemma, all we have to show is that the coefficients
m ,(i
VJ.
= 1,
2, ••• , v) in (3.2.16) satisfy the Noether condition
given in chapter II.
max 1m
'<'<
l_J._V
This is satisfied since
,I
VJ.
v
using (3.1.3) and
~
i=l
m2 ,
= -e'A
VJ.
e ~ g'~~ as v
-v~
----
+
00.
The next lemma due to Hajek [17] ensures the contiguity
of the sequence of probability measures {Q } to the sequence of
v
probability measures {p }.
v
LEMMA 3.2.2.
Under HO, the model (3.1.1) and the assumptions
(3.1.2)-(3.1.6),
LOv(~v)
is asymptotically
Nl(- }(X'~X)A2(F), (X'~)A2(F)).
REMARK.
p
=1
Hajek obtained the result under the model (3.1.1) with
(i.e. case of simple regression).
Hajek's result can be
trivially extended to our model once we observe that
h . = 0(1), i = 1,2, ... ,
VJ.
Next we
p~ove
V.
a result which gives the asymptotic distribu-
tion of Iv under the sequence of alternatives H '
v
Because of
'contiguity' it is possible to use LeCam's third lemma (see e.g. [17]
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or [19]) in proving the result.
proof with a view to extending it to the multivariate situation.
LEMMA 3.2.3.
Under the sequence of alternatives H , the model
v
(3.1.1) and the assumptions (3.1.2)-(3.1.6),
l v is asymptotically
2 (F,{I.}), AA2 (F,{I.}».
Np «Av)A
~
J
~
J
Proof.
We introduce ~e, mVl., WVl. (i
It is sufficient to show that
~'Xv
N1 «~ '1\X)A2 (F, {I }), (~'1\~)A2 (F, {I
j
j
(3.2.17)
E(W
.IHv )
= 1,2, ... ,
v) as in lemma 3.2.1.
is asymptotically
}».
Now
E ~.P.(l + h .~. + h .e.g.(h .»
· J J J
Vl J
Vl J J
Vl
JE
Vl
= hVl.A2 (F,{I.})
+
J
o(h .) .
Vl
2
E ~.P.(l + h .~. + h .e.g.(h .»
· J J J
Vl J
Vl J J
Vl
JE
(3.2.18)
E I~.I 3P.(1 + h .~. + h .e.g.(h .»
J
J
Vl J
Vl J J
Vl
(3.2.19)
jEJ
=
E 1~.13P. + O(h .) .
· J J
J
Vl
JE
In deriving the above results we have made repeated use of the fact
that for any positive integer r,
E I~.I
j EJ
I.
I
I
However, we present an alternative
r
P.:;;
J
J
l-r
and max P.
jEJ
J
<
00
independently of any i
= 1, ... ,
v and
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Thus,
v
\I
3
. I E( 1mV1. WV1. 1 1Hv )
1=1
since,
max 1m .1
1:::;i:::;\1 \11
I 1m .1 3 ( I It:, .13p . + O(h .))
V1
i=1 V1
je:J J
J
0(1), and
v
2
I m.
=
0(1) ,
0(1) ..
i=1 V1
Hence,
\I
(3.2.20)
I E(lm .(W . - E(W
. 1
1=
since E(W .IH )
V1 V
\11
V1
= O(hV1.),
.)1 3 IH
V1
\I
)
i = 1,2, ... , V.
Further,
v
(3.2.21)
I
i=1
=
Var[mV1.WV1·.IHV ]
=
v
I
i=1
(e'A e)A2 (F,{I.})
+ 0(1) ~ (e'Ae)A2 (F,{I.})
J
J
~
~v~
~
~
and
(3.2.22)
v
E[ I m .W .IH]
i=1
V1 V1
v
v
=
(I
i=1
m .h i)A 2 (F,{I.})
\11 v
J
+
0(1)
Using (3.2.20) and (3.2.21) we get the conditions of Liapounov
(see e.g. [11], p. 215) under which the central limit theorem
holds, are satisfied.
and (3.2.22).
II
I
:::;
The result follows now on using (3.2.21)
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Define the statistic
(3.2.23)
sv
We propose the following parametric test procedure for testing Ho
against the alternatives given in (3.1. 7) based-on -the· critical
1
(3.2.24)
1JJ
1 (~~)
0
> S
if S
= S
V
1V8'
0
if S
if S
V
V
< S
V,8
V,8
V,8
where S, and 0
are chosen in such a way that
v 8
IV8
I
E [",'1'1 (x*)
P v ]=
~V
8
8,
0 <
E <
1,
being the desired level of significance of the test.
that the above test is a similar size
8
test.
We may note
Further, under the
model (3.1.1) and the assumptions (3.1.2)-(3.1.6), S
v
is distributed
asymptotically under Ho as x~ and under the alternatives as x~ (n') ,
where
(3.2.25)
Noting also that 0
IVE
+ 0
and S
V,E
+
X2
P,E
,where X2
P,8
is the upper
1008% point of a central chi-square distribution with p degrees of
freedom, we can easily obtain the following theorem:
THEOREM 3.2.1.
Under the model (3.1.1) and the assumptions (3.1. 2).-
(3.1.6), the aSymptotic power of the test procedure described in
(3.2.24) is given by
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where thenon,;,.centralityparameternis defined in (3.2.25).
Let A be the likelihood ratio test criterion-for-the testing
v
Problem considered in (3.1.7) i.e. Av = (L(x*lv»
~v N
where
Xv
(m • .Q,.e.)
= (Y
.O/(L(X*/V»
. ~v N X=X
A
X=~
v
1v ,"', ypv )' is the maximum- likelihood estimator
Our next objective is to show S + 2 log A
v
e v
of X.
~
0
in P -probability, and hence also in Q -probability because of
v
v
the contiguity of the Q -probability measure to the P ,;,.probability
v
v
measure.
We make the following assumptions on the likelihood
function L(~~lx) as inWald [34].
ASSUMPTION I.
Denote by D . the set of all sample points E for
v
v
X
which the m. .Q,.e.
v
exists and the second order partial-derivatives
(d2:/dYkdYk,)L(~~lx) are continuous functions of Y l ' ' ' ' ' Y p '
It
is assumed that
(3.2.26)
lim P(D
V-700
v
Ix) =
1, uniformly in X .
If for a sample point E , there exists severalmaximmn-likelihood
v
estimators, we can select one of them by some given-rule.
Hence,
we shall consider y as a single valued function of E defined for
v
v
all points of D •
v
ASSUMPTION II.
For any positive
(3.2.27)
lim P [/ Xv
-
XI<
Ix v
-
xl
uniformly in X, where
E,
E / X]
= 1 ,
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Xo
=
IT
i=l
L(x*.lx).
V1
For any x t any 0(> 0) and any
= (YOlt ••• t YOp)' t let
(3.2.28)
and
~kk' , i (x tXo t o) = £.u.b.
(3.2.29)
where
Ix - xol
~
o.
ASSUMPTION III.
I
Ie
v
Write t L(~~lx)
lim X. I v = V-HlO
lim X2 V = X and V~
lim 0v
V-HlO
(3.2.30) lim E
V+oo
Xlv
1)Jkk"
,1
0t we have t
(x t X. 2V ' 0v) = lim E
~kk ' . (x, X2 ' 0 )
V~
XIV
,1
v
V
I
I
I
I
I
I
,.
I
I
uniformly in Xt where E stands for expectation.
(b)
There exists an
€
> 0 such that
are bounded functions of Xl' X2 and 0 in the domain DE defined by
the inequalities IXI (c)
x2 1
~
€
and 101 ~
€
•
The greatest lower bound with respect to X of the absolute
value of the determinant of the matrix
(3.2.3J.)
I
I
e
I
I
I
I
I
I
I
Ie
67
ASSUMPTION IV.
E [(3/3Yk)L(X*.lx)] = E [(3 2 /aY 3y ,)L(Xv*.lx)] = 0 ,
k k
X
V1. .
X
1.
(3.2.32)
for all k, k'
ASSUMPTION V.
1, ... , p;
There exists an n
for all k
1, ... , p;
i = 1, •.. , v .
Let the surface S (X) be defined by
c
(3.2.34)
and the weight function n(x) be defined by
(3.2.35)
n(x)
=
lim [A{w'(X, p)}JA{w(X, p)}] ,
p-rO
I
(3.2.36)
,.
I
I
0 such that
2
E I (3/3y k )logL(X*.lx)1 +n are bounded functions of X ,
X
V1.
where for any X and any p
I
>
0
(3.2.33)
I
I
I
I
1, ... , v
i
y
1'"0..1
>
0 ,
lies on the same surface Sc (y) as y, w'(y, p)is the image of
~
rooJ
("oJ
w(y,
B is a non-singular matrix
._ p) by transformation y* = ~ y and Ny
~
~y~
such that
(3.2.37)
B'B
~y~y
=
«EX (-32/3Yk3Yk,)logL(X*ly»),
~v ~
and for any set w, A(w) denotes the (p - 1) dimensional area of w.
THEOREM 3.2.2.
and (3. 2 . 35) .
(Wa1d) •
Let S (y) and n(y) be defined as in (3.2.34)
--c~--
.~
Then for testing H as defined in (3.1.7), under the
-
0 ------
I
I
68
I-
Assumptions I -
I
I
the surfaces S (X) and weight functionn(x);
(a)
I
I
Ie
has asymptotically the best average power with respect to
(b)
has asymptotically best constant power on the surfaces S (X);
(c)
is an asymptotically most stringent test.
c
We next prove the following theorem which establishes the
"asymptotic equivalence" of the statistic S to the likelihood
v
ratio test statistic -2 log A •
e v
Under (3.1,1)-(3.1.7), S + 2 log A
v
e v
THEOREM 3.203.
Proof.
-2 loge
A = -2[(log L(x*lv))
v e ~v ~ X=~0 - (log e L(x*lv))
~v ~
X=XA]'
v
(3.2.38)
By Taylor expansion,
(3.2.39)
I
I
I,e
I
I
0 in P v
We have,
logeL(~~lx) = (log e L~*lx))
. v . X=X
A
p
I
I
+
probability.
I
I
the likelihood ratio test
c
I
I
v,
+
A
L:
k=l
(Y k - Ykv ) (
where Xv* lies in the
alogeL(~~lx)
aY
alOgeL(~~IX)1
for all k
= 1,
x=l
)X=l
k
p~dimensional
aYk
v
=
v
rectangle [X, ""\)
Y].
Since,
0
v
2, ... , p, we have from (3.2.38) and (3.2,39),
I
I
,.
I
I
I
I
I
69
(3.2.40)
But, we can write,
v
logL(!~lx) = ~ log[Z . F(a' - h ,)
'i=l
v
vJ.O
VJ.
1
,Q,-l
+ '~lZ ,,{F(a~+l - h i) -F{a~-h ,) + Z , (l-'F(a~':'h
J=
vJ.J
J
v
J
VJ.
VJ.
~
,»].
VJ.
Hence, .
alogL(X*
l;x)
"'I)
~ Z . ,{f(a~ - h ,) - f(aJ~+l - h"J.')}
v
~
c* ,
i=l kvJ.
vJ.J
j~O
J
~
Z
,,{F(a~+l
j=O vJ.J
I
VJ.
v
J(;
O.
F(a~
- h ,) -
J
V1
J
- h ,)}
VJ.
Hence,
Ie
I
I
I
I
I
I
I.
I
I
,Q,-l
Z 'of'(a1'-h ,)+
VJ.
v
V1
Z .. {f'(a~+ -h ,)
J I
V1
* *
, ~ . ~vi~'vi
Jl
~
Z
,,{F(a~+l
,,{f(a~
- h .) -
j=O V1J
,Q,
~
Z
J
- h ,) V1
F(aJ~
- h
f(a~+
-
)}2
vi
* * J'=O V1J
J
V1
J 1
ckvick'vi .....
;
r
-i=l
JI:
~
-f'(a~ - h
J
+ ZvH (-f' (ai
1=1
v
~
j=l V1J
~ Z ,,{F(a~+l
, 0 VJ.J
J
J=
- h ,) V1
F(a~ - h
J
.)}2
V1
h,,~)}
v~
- h
,)}
V1
,»
V1
I
I
,e
I
I
I
70
Under H ,
(3.2.41)
.
Ie
I
I
I
I
I
I
,e
I
I
~
v
L: ZV1J
,,{i' (a~+ ) J 1
f'
j=O
L: c * ,c * ' .E
i=l kV1 k V1
I
I
I
9.,
~=0 ZV1J,,{F(a~+l)
J
F(a~.)}
J
-
-
vL: C * 'C * ' ,E( 9.,L: Z 'jl1,21 P )
i=l k V1 k V1 j=O V1 J \)
=
v *
* . 9.,L: {f'(a' '+1) - f'(aJ~)}
L: C
'C '
i=l k V1 k V1 j=O
J
-
( L:
v
•
C
1=1
*
,C \
k V1 k
(a~)j
J
,)A2 (F,{I.}) = J
V1
The above expression could also be obtained from the relation
E
X=Q,
(a 10gL )\1
[ fa\ 10gL)
aYk
aYk' LI
which holds under the stated regularity assumptions of Waldo
Now,
+
(3.2.42)
(~C~
,c~,
i=l
V1
9.,L:
v
= E[{ L: c~ ,c~'vi
i=l V1
j=O
Z "
V1J
{
")
f (a'+l
t
• 0
J=
-
v
L: c
*
,c
*, '
i=l k V1 k
V1
9.,
,)A2 (F,{I.})12
J
J
Z "P
V1J
J
V1
- f"
(a,) }
J
j
2
F
L: 11.(Z , , - p.)}2Ip
j=O J
V1J
J
V
I
I
,.
71
I
I
I
\J
i=l
\J1
\J1
j=o J
\J1J
J
\J
::; 2(max p~l)(max c~~.) ~ {f' (a~ ) - f' (a~)}2
jEJ J l::;i::;\J
\J1 jEJ
J+1
J
!1,
+ 2 ( max c *2.) ~ ~~ P.(l - P.) .
J
l::;i::;\J k \J1 j=O J J
But, we have already seen that I~.I is finite for all j = 0, 1, ••• ,
J
Further,
!1,.
~ {f'(a~+l) - f'(a~)}2 < 00, since the summation
jEJ
J
J
involves only a finite number of terms.
Using (3.1.4) and the above, we get now from (3.2.42),
Ie
I
I
~ c~2,c~~ .E[{ ~ ~~(Z .. _ p,)}2Ip ]
+ 2
I
I
I
!1,
+ (
~
c * .c *, .)A2 (F,a.})l2 = 0(1) •
i=l k \J1 k \J1
J
J
Thus, under HO,
I
I
I
I
I
,.
I
for all k, k' = 1, 2, •.• , p.
Also, using properties of maximum likelihood estimators,
we can show in this case, under HO,
N
P
X\J
is asymptotically
~~o.~, (i.E
(_ 02l0gL(~~IX))~)1
~. X=Q
oykoY ,
~
k
_1
i.e. NP (0,
A
~
~\J
-2
A
(F,{I.})).
J
I
Hence,
Yk"
v
=
°P (1),
for all k =
1,2~
••• ,p.
I
I
.-
I
I
I
I
I
I
72
Also, we have
converges to zero in P -probability for all k, k' = 1, 2, ••• , p.
v
Now, from (3.2.40), we get,
- 2log A - (v'A X )A2 (F,{I.})
e v
~~v v
J
(3.2.43)
I
I
I
I
I
I
.I
0
in P -probability.
v
But, again,
(3.2.44)
o
Ie
I
+
=
dlog e L(x*lv)
~v,.v
--:..----.:..--1
aYk
=
X=x v
OlogeL (~~ IX)
aYk
+
O
x=~
p
L:
k '=l
Yk ,
a2IogeL(~~lx)
v
aykay k ,
where Xv** lies in the p-dimensiona1 rectangle [Q,
(3.2.45)
so that
,
(3.2.46)
T' =
~v
,
v' : i
~,'
-
\\
But,
also converges in P - probability to zero.
v
Hence,
Xv].
x=xv**
Thus,
I
I
,e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
73
(3.2.47)
in P -probability by Slutsky's theorem.
The theorem now follows
\!
from (3.2.43) and (3.2.47).
Note that, since {Q } is contiguous to {p }, S + 2log A
\!
\!
\!
e \!
~ 0
in Q -probability as well.
\!
The above theorem implies that Sand - 2log A have
e \!
\!
asymptotically the same distribution under Hoand also, under the
sequence {H } of alternatives.
\!
Thus, under the given model and the
assumptions, the test procedure based on S
\!
possesses the asymptotic
optimal properties of the likelihood ratio test as given in Theorem
3.2.2.
3.3.
Asymptotically Optimal Nonparametric Test.
Define as in Sen [33J,
(3.3.1)
\!
I Z .. = \!. for j = 0, 1, ••• , 1; \!
i= 1 \!l.J
J
(3.3.2)
F
\!, 0
=
0, F\!,j+l
=
j
I \!
m=O
lv,
m
j
I\!.
j=O J
= 0, 1, .•• , 1 •
I
I
.-
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
,I
74
Define
A
(3.3.3)
IJ.
vj
(",j+l
~(u)dul
F .•
F
v,.J
(j = 0.1, ••. , .Q.).
(",j+l
v,J
>
0, = 0 if v. =
J
°
9"
L C* . L IJ. .2 .. ,
i=l kV1 j=O vJ V1J
A
and U = (U ,
v
1v
J
Let
v
(3.3.4)
du, for v.
•
••• ,
U )'
pv
9"
L
(3.3.5)
k=l, ••• ,p,
A2
IJ.
.(v.lv).
VJ J
j=o
We propose the test statistic
(3.3.6)
M
v
The rationale behind using this test statistic is as follows:
We may first note that (as in Sen [33]) the distribution of S
v
will depend on the unknown IJ.. (j = 0,1, ••• ,9,,) even under H in
J
(3.1.7).
S
v
0
This is because we are dealing with grouped data.
However,
provides a distribution-free test under the same permutation
argument as of Sen.
ness.
We repeat this briefly for the sake of complete-
Under Ho in (3.1.7) X*. (i = 1,2, ••. , v) are Lied.r.v. and,
V1
hence, the joint distribution of X* remains invariant under any
''''V
permutation of its v arguments.
A permutational probability measure
denoted bY~ is defined on the set of v! equiprobable points
(actually there are v!1 II v.! distinct equally likely permutations
j=O J
)~
of X ) on the v dimensional real space rev.
~v
UnderCY , all the v!
v
I
I
,e
75
equally likely permutations have the common probability (vl)-l.
v. in (3.3.1) and
J
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
j = 0, 1, •.• , £.
~
. in (3.3.3) are permutation-invariant for all
VJ
We can easily verify that
= v.lv
J
(3.3.7)
E(Z .. I6=> )
(3.3.8)
E (Z .. Z .., I(?)
(3.3.9)
E (Z ., Z .,., 1(9)
V1.J
V1.J
V
V1.J
V1.J
Also
V1.
J
v
(j, j' = 0, 1, ... , £;
Kronecker deltas.
=
V
(i
=
°
(i
=
i, i'
1,2'00', v; j
=
=
0,1,00', .Q,);
1, 2,..., v; j, j'
= 0,
1,..., £, j ~ j ') ,.
(v. (v ., - 0 .. ,)) I (v (v - 1)),
J
=
J
JJ
1, 2, ..• , v,
i
~
i'), where 0 .. , are
JJ
Using these, we get,
(3.3.10)
v
(3.3.11)
=
Cov. (Ukv ,Uk' v 16'»)
v
(v/(v-l)) ( ~ c~ iC~' .)A2 (F ,{I.}).
i= 1 v
V1.
J
Thus,
(3.3.12)
(v/(v - 1))A2 (Fv ,{I.})A
•
J ~v
E(U ICP)
v = 0, Var(U Ie?)
v
~v
~
~v
Let
(3.3.13)
M
v
The following nonparametric test is proposed:
1
1JJ2(~~)
=
0
2VE:
°
if M
>
ifM
= MV,E:
if M
<
V
V
V
M
V,E:
M
V,E:
Mv, E: and O2VE: .being so chosen as E[1JJ2 (X*)
I~]
~
v
cance.
test.
This implies E[1JJ2(X*)/P ]
~
v
= E:
= E:,
the level of signifi-
i.e. 1JJ2(X*) is a similar size E:
~
I
I;e
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
,e
I
76
We may remark again in this chapter the limitations of the permutation
test statistic.
The task of finding the percentage points of the
actual permutation distribution becomes tremendous even with moderately
large sample sizes.
This leads to the study of the large-sample
distribution of the permutation test statistic.
Next we prove a lemma which is due to Sen [33].
An alternative
proof is, however, proposed with a view to extending it to the multivariate situation.
Sen's method consists in proving the result under
the null hypothesis and then extending it to the sequence of
'contiguous' alternatives.
Our proof involves a direct argument
and does not appeal to contiguity.
Under the assumption (3.1.6), A2 (F ,{I,})
v
J
either in P - (or in Q -) probability.
v
v
LEMMA 3.3.1.
as v
+
00
+ A2 (F,{I,})
J
Proof.
First we prove the result under P -probability. Write
v
2
_
A2
A2
A (F ,{I.}) E IJ. ,(v.lv) + E IJ. ,(v,lv). Now, for any j ~ J,
v J
j8J vJ J
jJJ vJ J
FV ,j+1
A0
(3.3.15)
IJ."- ,(v.lv)
vJ
J
JFv,J,
¢2(u)du -
(since P. = 0 for j ~ J).
J
_1
But using a well-known result, under Ro, F
, - F, = 0 (v 7) for
v,J
each fixed j.
Since J>2(U)dU = A2 (F)
<
00,
J
we have
p
I
I
,I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.-
I
77
F
I
.
v,J
=
¢2(u)du
0
F.
P
= a,
(1) for all j
1, ••• , ~.
J
Hence,
A2
L: b, .(v.lv) =
j~J vJ
J
(3.3.16)
(1) ,
0
P
since the summation involves only a finite number of terms.
(3.3.17)
A2
L:
b,
jsJ
. Iv)
J
vJ
L:
=
• (v
jsJ
Also,
(E 2 •
b, ~) (v . Iv) +
-
vJ
J
Now, under P , (vjlv) - P. = (F
v
for each fixed j, F
,~.
b, ~ «v. Iv) J
P .)
J
J
'+1 - F.+) - (F
v,J
J
for all j = 0, 1, ...
L:
jsJ
J
L:
jsJ
b, ~ p.
. - F.) =
V,J
J 1
+
-
J
. - F.
J
=
°P
_.l..
2)).
(v
As already observed in Section 3.2,
~ (max P:1)A(F) for all j s J, and we get,
J
(3.3.18)
L:
jsJ
f:,~«v.lv)
J
-
P.) =
J
J
Further, for any j s [0, 1, ... ,
(3.3.19)
Pr·[(v.lv)IE2. - b,~1
>
s
Ip ]
Pr·[(v.lv)I~2. - b,~1
>
s
J
=
vJ
J
aP (v
_1
2'),
(Using Chebyshev inequality, it follows that
V,J
jsJ
•
J J
J
VJ
J
0
P
~],
(1)
given any s
v
v.
'J
>
•
alp]
v
>
a,
Ii
I
,e
78
+ Pr·[(v,/v)IE2, -1I~1
J
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
=
VJ
> e:,
J
v,
J
=
olp ]
v
pr[(Vj/v)I~2, - 1I~ I >e:, v, > olp ]
VJ
J
J
v
But for any j e: J and v. > 0, we can write
J
+ Pj
l{[fFV,j+l
Fv,j
]2
~(u)du
2
-2
+ P j (P j
v,
- -;.)
fF'+l H u) du1
[
J
F,
J
•
)
Hence, for j e: J and v, > 0, after a few steps, we get.
J
I1IA2VJ•
- lI,21 (v,/v)
J
J
I
I
,.
I
I
I
I
I
I
79
1 2
:;; 2P,-1 A2 (F) I (V.!v) -P, I+ 2P,-1 A(F)[ IF ,-F,. 1 / +
J
J
J
J
v,J
J
Using the above inequality, given any
E
+ Pr'[2P,-1 A(F) IF '+1
v,J
J
cP,
:;; Pr·[1 (Vj/V) - Pjl> 6A2~F)
I
I
I
I
I
I
I
+ pr'[IFV,j - Fjl> 36A2(i) IP v ]
I
1 2
'+1 - F'+l 1 / ].
v,J
J
> 0, we get from (3.3.19),
Ie
••
IF
IP v ]
E2p 2
E2p~
+ pr'[IFV,j+l - Fj+11>
36A2(~)
!Pv ]
0(1) •
(3.3.16), (3.3.17), (3.3.18), (3.3.20) ,show that
R,
(3.3.21)
~ ~2;(v,/v)
j=ovJ
J
+
~
j£J
~~p, = A2 (F,{I,}) in P -probability.
J J
J
v
To prove that the result holds in Q -probability, we proceed as
v
follows:
Let
(3.3.22)
Fvi, j = F [(J
-1
(aj -
(3 0) -
hv i ], i = 1, 2, ..• , v, j = 0, 1, •.. , R,.
80
From (3.3.12), P 'j = P, + 0(1), uniformly in i = 1,2, ••• , v,
J
Vl.
j = 0,1, ... , g,.
H*(x)
(3.3.23)
Let
v
= v -1
F , (x)
L:
Vl.
i=l
I
I
I
I
I
*
(3.3.24)
H . , =
V,J
b. vJ'0 =
(3.3.25)
v -1
v
F i ' (j = 0, 1, ••• , g,)
v ,J
L:
i=l
H*
v,j+1
JH*
~(u)du/
fH*v,j+1
H*
v,j
v,j
du, i f H* '+ - H*
v,j >
v,J 1
= 0, otherwise.
Let, H**, = H*
- H*,
V,J
v,j+1
v,J
(j = 0, 1, •.• , g,).
Define
g,
b.~J'O H~:j
Ie
(3.3.26)
I
I
II
I
I
I
Arguing as before, it can be shown now that
I.
I
I
°,
L:
=
j=O
A2 (F ,{I,}) - A2 (F*
o ,{I,})
(3.3.27)
v
v
J
J
+
° in Qv -probability.
g,
b.~)H*~ + L: ~~(H*~ - p,) •
J vJ
j=O J VJ
J
But, H** - P
vj
. j = v
-1
v
L: (P
P ) = 0(1), for all j = 0, 1, ... ,
i=l vij - j
g,
and
L:
j=O
b.~(H*~ - P,) =
J
VJ
J
L:
je:J
,b. ~ (H *~ - p,).
J VJ
J
Using now the remark
g,
immediately before (3.3.18),
L:
j=O
b.~(H*~ - p,)
J
VJ
J
= 0(1).
Also,
g"
I
I
I·
II
I
I
I
I
I
Ie
I
I
II
I
I
I
81
oA 2.
J
=
1:::,.
J
"(
A2
!.J
0
'0
VJ
jE:J
° and as in (3.3.15) we can show that
L: (1:::,2'0
j¢J VJ
-
I:::,~)H*~
J vJ
= 0(1).
Proceeding now as in (3.3.20) it can be shown that
L:
. J
JE:
(1:::,2'0
1:::,2j)H*~
VJ
-
VJ
= 0(1).
Thus
°
(3.3.28)
A2(Fo
* ,{I.}) - A2(F,{I.}) +
in Q -probability.
v
J
J
v
(3.3.27) and (3.3.28) prove the result. Q.E.D.
Define
~
W*. =
(3.3.29)
V1
Under C? ,
v
i
L: I:::,v" Z •. ,
j=O J V1J
1,2, .•• , v.
6vJ.
are all invariant, while Z .. are stochastic. v of
j
V1J
v
*
W*. are equal to E . (j = 0, 1, ... , ~). Also U = L: c~ .W . ,
kv
V1
vJ
i=I V1 V1
k = 1, 2, •.. , p.
The following theorem gives the permutation limit
distribution of U •
~v
THEOREM 3.3.1.
Under(f' and (3.1.1)-(3.1.6), U is asymptotically
v
~
Np (O,A A2 (F,{I.}»
J
in -probability.
-
~.~
Proof.
It is sufficient to show that for any
d
d real and fixed,
p
I
, .•• ,
N1 (O, (d'Ad)A2(F,{I.}»
J
"'~
V1.
~
=
~v
*
p
L: ~ckvi
k=l
dIU
~
~v
E = (d I , ... ,
d )'
p
~
Q,
is asymptotically
underU> in probability.
v
v
dIU
(3.3.30)
where e .
I.
I
I
)H**.
VJ
Write,
*
L: e .W i
i=l V1 v
(1
1,2, ... , v).
Using (3.3.10) and (3.3.11),
I
I
I·
I;
I
I
I
I
I
Ie
I
I
I
I
I
I
I.
I
I
82
E(d'U
1,\') = 0, Var(d'U
16»v
~ ~v ' v
~ ~v
(3.3.31)
= (v/(v - l»)(d'A
d)A2 (F.{I.}).
~ ~v~
v· J
To prove the theorem, we shall make use of the Wa1dWo1fowitz-Noether-Hajek permutational central limit theorem again.
We shall show that the sequence {e .} satisfies the Noether conqiV1
tion, while the sequence {W .} satisfies the condition Q of Hajek
V1
(see chapter II) in probability.
.
Slnce
v.
0
J
~
W = v
v
max
•
l:::;l:o>V
We note first that
f th e W*. are equa 1 t 0 0A
.,
vJ
V1
j = 0, 1, ••• ,
ev
= 0 and
Q,~
Q,
-1
A
I v.6. . = O.
j=O J vJ
Since,
IeV1.1
and
d'A.......,'V.......,d
~
~ .......,.......,.......,
dIAd
(non-zero and finite).
So,
v
max e 2 ./ I e 2 . = 0(1), i.e. the sequence {e .} satisfies the Noether
V1
l:::;i:::;v V1 i=i V1
condition.
As in chapter II we can show that
max
v
-1
W*~
V1
<·1 1
< ' •• <.1~ <
1-v
v
v -1 I w*~
i=l V1
a
-+- 0 in ('.I
. v probability and
A2 (F ,{I.}) -+- A2(F~{I.}) in probability by lemma 3.3.1.
v
J
J
Hence, the theorem.
Next we prove that the test procedure based on ¢2(~~) is
power-equivalent to the one based on ¢1(~~).
To prove this~ we need
the following lemma in addi tion to lemma 3.3.1.
LEMMA 3.3.2.
Under (3.1.1)-(3.1.6), U
probability as v -+-
~v
00.
T-+-O in Pv - (or in Qv -)
~v
~-
I
I
I·
83
Proof.
It is sufficient to show that U
kv
for each k = 1, ... , p.
I,
We have,
]I,
• 0
Ie
I
I
I
I
I
I
I.
I
I
V
A
L (~ . - ~.) L ck*.z
V]
]=
I
I
I
I
I
- T
+ 0 in probability
kv
•
V1 Vi]'
].
1=1
But,
V
V
L c * .(P. + h .~.P. + h .p.e.g.(h .))
i=l kV 1 ]
V1 ] ]
V1 ] ] ]
V1
IQ)
V1]
V
E(LCk*·Z ..
i=l
V1
V
=
V
L c * .h . + p.e. L c * .h .g,(h ,) = 0(1),
] ] i=l kV1 V1
] ] i=l kV1 V1]
V1
~.P.
V
since
L c* .h . =
i=l KV1 V1
uniformly in 0
v
Var( • L c k*V1.ZV1]
..
1=1
~
j
IQv )
0(1), and g.(h .) = 0(1)
]
~ ]1"
1
~
i
V1
v.
~
v
= L ck*2. Var(Z ..
i=l V1
V1]
IQ )
v
vL c *2 , P .. (1 - P .. ) ~ 1/4
i=l k V1 V1]
V1]
1/4.
inequality, we can conclude, for each j (0
~ j
~
Then, using Chebyshev
]1,) ,
*
v
L c . Z .. = 0 (1)
k
P
1=1 V1 V1]
•
under the sequence of alternatives H. The same conclusion follows
v
v
under Ho' since E( L c~ .Z .,Ip ) = 0, and
i=l V1 V1] v
v
Var( L c * ,Z .. Ip ) = P.(l - P,), 0 ~ j ~]I,. Next we have to show
i=l kV1 V1] v
]
]
that for each j = 0, 1, ••• ,
in Q - probability.
v
P
j
= 0,
~j
=
o.
]1"
I~
.-
V]
~.I =
]
0
P
(1) either in P - or
We first prove the result under HO'
Then,
v
For j ~ J,
I
I
III
I
I
I
I
I
84
= 1 -
that is,
For j
II
I
E
0,
•
J, given any
E > 0,
Pr·(I~ , - ~,I> Elp)
(3.3.33)
J
vJ
= Pr· (12
+ pr·(I~
.-
v
~,I >
E,
v, > 0 IP )
J
v
, - ~,I>
J
E,
v.
VJ
J
VJ
J
= olp )
v
•
But,
Pr·(I~ , - ~,I> E, v,
(3.3.34)
J
VJ
Ie
I
I
I
I
I
I
= 0p(1)
~Vj - ~j
(1 - p,) v
J
as v
+
00
J
since P,
J
>
~ Pr·(v,
= 0)
= 0) =
J
(1 -
P,)v
J
+
0
o.
A
~
,
VJ
Hence,
~
-
,
VJ
~,
J
= (~_
v.
J
J:...)
P,
J
Fj + 1
fF
J.
</>(u)du
v fF j
v. F
+ -. ..
J
<p(u)du
v fF j + 1
ep(u)du •
v, F
J
v ,J'+1
+-
V,J
·
Thus,
I~
, - ~ J,I
vJ
+ ~IF
Vj v,j+l
-
F
1 2
j+l
1 /
A(F)
. •
I
I
I·
I
I
I
I
I
I
85
Hence,
(3.3.35)
J
I.
I
I
alp )
v
> E, V. >
J
::; pr·[I~-J:...IA(F)p~/2
v,
P.
J
> E/3,
J
1
+ Pr[...L A(F)IF . - F,1 /
vj
V,J
J
2
vj
> E/3,
v A(F) / F .+ - F.+ 1
+ Pr[v,
V,J 1
J 1
1 2
/
> alp]
v
V
, > alp]
v
J
A(F) > E/3,
V
J
l
::; Pr' [ IVV - p
j
j
v 21
+ pr'[(~)
I>
' > alp]
v
J
;-}r p)
3A(F) P
2
E
FV,j - Fjl> 9A2(F)/P)
J
2
v 21 F ,j+l - Fj+I I> 9A2E (F) IP)
+ Pro [(~)
v
Ie
I
I
I
I
I
I
Pr' (I ~ . - t..1
VJ
J
J
.
But, since, under P , (v, Iv) - P =
j
v
J
for j = a, 1,.".,
1
(v /v .)
)1"
- -=
P,
J
o (v
P
o (v
P
J
_1
2),
=
F
v,j - F.J
o (v
P
_1
2)
_1
2) for j E J.
Hence each term on the right hand side of (3.3.35) can be
made arbitrarily small for j E J.
we get from (3.3.33),
shown, ~ . - t. =
j
VJ
0
p
EVJ, -
t., =
J
(1) for j
Using this along with (3.3.34),
0
E J.
P
(1) for j
E J.
We have already
Thus, the lemma is proved under Ho.
Under {H }, we prove the result by writing
v
~ . - t.,
VJ
J
= ~v j
- t.
'0 + t. '0 - t.., and then showing ~ . - t. '0 =
VJ
VJ
J
VJ
VJ
0
P
(1),
t. '0 - t.. = 0(1) .. The proof is analogous to the one earlier and
VJ
J
hence details are omitted.
We may, however, point out for later
reference that in this case F , - H* .
V,J
v,J
= 0 (1), and H* , - F. = 0(1),
P
V,J
J
I
I
I·
I
I
I
I
I
I
Ie
I
I
I
I
I
I
86
each for all j = 0, 1, ••• ,
. - F. =
so that F
J
v,]
Hence, (v. Iv) - P
j=O,l, ... ,£.
J
=
j
0
P
0
p
(1) for all
(1), for all j
=
0, 1, •.• , £.
Using the above lemma, it now follows that under HO and also under
the sequence of alternatives {H }' ]v and
v
l v have asymptotically the
This along with lemma 3.3.1 implies that the
same distribution.
quadratic forms M and S as defined in (3.3.13) and (3.2.23)
v
v
respectively have asymptotically the same distribution.
We conclude
that under the sequence of alternatives {H }, M is distributed
v
v
asymptotically as x2 (n), n being defined in (3.2.25).
p
3.4.
Extension to the case of countable set of class intervals.
Suppose, instead of a finite set of class intervals, we have
a countable set of class·intervals
I. = [b., b .+1)' j = ""
J
J
J
-2, -1, 0, 1, 2, ...
We introduce the notations
(3.4.1)
F'
.
v ,J
(3.4.2)
F(a
-1
(b. - So)), j = ... , -2, -1, 0, 1, 2, . . . .
J
F'
= F'
v,j+1
(3.4.3)
(j
= •.• ,
A '0
:J
J
Hu)dul
F'
j
v ,j
F'
v,j+1
F'
V,J
du
.
-2, -1, 0, 1, 2, •..
if P jO
~
0,
- 0, otherwise,
V,J
Let
-2, -1, 0, 1, 2, ... ).
z'vJ.J
..
= ••• ,
JF'v,j+1
.
1
(3.4.4)
I.
I
I
II,
if X .
VJ.
E
I.
J
j
=
°
otherwise
••• , -2, -1, 0, 1, 2, •••
I
I
I·
I
I
I
I
I
I:
Ie
I
I
I
I
I
I
87
Consider the statistics
~ c
i=l
*
'
k V1.
00
~
j=-oo
11, OZ' • , "
J V1.J
k= 1,2, ... , p.
Our objective is to show that in this case, if classes are coalesced
on either end, thus reducing the problem to the case of finite number
of class-intervals, and consider test procedures developed earlier,
by proper choice of the terminal class intervals, the resulting test
procedure will be asymptotically power equivalent to the one based
on the statistics T
(k = 1, 2, ••• , p), (Le. one which uses all
kvO
the class intervals).
More specifically, if we introduce the statistics
v
(3.4.6)
,
~
1.=1
where 11 , = 11 j 0
J
11
00
~
s2
(j
-sl
Z "
V1.J
~
'
k*V1.,
s2
~
(k = 1, 2, ••• , p),
I1,Z,.
J=-sl
J
V1.J
-Sl + 1, ... , s2
-
1)
00
JO JO j=s2
j=s2
=
c
l1,p,/~
P jO
-sl
-sl
11
l1'op' o I
~ P
J j=_oo jO
j=_oo J
= Z' "
V1.J
(j
-s 1 + 1, ... , s2 - 1),
sl
,
~
Z'"
J=-oo
we want to show that T
- T
kvO
kv1
+
V1.J
0 in P - (and because of contiguity
v
in Q -) probability for all k = 1,2, ... , p.
v
that
I.
I
I
v
(3.4.5)
It is sufficient to show
I
I
.e
I
I
I
I
I
I
88
Now,
(3.4.8)
v
=
'I"'
L.
-I
~
L.
V].._
c*k ' [(
i-
J- S 2
.e
I
Z ' •. jO V].J
A
U
Z •
)
S2 V].S2
-SI
+ (j:-ooD.jOZ~ij
-
D.(-SI) ZVi(-SI))]
*
v
00
L: C . [L:
(D. • 0 - D. ) Z .•
k
i=1 \,l]. j=S2 J
S2 V].J
+
-Sl
'L: (D.. 0 - D. (
) ) Z .. ]
J=-oo J
-SI
V].J
•
Then,
v
(3.4.9)
=
+
(3.4.10)
E[(T
k vO
v
*
• 00
L: C
L:
(D. • 0 - D. ) P •0
i=1 k \,l]. j=S2 J
S2 J
v
Ie
I
I
I
I
I
I
I
A
U
L:
i=1
C~.
\,l].
-SI
L:
(D.. - D. (-S 1 ) ) P . 0 =
J
j=-oo JO
- T 1)21 p ]
k'J
V
=
L: C *2 . E [{ 00
L:
(D. • 0 - D. ) Z· •. '
i=l k v].
j=s2 J
s2 V].J
~
2
V
0
11
+
'I"'
L.
c *2 . 00L: (D.. - D. ) 2 P ' O
J
i=1 kv]. j=s2 JO
s2
L:
(D. j 0 - D. (
) ) Z .. } 2/ P]
j=-oo
-sl
V].J
v
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I'
89
.-
I
L
j=S2
~
PJ'O' P(-Sl) =
2
tJ.,2 P,
,
J a JO
J=-oo
00
Since,
uniformly in j, given any
L
€
>
0,
we can find sl and s2 such that
-sl
L tJ.,2 P,o
00
<
j=S2 JO J
2
L tJ.jOP
jO
j=_oo
€/2,
<
E/2 ,
so that from (3.4.10),
- T 1)21p ] < 2€ .
kv
v
E[(T
k va
00
Since,
L
2
tJ.'oP,o
j=O J
can find an
3.5.
J
~
This proves (3.4.7).
uniformly in j, given any
~
such that
2
L fj" oP 'a <
j=~ J
J
00
€.
€
> 0,
we
This proves (3.4.7).
ASymptotic Relative Efficiency (ARE).
Suppose the true distribution function is G(x) instead of
F(x) and let g(x) = G'(x).
Let
(3.5.1)
Further, we define,
(3.5.2)
I
I,
00
=
(where,
G[cr
-1
(a, -So)],
J
j* = Gj+1 - Gj'
=
j = 0, 1, ... , ~+ 1;
(3.5.3)
P
(3.5.4)
~*(u) = _g'(G- 1 (u))/g(G- (u)),
j
0, 1, ... ,
1
~;
°
<
u < 1
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I·
.I
90
G'+ l
JG.J
*
~. ~
(3.5.5)
J
~
*(u)du/P *.•
if p~
>
J
J
O,~ 0, otherwise,
J
~
(j
0,1, .... .R-).
Gj + 1
~~* ~ J
(3.5.6)
J
G.
~(u)du/P~,
j
J
~
0, 1, ... ,
.R-;
J
(3.5.7)
B2 (F, {I.}) ~
(3.5.8)
J
.R-
~ ~**2 P*
j~o j
j
.R-
e(F
(3.5.9)
, .
p(F, G,{I.})
J
(3.5.10)
'" A*.A*.*P*.
j=O J J J
G {I })
J
,
L,
'-'
'-'
= e(F, G,{I.})/[A(G, {I.})B(F,{I.})];
J
J
J
(3.5.11)
- g' { (a. - 13 0) / C5
J
where 0
<
e.
J
1 for all j
<
i = 1,2, .•. , v.
=
J
{jlO ~ j ~ .R-, p~ > O},
J
.R-
*
~ Z .. ~~, k = 1, 2, •.• , p.
kv = ~ ckvi j=OV1J
J
i=l
T*
Let T* = (T~v"" , T*pv )'.
~v
(3.5.13)
J*
e. y }] dy
Define the statistics
v
(3.5.12)
E
-
Then,
E(T*lp v ) = 0, Var(T*IP v ) = A A2 (G,{I.})
,
J
~v
~
~v
~v
,
\1
I
'I·
'I
I
I
I
I
I
,Ie:
I
91
Let,
s*v
(3.5.14)
Further, let,
T**
kv
0.5.15)
T**
~v
S**
(3.5.16)
v
v
*
c kvi
L:
i=l
1v '
(T**A- 1T**)B- 2 (F
~v
k
1.J J
1, 2, ... , p,
~v
~v
' {I.})
J
Then,
E(T**lp
A B2 (F,{I.})
•
~v
v ) = ~0, Var(T**IP
~v
v ) = ~v
J
(3.5.17)
If the permutation test procedure as described in Section 3.3 is
now used, it is asymptotically power-equivalent to the test procedure
(based on the critical function ~3(X~»
I
I
j =0
Z. . L'L** ,
,
... , T**)
pv
(T**
.-
I
I
I
I
I,
I·
Q,
L:
i f S** > S**
1
(
defined as follows:
V,E:
V
\
~3 (fS~)
i
-
1
0
-GVO'
o ,
if S**
v
E(e'T**IP v )
~
=
~v
V
P
L: ( L:
v
V,E:
i f S** < S**
V,E:
V
We may note that for any
(3.5.19)
S**
p
L: ( L:
i=l k=l
~ ~
Q with real and fixed elements,
*
ekc k .)
V1.
Q,
L: 6~* E(Z .. /H )
j=O J
V1.J
V
.»
*
ekc * .) Q,L: 6.**P.*( 1+ h .6.* + h .B.k.(h
k
i=l k=l
V1. j=O J
J
V1. J
V1. J J V1.
II
I
92
.I
I
I
I
I
I
p
v *
~ * ** *
L e k L ck .h .) L ~Jo~J' PJo + 0(1)
k=l
i=l Vl Vl j=O
=(
(3.5.20)
Var(e'T**lp v )
~
=
~v
=
(e'A e)B 2 (F,{I.})
+ 0(1) ~ (e'Ae)B 2 (F,{I.})
.
J
J
~
~v~
~
~
It can be proved in the same way as lemma 3.2.3 that under the
sequence of alternatives {H }, the model (3.1.1) and the assumptions
v
(3.1.2)-(3.1.6) with G and g replacing F and f respectively, T**
~
is asymptotically Np «Av)C(F,
G,{I.}),
AB 2 (F,{I.}».
.~
J
~
J
Hence, S**
Ie
is, under HO, distributed asymptotically as x~ (no) , where,
I
(3.5.21)
I
Also, ~v
T* is under Hv asymptotically
Np~'<'
«Av)A2 (G,{I.}),
AA2 (G,{I.})).
.
J
~
J
I'
I
I
I
I
e
I
I
v
Hence; S* (which provides now the asymptotically optimal test in the
v
sense described in section 3.2) is asymptotically x~ (no) with
(3.5.22)
Using the definition of ARE as given by Hannan [20] or
Andrews [2] as the ratio of non-centrality parameters the ARE of the
permutation test procedure as given in (3.3.14) relative to the
asymptotically optimal test as given in (3.2.24) with S~ replacing
Sv is given by
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I
I
I
I
a-
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93
(3.5.23)
e
F,G,{I.}
J
The above expression turns out to be the same as in Sen [33].
We may note that the expression for ARE depends on the true distribution, the assumed distribution as also on the grouping structure.
However, this is unavoidable and the optimum way of grouping in a
particular context will depend heavily on the parent distribution.
In case of ungrouped data where the X . 's are observable,
V1
while the model (3.1.1) and the assumptions (3.1.2)-(3,1.6) remain
the same, for testing HO against {H }, one proceeds on lines analogous
v
to that of Hajek [17].
Hence, if S
denotes the test statistic for
vo
ungrouped data, in the situation where both the true and assumed
distribution functions are F, Svo is asymptotically x~(noo) with
non = (X'~X)A2(F).
In that case the relative loss of efficiency
due to grouping will be
1
Q,
(3.5.24)
-
2:
j=O
~.]2dU/J
¢2(u)du
J
0
I
I
I
I
I
,e
I
We can determine the class-intervals in such a way that given any s
max P. = max (F
j
J
j
j+l
- F.) < s.
J
This can be achieved
independe~tly
proper choice of class-intervals.
Again, in situations where the true distribution function G
of efficiency is
0,
of v.
Thus, the loss of efficiency can also be made arbitrarily small by
and the assumed distribution function F differ, the relative loss
+
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94
.I
I
I
I
I
I
= 1 - [p(F, G,{I,})/p]2[A(G,{I,})/A(G)]2,
J
J
which also can be made arbitrarily small as in the preceding case.
Also, the expression (3.5.25) may be greater than, equal to or less
than (3.5.24) with G replacing F, depending on p(F, G,{I.}) and p.
J
We give expressions for
~(u)
and
~,
J
known distributions:
TABLE 3.5.1.
(a~
J
f(x)
Density
~(u)
Some expressions for
=0
-1
and
.I
J
-f'(x)jf(x)
~,
~(u)
J
(when P ,~O) ,
J
j=O,l, •.• ,!/'.
Normal
x
~-I(U)
[f(aj)-f(aj+l)]/
[~(a~+ )-~(a~)]
J 1
Double
Exponential
2"1
expo (- Ix .I)
sgn x
J
sgn(2u-l) i -1 if a' ::;;0
j+l '
\ +1 i f a~2:0,
l
,/
'I
J
,
,
(e aLe-aj + 1 )
Logistic
e- x (1+e- x )-2
(l_e- x ) (l+e-X )-l
2u-l
/
(2-e a'j -e-a'j+l)
i f a~<O<a~+
J
J I
I
I
I
~,.
(a, - So))
J
Ie
I
I
I
I
in cases of some wel1-
(l+.e
-a''
+(l+e
J)
-a'
.
-1
j+1)-1_ 1
.
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I
.I
I
II
I
I
I
From the above table, it is clear that the Normal Score Test, Sign
Test or Wilcoxon Test are asymptotically optimal according as the
parent distribution is Normal, Double Exponential or Logistic
respectively.
3.6
Applications.
The model considered includes as particular case the
p-sample (p
xV1..
(i
.-
I
2) problem.
Note that v =
P
E n
k=l
(3.6.1)
Consider the situation when the
1,2, ••• , v) are from p populations IT l
of the v observations n
kv
'
lim nkv/v =
V-7<>o
Ie
I
I
I
I
I
I
I
~
kv
are from population IT
TI
k (0
< TI
k
<
1, for all k
k=l
The regression constants c
(3.6.2)
c
k
(k
=
p
1, 2, ••• , p).
TI
k
1,2, ... , p;
1) •
. are given in this case by
k V1
(nkv/v) if the i
th
observation is from k
kvi
otherwise,
(i = 1, ... , v; k = 1, 2, ... , p).
v
E c .
i= 1 k V1
0,
and out
Assume
P
E
1 -
IT
, •.. ,
Then,
v
E c2 .
i=l kv1.
1, ... , p.
th
sample,
I
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I
I
I
I
I
Hence,
(3.6.3)
1
nl v n pv
~
n2v n pv
~ 1/2
G
_
1
e ,. •
(v-n 2 ) (v-n )
v
pv
_
.............................................
n Iv n pv
~ 1/2
-
n 2v n pv
~. 1/2
1
(v-n ) (v-n )
2v
_
pv
G
Under (3.6.1), ~v +~, where,
1
Ie
I
I
I
I
I
I
I
,I
1/2
1
• " ••••••• e •••••••
G ••••••••••••••••••••••••
1
A is a p x p matrix of rank p
"'v
A
~11v
A =
"'v
(p-1) x(p-1)
A
~21v
1 x (p-1)
A
~12v
(p-1)x1
A
22v
-
1.
Writing
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97
We notice that
I
AIIV
is of full rank.
Let
D
= (I 0), where
(p-I)xp
p-l~'
is the identity matrix of order (p-l).
p-l
the same distribution as
of alternatives {H }'
v
QIv
DU
"""'v
has asymptotically
under HO and also under the sequence
Thus,under H ' QQ
v
v is asymptotically
Np«~)A2(F,{Ij})' (~D')A2(F,{Ij}))' But ~Q' = ~ll where
~ll
!l
=
~12
(p-l) x (p-l)
(p-l)xl
1\.22
t:.21
1- P(p-l)
and thus by reparametrization get a non-singular distursion matrix for
transformed variables.
=
-1
(D
U )'1\.
(D U) where ~v
1\. *
~v~v
~11v ~~v '
is a generalized inverse of 1\. , Mv remains invariant under repara~v
I
I
I
I
I
I
I
.I
metrization.
p. 20.
..:.1
~llv
can be obtained by using a formula given in Rao [32],
In the case of ungrouped data,
~he
p-sample problem was
considered by Puri [28(a)] under essentially more stringent conditions.
We might remark that in the p-sample problem, Basu [3] has
considered the censored case where only v *«v) of the ordered variables
in the combined sample are observable, while v*/v
+
p, (0
<
p
<
as v
+
00.
IO:x
<
xc' while probabilities P. (j = 1, ••• , £) of belonging to
II, 12"'"
1)
A similar problem follows as a special case of ours when
J
1£ are sufficiently small.
However, while in Basu's
case v* is given, while the corresponding truncation point is random,
in our case, the truncation points are fixed, while v. (j = 0, 1, ••• ,£)
J
are random.
In spite of these basic differences, the A.R.E.'s of
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98
the two tests will be the same. and further. we can study the
optimality properties of Basu's test according to the criteria
described earlier.
test is optimal.
As such. for the logistic distribution. Basu's
Similar comparison can be made in particular with
the test proposed by Gastwirth [14] for the two sample problem in
the censored case by taking p = 2.
(See also [33]).
The above analysis is also applicable when the observable
r.v.'s are grouped in several ordered categories (the case of Categorical Data) if the underlying parent distribution is continuous.
Concluding Remarks.
3.7.
The present chapter includes. as a particular case (when
p
= 1). Sen's model where the null hypothesis has been tested
against one-sided alternatives.
This also includes a similar two
sample problem as considered by Gastwirth [15].
Unlike the case of
one-sided alternatives. we do not get here an asymptotically most
powerful test. but a test asymptotically optimal in Wald's sense.
Also when the class intervals are sufficiently small. the efficiency
of the M -test reduces to that of a similar test (for ungrouped
\!
data) considered by Jogdeo [24].
We may also remark that unlike
Hajek [17] and Sen [33] we need only impose the Noether condition on
\!
the c
k
=
.'s. but no other assumption like
k \>1.
1 ••••• p.
~
0(1), for all
This helps avoiding some artificiality in taking the
regression constants c
\>
~ c2 • =
• 1 k\>1.
1.=
. = 0(1)
k\>1.
(k = 1, ••• , p; i = 1 •••••
1); here. for example. we may consider the case
\>;
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c
kvi
=
i -
V;l
(k
= 1, ... ,
p; i
= 1, ... ,
hand, we have considered a sequence
~
v
v; v
~ 1).
On the other
of alternatives which are
"local," while Hajek and Sen have regarded the regression coefficient
S as fixed and still, in effect, have considered local alternatives
v
by imposing the condition
condition.
L c2 .
•
1 V1.
1.=
= 0(1)
along with the Noether
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CHAPTER IV
ON TESTING THE IDENTITY OF MULTIPLE
REGRESSION SURFACES FOR GROUPED DATA
4.1.
Notations and Assumptions.
Consider a double sequence of LLd.r.v. ~v
X = (XVI. ,"', XVV )',
v
~
1, satisfying the model (3.1.1) and the assumptions (3.1.2)-
(3.1.6).
In this case, however, we want to test
(4.1.1)
HO:.@.*
Sp )' = Q
(So, S1,""
We assume in addition,
F(x) + F(-x) = 1, for all real x.
(4.1.2)
The usefulness of the symmetry assumption about F will be revealed
in section 4.3.
A particular situation where this model is realistic
is as follows:
Consider a double sequence of i.i.d. stochastic vectors
X.
~1
( 1) X(2.
(Xvi
' V1
(4.1.3)
~
i
»' . =
X(~) =
V1
= 1, 2; i = 1,
,1
i
k=O
1, ••• , v,
Sk(~)
c
.
kV1
v, v
v with cO ..
V1
~
\>
~
1, satisfying
+ 0E(:)
\>1
1, where, c.
= (co .,"', c . ) ' ,
V1
V1
P\>1
= 1 (i = 1,2, ... ,
v), v ~ 1, are vectors
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101
of known constants, 0(>0) is a nuisance parameter and S(~)
k
k = 0, 1, ••• , p,
~
= 1, 2, are unknown regression parameters.
E(~»
\)1
We assume as in [30], (E(:),
V1
are either interchangeable
or their joint distribution is diagonally symmetric about (0, 0).
We want to test
,·o(l)
Ho·
lJ k
(4.1. 4)
0(2) ' for all k =
IJk
°
, 1 , ••• , p,
against the alternatives that at least one of the (p + 1) equalities
in (4.1.2) is not true.
= HO
Then Ho
E
•
S~l) - S~2), k =
1, 2, ... , v, v
EVi ' i
0,1, ... , p.
Writing X . = X(l)
\)1
vi
given in (4.1.1).
(2)
(I)
V1
Let Sk =
~
1, the model can be rewritten
as
(4.1.5)
X.
\)1
1, ... , v, v
~
1.
I f E • (which are i. i. d.) have a commond. f. F, (3.1. 1) and
\)1
(4.1.2) follow.
Note that, C~v
v
L c~ . = v.
i=l
So,
\)1
-.1..
cOVi/C o\) = v 2, for all i = 1,2, ... , v.
v
.
v
= L C .
.
V1' k V1
i=l k V1
* *
*
L Co • c
1=
I
= 0,
= 1,
k
2, ••. , p.
We introduce the
notations
v
(4.1.6)
where,
A
""Va
=
« . L I ck*
1=
.»
.ck* ' \)1
V1
k, k'--o,l, ••• ,p
-+
~O'
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102
(4.1.7)
(3.1.5) implies
(4.1.8)
~vo
and 1::,0 are positive definite.
°
<p (u) = -
(4.1.9)
1/J(u) = - f' (F
-1
«1 +
.
Let
< u < 1
-1
u)!2»!f(F· «1 +u)/2»,
°< u < 1
then,
f
1
1
2
2
<p (U)dU= !o1/J2(U)dU =A (F) <
O
(4.1.10)
00.
In our case, the X , (i= 1, ... , v) are not observable.
v~
We
have a finite-set of class intervals
(4.1.11)
I,:a,
[where
= a_.Q,_l
-00
J
1::;
J-'2
x < a
<••• < a
2 .
1 (j =.;....Q"
j +2'
1
-2
... .Q,+1,,,.,-1,
< al <••• <a 1
'2
.Q,+2'
=
0,1, ... , .Q,-1,.Q,)
denote a finite set
00
of ordered points on the real line with a , = a, ; aO = 0].
-J
I
~~ = (X~I"'"
observable stochastic vector is
J
The
X~v)!,where
.Q,
(4.1.12()
X*,
(i = 1, ... , v)
L: 1,2"
_.Q, J
V~
v~J
and
(4.1.13)
for all i
z ..
v~J
= 1,
1
if X i
v
I,
o
otherwise
E:
J
=
2, ••• , v, v
~
1
(j
-.Q, , •.•• , -1,0, 1, ••• , .Q,).
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103
For the testing problem considered here, a class of permuta.,..
tiona1ly (conditionally) distribution-free nonparametric tests has
been proposed which is comparable to one in the previous chapter
where the problem was one of testing the hypothesis HO : 13
(k
= 1,
2, ••• , p), while 130 and
=
were nuisance parameters.
°
As it
has been remarked a1ready,:'inchapter III, statistics analogous
to rank statistics were considered, while here statistics analogous to
signed-rank statistics have been used.
We may also comment at this
stage that in what follows, the assumption (3.1.2) namely
v
L c
*
.
i=l k V1
=
° (k = 1,
2, ••. , p) is unnecessary, and the same results
without any modification will hold once we assume that /::;vo and /::;0 are
positive definite.
This assumption,however, can be made without
any loss of generality as already observed in chapter III.
4.2.
Asymptotically Optimal Parametric Test.
We define,
(4.2.1)
F. = F(aj
J
_11 0) ,
j =
-~,
... ,
-1, 0, 1, .. " ,
9.,
-1, 0, 1, ... ,
9.,;
+ l',
2
(4.2.2)
P.
(4.2.3)
F~ = F.
Fj + 1
J
Fj' j =
-.R,' ••
~"
= 2F. -1 (j = 1,'2, ••• ,
F
-j+1
J
J
J
-
so that, P* = F*+ 1
j
J
F~
J
2P
=
j
9.,
+1),
(j = 0, 1, ... , 9.,)'.', .
Fj + 1
(4.2.4)
1::.. =
J
J
F.
Hu)du/P j ' if p.
J
J
I.
I
I
0
k
= 0,
otherwise
it
°
j = -9., , ••• , -1, 0, 1, ... ,. Q,.
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104
(4.2.5)
*
~.
J
*1
j
.
*
= fF + $(u)du/P
j
F~
P~
, if
0
;t
J
J
j
;t
= 1,2, ••• ,
Q"
otherwise
= 0,
When P
.j
0, we can write,
It follows immediately that,
(4.2.7)
P.=P.,~.=~.
So,
0 .. Also, if P.
~o=
(4.2.8)
-J
J
-J
;t
J
0,
*
~2 f(1+Fj+l)/2
J
_
~. =
(j=O,l, ••• ,t).
J
IJ(1+Fj+l)/2
~(u)du
(l+Fj)/2
2
.
(1+Fj)/2
Fj +1
=
JF.
du
$(u)du
=
~.
J
(j
=
1,2, ... , t).
J
Let,
t
t
(4.2.9)
where J+ = {joe (1,2, •.• , t)IP
(4.2.10)
2
2
A~(F,{Ij}) =:~' ~2.p. = 2 ~ ~.P. = 2 ~ ~.P. =
j=-t J J
j=l J J
jeJ+ J J
0 < A5(F,lI.})
J
~ 2"
j
jeJ+
>
O}.
Hence,
f(j+l,j>(U)dU
~ Fj
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105
Assume t
(4.2.11)
P,
max
j=Ot 1 t'''tJ/, J
< 1.
Consider the sequence of alternatives t
Lett
(4.2.13)
(4.2.14)
Yk = a
-1
= a-1
*
'k (k = 0t 1 t " ' t p) and let Y = (YO"'"
*
P
k=O
r ck
Yp )';
"k (i = It 2 t ' ' ' t v).
V1.
Then the likelihood function of X* under H is given by
"\)
v
(4.2.15)
v
J/,
L(~~I~*) = IT {r z , ,P(X i
vi
=1'J=-x,n V1.J·~
E
I,)}
J
v
J/,
= IT { r Z, ,(F[(a'+I!a) - h ,] - F[(a,/a) -h ,J)}
i=l j=-J/, V1.J
. J ..
V1.
.J.
V1. ,
v
=
J/,
{r z
i,P,(l + h .6, + h iSjg.(h,))}t
'-I J--x,
'- n V J J
V1. J
v
J V1.
1.IT
°
< S, <
J
1,
where t
h ,
(4.2.16)
-If V1. {f ,
gJ,(h V1.') = p.
J
°
[(a'+lfa) - S,y] - f'[(a,fa) - S.y]}dYt
J
J
for j E J+ U J_ , where J_ = {j E (-1, -2, ••. , -J/,)
J.
Ip.J
J
> o} •
Let
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v
=
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s/,
c* . L: Z .• b. •
' - S/, vJ.J J
i=l kvJ. J-L:
v
S/,
c* . L: (Z ..
=
• 1 v J.J
i=l kvJ. J=
L:
(as in chapter III)
-
ZVi(_j»b. j , k = 0, 1, ... , p.
j
i.e. X i
v
Let
(4.2.18)
IXVJ.. I
1
if
o
otherwise
E:
I
8
Ij,U I -J.
U •.
VJ.J
(i = 1, ... , v; j = 1,2, ... , S/,).
Then,
(4.2.19)
Z
- Z"
= U
sgn(X)
vij
vi(-j)
vij
vi '
j = 1,2, ••. , s/', i = 1,2, ... , v,
where,
·1
(4.2.20)
sgn X =
vi
0
-1
if Xvi > 0
if X = 0
vi
< 0
if X
vi
i=1,2, ••• ,v.
Thus,
v
(4.2.21)
T
kvO
L:
i=l
S/,
c{:v i ( j=l
L: U. ; b. . )sgn
\)J.] J
X ., k = 0, 1, •.• , p.
VJ.
Let
Next we state three lemmas the proofs of which are exactly the same
as lemmas 3.2.1 - 3.2.3.
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LEMMA 4.2.1.
(3.1. 6),
Under HO' the model (3.1.1) and the assumptions (3.1.2) -
l vo
LEMMA 4.2.2.
is asymptotically Np~l (Q~ ~oA~ (F ,{I
j
}» .
Under HO' the model (3.1.1) and the assumptions
(3.1. 2) - (3.1. 6), LoV<~) is asymptotically
N1
(-'t
(;/'fjoX*)A2 (F) ,
LEMMA 4.2.3.
(X*'~oX*)A2(F».
Under the sequence of alternatives {H }, the model
v
(3.1.1) and the assumptions (3.1.2) - (3.1.6), !vO is asymptotical}.7
Np+l(,(~ox*5A~(F, {I j
}),
~oA~ (F ,{ I j
}) ) .
If {P) and {Qv} denote two sequences ofprohahility measures
as in chapte r I II, lemma 4.2.2 ensures that "Q
P ."
v
v
are contiguous to
Define now,
(4.2.22)
8
voo
-2 .
-1
= Ao -(F, {I. }) T' 011. 0 T 0
J
"'v "'v "'v
The optimal parametric test procedure in the sense described in
chapter III is the same as in (3.2.24) with Svoo replacing 8 '
v
4.3.
A Class of Nonparametric Tests.
We introduce the following notations:
v
(4.3.1)
~
Z"" = v ", (j = -5/" ••• , -1, 0,1, ... , JI,).
i=l V1J
J
80,
v
(4.3.2)
~
U ""
i=l V1J
= v" +
J
v
= 1,
" (j
-J
2, ... , 5/,).
Let
I.
I
I
(4.3.3)
J
=
0, FV 'J"+l
=
~
v lv,
m=-JI, m
j = -5/" ••• , -1,,0, I, •.. ,5/,;
I
I
I·
I
I
I
I
I:
I
Ie
I
I
I
I
I
I
108
(4.3.4)
F*
= F
- F
,
V,j+1
v,j+1
V,-J =v
(4.3.5)
=
=
J
F~ ,j+1
~(u)du
E
F* ,
v,J
',V
m=-J
/JF~ ,j+1
m' j = 0, l, 2, ... ,
du
(l.+F* '+1)/2
V,J
J(l+F*
/J..
qi(iJ)du
)/2
J( l+F*j+
V.
1
du
.(l+F*,)/2
V,J
~*
1,2, •.• ,
, = 0, otherwise (j
vJ
J/,).
Let
v
J/,
E ck* ' ( L: U ,,~A* j) sgn X ."
i=l V1. j=l V1.J v
V1.
(4.3.6)
k = 0, 1, ..• , p,
*
and .1J * = (Uov"
•• , U*pv )'.
We may note that in the case Xvi
regarding the value of sgn X ,.
V1.
£
la, we have no information
This, however, does not" affect the
test procedure (based on the statistics U:v,k = 0, 1, ..• , p), since,
if Xvi
£
la, UVij =
° for all
j
= 1,2, .•• ,
J/,
and, as such,
contribu~
tion of such terms to the calculation of Uk*V (k = 0,1, •.. , p) is
zero.
Since, we are dealing with grouped data,even under HO, the
joint distribution of the U* 'swill depend on the ~~'s.
kv
.
J
But, using.
a permutation argument, we can get here apermutationally (conditionally) distribution-free
test~
Under HO in (4.1.1),
IXvi I,
J/,
I.
I
I
J/,;
F* ,
v,J
,}/2
V;J
~
-1 j
i = 1, 2, ... , v are LLd.r.v. and so are x*
via
L: I~U " ,
j=l J .. V1.J
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
109
j = 1, ••• , )/" i = 1, ••• , v). Defining now,
(I' = 1. U I
-j'
j
J
J/,
A*
W = L: U .. f:::, • (i = 1, ••• , v), we see that. -v
W = (WV 1"'" Wvv )'
vi
.
V1.J v J
J=1
has a joint distribution which remains invariant under any permutation
of its v arguments under Ro.
Further, under RO, sgn X . can assume
V1.
two values +1 and -1 each with prob. 1/2 independently of the W . 's;
V1.
sgn X, •.. , sgn X
vv
VI
X*
v
vo = (X*10""
are independently distributed.
X*
vvO )' and sgn XV = (sgn XI"'"
V
Use the notations
sgn Xvv ).
If we now consider a finite group G of transformations
v
{gv}
which maps the sample space onto itself, where a typical trans-
formation gv is such that
m
m
g)~v = «-1) 1 W .-, •.• , (-1) v Wvi ), mJl, = (0, 1),
v 1.1
v
= 1, 2, ••• , v and (i}, ••• , i ) is any permutation of the integers
v
(4.3.7)
J/,
v
(1, ••• , v)~ then, there is a set of 2 vI transformations in Gv '
v
Under RO, the conditional distribution over the set of 2 vI
realizations (generated by G ) is uniform, each having common condiv
v
tiona1 probability (2 vl)-l. We denote this conditional probability
measure by CY'.
V
remain fixed.
(4.3.8)
We may note that under (j)', F* .' s and hence ~*.' s
v
v,J
vJ
It is now easy to verify that
E(U* If,:-,.,) = 0
kvl\T v
(k = 0,1, ... , p) ;
v
E(Uk*"Uk*,,,!(Y,')
=
(L: c * .c *, .)A2 (F*,{I.})
v
v
v
•
J
kV1. k ·V1. 0 . v
1.=1
(k, k' = 0,1, ... , p), where,
J/,
(4.3.9)
A20
(F*,{I.})
v
J
=
A*2
L: f:::,
.(v. + v .)/v •
j=l vJ
J
-J
I
I
.e
I
110
Consider the quadratic form,
(4.3.10)
As in the earlier case, the following nonparametric test is proposed:
I
I
I
I
I
where M
Ie
score" statistics instead of the ordinary "score" statistics as in
I
I
I
I
i f M* > M
1
(4.3.11)
<5
4VE
0
V,E
and
<5
4VE
E,
that E[~4(X* , sgn X )Ip ]
"'v 0
"'v
v
E
i f M*
V
V,E
M
V,E
i f M* < M
V
V,E
are chosen in such a way that
E[~4(~~O' sgn ~v)I(P~] =
size
V
the level of significance.
=E
This implies
i.e. ~4(X* , sgn X ) is a similar
"'v 0
"'v
test.
We may remark that it is necessary to introduce the "signed-
chapter 3 because here we are interested in testing for So along with
Sp •
Had permutation-test statistics as in the earlier chapter
been introduced without consideration of signs, the one for So would
have reduced to a constant under the permutational probability
measureO) '. A similar situation was faced by Adichie [1] and by
v
Puri and Sen [30] in case of ungrouped data and they introduced
"signed-rank" statistics instead of "rank"-statistics for similar
testing problems.
To study the asymptotic permutation distribution of the
I
I
.e
I
test statistic, we first extend a result
0t
Hajek [16] involving
the asymptotic distribution of the "simple linear rank statistic" to
that of the "simple linear signed-rank statistic."
To forndate the
I
I
I-
I
I
I
I
I
I
III
theorem, we proceed as follows:
Consider double sequences {b "
V1.
~
{a ' 1
vi
v
max
l~i~v
I
I
I
I
I
.I
b 2 ,/( E
V1.
i=1
2 )
b "1.'
v
i
v, v 2 I} and
~
= 0,
k
<max<,
] <'1.1 -.
•• ·1.
•
q
<v
-
v
E- (aV 1.,
k
.
a=l
a
lim[k Iv = 0] => lim --'-----~v_v--------v-700 v
V-700
E (a , - a )2
'1
v J.
V
1.=
(4.3.14)
o.
Let XI'"''
Xvv be v LLd.r.v. having a continuous d.L F(x)
v
satisfying F(x) + F(-x) = 1 for all real x; also let (R "'"
V1
R
vv
)
be a random vector which takes v! possible permutations of
(1,2, ... , v) with equal probabilities and distributed independently
Ie
I
I
~
i 2 v, v 2 I} of real numbers satisfying
lim
v-700
(4.3.13)
1
v
sv
(4.3.15)
=
E
i=1
bvia vR .· sgn Xvi'
vi
where sgn XVi's are defined in (4.2.20).
THEOREM 4.3.1.
Under (4.3.13) - (4.3 .14)
.;;.;a..:.;;n..:.;;d---"-'tho:..e'-....:a..:.;;s:...::s:...::um=p-=t-=i.;:.;on;:;s~f:..:o:...::l:;.;::l:;.;::o...;.;w..:;:i:;;.::n=g
it Sv defined in (4.3.15) is asymptotically Nl(O, v
Proof.
v 2
v 2
E b, E a ,) a.e •
• 1 V1. , 1 V1.
1.=
1.=
It can be assumed without any loss of generality that
avI
~
a
1
i
~
~
-1
V2
v.
::;
...
a vv and put a (A) = a vi for (i - 1) Iv < A ~ i/v,
v
Let T = E b ,a (U,)sgn X ., where Ul, U2, ••• is a
v
, 1 V1. V 1.
V1.
1.=
~
sequence of independent random variables distributed independently
of the: X , 's, each U, uniformly distributed over (0, 1], and the
V1.
1.
rank of U in the partial sequence Ul,""
i
Uv is denoted by
I
I
••I
I
I
I
I
I
Ie
112
Ri
(1:::; i :::; v, v ;: ;: 1).
Hence, the vector (R' , .•. , R' ) satisfies
vI
the same conditions as (R
VI
(4.3.16)
, ••• , R
vv
vv
) in the theorem, and we get,
E(T - 8 )2
v
v
using lemma 2.1 of [16].
v
Also, E(8 )
v
o
and
v
-1
Var(8 ) = v
L: b 2 . L: a 2 ..
V1
V1
v
i=1
i=1
Hence,
E(T -8 )2/Var(8):::; 212 [max la
(4.3.17)
+
I
I
I,
•
V1
v
0 as v +
00
v
v
1<'<
-l-V
from (4.3.14) putting k
is sufficient to show that (i)
N1 (0, 1) a.e.
in probability.
and (ii)
(8
v
(T
v
v
,-a
V1
= 1.
v
v
1!(L:a2 ,)1/2]
1:
V1
To prove the theorem, it
- E(T »/IVar(T) is asymptotically
v
v
- E(8 »;.IVar(S) - (T - E(T»/IVar(T )+0
v
V
V
V
The proof which is based on (4.3.17) is given in
Hajek [16].
To prove (i), we use Theorems 4.1 and 4.2.of Hajek [16].
I
I
I
I.
I
I
V
need observe only that the Lindeberg condition
where TV1. = (b V1. - b)a
v V (Ui)sgn XV1, (i = 1, 2'0'0' v), and
otherwise
We
I
113
I
,.
1
~
i
~
v, is equivalent to the condition
1
v
.L:IE[T~i I(IT' .1;:0:0)]
lim Var(T')
I
I
I,
I
I
,I
Ie
I
I
I
I
I
I
I.
I
I
v-?<Xl
V
V~
~=
v
where T'. =
v~
bv )av (U.),
1.
L: (b . i=l v~
=
0,
v
T' = L: T'
V
and
i=l vi
otherwise
.1 = 1
The last assertion follows from the fact Isgn X
V1.
bility 1 for all i
= 1,
2, .•. , v and sgn Xl"'"
v
with proba-
sgn X' , U '
vv
1
••• ,
U
v
are mutually independent.
The above theorem will be utilized in deriving the asymptotic
permutation distribution of
Q:'
In order to prove the result, the
following lemma is also needed.
LEMMA 4.3.1.
2
*,{I.})
Ao(F
v
J
Under the assumption (3.1.6) and (4.1.2)
A2 (F,{I.}) as v
+
=
in probability as v
+
2 * ,{I.})
Ao(F
v
J
00
either in P,,- (or in Q,,-) probability.
v
v
£
2 * ,{I.})
Ao(F
v
J
Proof.
+
J
0
L: u;*2("
' v. + v. )1 v.
j=l v J J
-J
00,
Noting that (v. +v :) Iv + 2P.
.
J
-J
J
we get exactly as in lemma 3.3.1,
JI,
JI,
2
L: L\~2(2P.) = L: L\. (2P .) =
j=l J
J
j=l J
J
+
Qv-) probability as v
THEOREM 4.3.2.
+
00.
Under G)', (3.1.1) - (3.1.6) and (4.1.2) U* is
v
--
"'v -
asymptotically Np+l(Q; iboA~(F,{Ij})) in probability.
Proof.
Consider any linear combination f"'W""'v
d'U*, .9.
dO, dl"'"
d
p
are fixed and finite.
=
(d 0"'"
d p)
"
;i!,Q,
I
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114
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
v
p
L ( L dkc
i=1 k=o
where m . =
V1
v
£
A*
.)(
L U i'~ .)sgn X . = L m .W . sgn X . ,
kV1 j=1 V J vJ
. 1 V1 V1
V1
V1
1=
*
*
p
L dkck i' i = 1,
k=O
v
Now,
max 1m .1 = 0(1) and
1<.<
-1-V
V1
v
L m2
i=l vi
= 0(1), (because of the assumptions (3.1.4) and (3.1.5)).
So the sequence {m .} satisfies the Noether condition.
Further,
V1
v-
v
1
L
i=1
W2
=
•
V1
2
AO(F* , {1.})-+- A5 (F, {1.})
J
v
J
in probability.
It
can also
be shown as in chapter II (see also chapter III) that
kv
2
wvi
-+- 0 in@'-probability. So the sequence
v
<
a=l
a
-v
k
v
{W .} satisfies Hajek's condition Q in probability (see [16], p. 519).
max
v-I L
<· < .-1
<.
1 -11-"
V1
Now, we refer to Theorem 4.3.1 to get the result.
LEMMA 4.3.2.
Under (3.1.1) - (3.1.6) and (4.1.2) ,...,v
u* - rlJvO
'T'
--r ,...,
0 -in
P - (or in Q -) probability.
v
v
Proof.
Write
£
(4.3.18)
U* - T
kv
kvO
=
~ (~*.
- LlA*.)
Ll
'-'
j=l
VJ
J
v
~ c * . U . j sgn X . , k = O,l, ..• ,p.
'-'
i=l kV1 V1
V1
The rest of the proof is the same as in lemma 3.3.2 and, hence, is
omitted.
From the above results, it is now obvious that M~ and Svoo
have asymptotically the same distribution under HO or {H v }.
It is
immediate on the basis of lemmas 4.2.1 - 4.2.3 that Mv* (or SvOO )
is asymptotically X~+l under Ho and X~+I(SO) under {H v }' where,
(4.3.19)
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115
The extension of the above results to the case of countable
set of class-intervals is similar as in chapter III since the
sequence {Q } of probability measures is contiguous to the sequence
v
{p } of probability measures from lemma 4.2.2.
v
Again, in this case,
since the asymptotically optimal parametric test is similar as in
chapter III (involving only the additional statistic corresponding
to po), the ARE of the proposed test procedure compared to the
asymptotically optimal test procedure will be similar as p2(F, G,{I.})
J
given in (3.5.23) where p2(F, G,{I.}) is defined similarly as in
J
section 3.5.
The loss of efficiency expressions will also be similar
as (3.5.24) or (3.5.25).
4.4.
An Important Application.
The test procedure developed in earlier sections can be used
in the particular case of 'paired comparison' problem.
The problem
(which has been considered by Mehra and Puri [28]) is as follows:
. a sequence
ConSl'd er p treatments ln
'
0f
experlments,
t h e v th
sequence yielding paired observations (Y vkm ' Yvk'm), m = 1,2, ... ,n vkk "
1 :::; k < k' :::; p,
=
H
1 :::;k<k':::;p
(k k')
scores Yvm '
= Yvkm
n kk'.
v
- Yvk'm' (m
continuous d.f. Fvkk'(x).
Assume that the n vkk ' difference
= 1,2, •.. ,
n vkk ')' have a common
This is the situation, for example if in
the analysis of an incomplete block experiment, with each block of
size two, one makes the assumption of additivity as in the usual
analysis of variance model.
We want to test the hypothesis
I
I
.e
116
for any two pairs (k, k') and (k", k"'), where
1
~
k, k', k", k'"
~
p, k
I
of alternatives
I
(4.4.2)
I
I
for each pair (k, k'), 1
I
I
Ie
I
I
I
I
I
I
I
.e
I
~
~
k', k"
~kk'
not all zeroes satisfying
~
k < k'
=-
~
k'"
against the sequence
~kk'
p, where
~k'k
(1
~
k
~
are certain constants
k'
~
p).
The test statistic used by Mehra and Puri can be introduced
as follows:
Let {J ., 1
Vl.
~
~
i
v, v
~
I} be a double sequence of numbers satisfying
either the assumptions (on the "score" functions) by Chernoff and
Let R(k,k') be the rank of
Savage (1958) or by Hajek (1962).
(k,k')1
lyvm
m
= 1,
wh en t h e v =
2, ••• , n
vkk
'; 1
n kk' values of
L:L:
l<k<k'<
-p
~
k < k'
v,m
v
~
p, are arranged in ascending
order of magnitude in a combined ranking.
T
(k,k' )
V
=
~vkk'
~
m=l
(k k ')
IY'
I'
Vm
Define
(k k')
(k k')
J (R . '
/(v + l»sgn y '
,
v ~,m
vm
where J (u) is a step function defined over (0, 1] taking constant
v
values J . over the interval
Vl.
J (u)
v
= J Vl.. = J(i/(v + 1»
«i -
for (i
~
l)/v, i/v], i.e.
l)/v < u
~
i/v.
Then the test
statistics are of the form
(4.4.3)
It has also been shown that under HO defined in (4.4.1) and
under the conditions given in [28], Kvo is under HO asymptotically
I
I
.I
I
I
I
I
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I
I
I
I
I
.I
117
X2 _ 1 and under the sequence of alternatives defined in (4.4.2) is
p
2
asymptotically X l(o)t where
p(4.4.4)
a
p
= p-l ~ { ~ :(/pkk , V . · ,)}2
k=l k'~k
-kk
Pkk , = lim nvkk'/v is assumed to exist and is
V-7<lO
(k, k')t (1
~
Let n
~
k < k'
vk
p).
P , =
kk
p
~
nvkk "
Invk '
- --.-Invk
=
k'=l
Pk'~
(1
~
> 0
k
~
for each pair
k'
~
p).
To reduce the problem to our model t
we first set
(4.4.5)
. ].lkk'
----
IV
x=
(n
t ••• t
observations
Yk ,
=
Yk
t 1 ~ k ~ k' ~ Pt
Y )' being defined in (3.2.10).
We pool all the
p
y~~tk')
(m = It 2 t " ' t n
"
vkk
1
~
k < k'
this pooled set of observations by X It"'t X •
v
problem belongs to our model with So = 0t c
a block where the k
th
vv
~
p) and denote
Then this testing
. = +1 if XVl. is from
kVl
treatment is paired with a treatment k'
(k' = k + It'''t p)t c v i = -1 if XVl. is from a block where the k
k
treatment is paired with a treatment k' (k'
c
=
~t
th
2 t " ' t k-l)t
. = 0t if X . is not involved in the block at all.
kVl
Vl
We consider
the situation when the observations are classified itt several
groups.
Define test statistics T
as in (4.2.21) or U* as in
kv
k va
(4.3.6)t k = It 2 t " ' t p.
Then the quadratic form M~ (or Svoo) is
asymptotically X2
under HO and X2 (s) under the sequence of
p-l
p-l
alternatives with
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e
I
I
·118
(4.4.6)
A and A20 (F,{I.}) being defined in (3.1.5) and (4.2.9) respectively.
""
J
We may also mention here that since, under the model So is absent,
even in the process of finding permutationally distribution-free
tests, statistics (as in chapter III), regardless of signs, may be
used.
We may remark at this stage that the test proposed by
Mehra and Puri, considered as such has some limitations owing to
the fact that it allows a judge only to give two possible verdicts
"better" or "worse" in comparing one treatment with another.
However, if we want to allow greater flexibility in the judgment
of a judge by making the judgment level classified into several
ordered categories, the above test procedure will fail.
For example,
in comparing the performances of two musicians at a music competition,
the performance of Musician 1 compared to that of Musician 2 may be
classified as "Much Inferior," "Slightly Inferior," "More or less
Parallel," "Slightly Better" and "Much Better."
In such cases, it will
not be possible to use the test of Mehra and Puri, while our test procedure can be used if these ordered categories are represented as
class-intervals on the real axis symmetric about the origin.
It should be possible to introduce, however, statistics
analogous to that of Mehra and Puri for grouped data by replacing
"score" functions J (R(~,k') /(v + 1)) by the corresponding "scores"
v v,m
for grouped data.
We have already mentioned that their tests do
I
I
.-
119
not agree in general with our tests.
To show this, we first define,
statistics analogous to that of Mehra and Puri for grouped data.
I
I
We have a finite set of ordered points on the (extended) real line
I
I
I
I
Ie.
I
I
I
I
I
I
I
1
.I
giving a finite number of class-intervals in which the observations
y(k,k') belong.
vm
~
~
k < k'
p.
as in (4.3.5).
These points may vary for different pairs (k, k'),
;*(k,k')
For each pair (k, k') we define scores 0Vj
Also, we define
if y(k,k')
vm
U(k,k') =[1
vmj
(4.4.7)
o
E
I~k,k')
J
otherwise
(k,k')b e~ng
.
defined
(m= 1,2, ... , nvkk '; j = 0,1, ... , '!/'kk' (say) I .
J
similar to (3.1.10) for each pair (k, k'), 1 ~ k < k' ~ p). Then
the grouped data analogue of the statistics of Mehra and Puri will be
(4.4.8)
1
~
T
k < k'
(k ,k')
V
= ~Vkk'
m=l
~
p.
(
~kk' ~*~k,k')u(k,k'»s
j=O
vJ
vmj
n y(k:,k')
g
vm
'
The appropriate quadratic form (see [28]) will be
It can be proved as in [28] that under HO' M is asymptotically
vO
x~,
while under the sequence of alternatives, MyO is asymptatically
X2 (o), where
p
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~ (F ,{ I.})
is = -
(4.4.10)
P
kk
, (1
~
I
~
k
p) being already defined in (4.4.4).
~
k'
In the particular case when each pair of treatments is
compared the same number of times 1. e. n
vkk ' =
(k, k' ) such that 1
~
k < k'
~
(~) -Iv, for all
p, we have P , = (~)~lv,
kk
1
I
~
k < k'
~
p.
From (4.4.5), ],lkk' =
1
~
k < k'
~
p.
Then from (4.4.10) ,
2
(4.4.11)
is =
Ie
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]
p
.-L
p
P
-I
\)
o
L Y •
k
k=1
(Y k , - Yk ),
p
L { L (Yk' _ Yk )}2
k=l k'~k
2
p
~1 A (F, {I.}) L (Y
=
where y = p
A (F, {I })
j
O
p(p-l)
(*J~
k=l
J
k
- Y)
2
Also, in this case,
P
L c . =
Ln
'
1=1 kV1
k'=l vkk
\!
,2:.
1=1
ckvick'vi = - n vkk ' = -
v
So,
L:
i=l
c~ ,c~,
V1
Hence,
c
-?J
-
\)1
A
_ ~:l
I~
V(P2)-I,
'v (P J -1
1
~
k
~
= - (p -
k'
~
p.
1) -1, 1
2
P
1
p~J~
, where p
~
k
~
- (p - 1)
~
k'
-1
p.
•
P
(Note that
~
is not of full rank.
But we can take a principal
submatrix of order (p - 1) which is of full rank) •
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From (4.4.4) we get,
P
1
k=l
p-
2
2
- ---1
~ = AO(F,{I.})( E Yk
(4.4.12)
J
E YkYk· ,)
k~k'
2
P AO(F , { I } )
j
p-1
(4.4.11) and (4.4.12) justify our assertion.
If however all possible
pairs of treatments are not compared the same number of times, the
two test procedures are not necessarily asymptotically power equivalent
since the noncentra1ity parameters
~
and 8 may differ.
The following
example will justify the statement:
Consider the case p = 3, n
Then, P12
"13 = 16
=
P 13
=
[~ -~]
, "23 • 16
11
flo
V13
1
1
110
110
4
5
1
4
5
1
= v/6,
Y2·
= -- - -12
Also, here,
~~
= n
Y1
2/3;]J
1/6, P 23
algebraic simplifications,
~
V12
[~ - ~] "
/2/6
15/6
Hence, we get, after some
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22217"
_2
Hence, ~ - Ao(F,{I j }) [Y 1 + Y2 + Y3
Thus,
1'; -
-y
t
8
{YI Y2 + Y1Y3) - ~ Y2Y3] •
12
8 = 15 Ao (F,{I })(Y 2 - Y3) 2 > 0, when y 2 ;t Y3 •
j
The above example also shows that in this particular case
our test procedure cannot be less powerful than that of [28].
We
may, thus, conclude that the test proposed in [28] does not have the
asymptotic optimality properties as described in section 3.2.
Also,
computationally, our test statistic seems to be simpler than its
rival.
It is also obvious that the univariate one-sample location
problem for grouped data follows as a particular case of our model,
since then we have only to set SI
= S2 = ... =
Sp
= o.
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CHAPTER V
NONPARAMETRIC TESTS FOR GROUPED DATA IN THE
MULTIVARIATE LINEAR REGRESSION MODEL
5.1
Introduction, Notations and Assumptions.
Consider a double sequence of stochastic matrices
~= (~V1"'"
i
= 1,
, _
(1)
(q)
~VV), V ~ 1, where, ~Vi - (XVi , ••• , XVi ),
2, ••• , V are independent stochastic vectors having
continuous d.f.
FV.(X')
1 ~
(5. L 1)
and ~-
= (xl"'"
=
S' - cv' .S'),
F(x' -
~O
~
= (S 10 "'"
x)'
S
q
, ~O
i = 1, •.• , v, V ~ 1,
1~
Sqo )' is the nuisance
parameter (vector), S = «S sk)) is the parameter (matrix) under
qxp
test and
~Vi
=
(c 1Vi ' •• ·, cpVi ) , , (i
known regression constants (vectors).
= 1,
2, ••• , v, V ~ 1) are
We want to test the null
hypothesis of no regression, that is,
(5.1 •• 2)
H
o
S= a
against the set of alternatives
S# a •
The notations and assumptions in (3.1.2)-(3.1.5) regarding
the regression constants c
.(k
k Vl
= 1,2, •.• , p; i = 1,2, ••• , v,
V > 1) are the same in this case.
Let the univariate marginal d.f.
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124
th
of the s
be F.
v ~
v~,s
v
~ 1).
variate corresponding to the d. f. F . (x_ I) and F(x_ I)
(x) and F (x) respectively.
s
= 1, ••• ,
(s
v
~
= 1, 2, ••• , v,
th
th
The corresponding bivariate joint d.f. for the sand Sl
variates are denoted respectively by F"i
(1
q; i
~
s # Sl
q; i
= 1,2, .•• , v, v
= v- l
notation H (Xl)
V -
for X 1' •.• ' X ).
-v
-vv
,ss
> 1).
I (x, y) and F
ss
I(x,y) ;
We also introduce the
v
~ F i(x ' ), (the average q-variate d.f.
i=l v -
The
correspo~ding
univariate marginal d.f.
for the sth variate and the bivariate joint d.f. for (s,s,)th
variates are denoted by H
v,s (x) and HV,SS I(X, y) respectively;
(1 ~ s
:f: s I
.::.
Let
q) •
00
(5.1. 3)
f (x)
s
f
2
= F'(x)
exist and A (F ) =
s
s
_
for all s
= 1,
(f'/f )2 f dx <
s s
s
00
,
00
2, ••• , q.
Consider the sequence of alternatives
(5.1.4)
H
V
s=
-
T =
S = «T Ic )) =
-v
sk kv
Tn
...
T
T
...
T
q1
1p
qp
T
where
~v
qxp pxp
D
IC
1V
0
.1Ic
0
pv
The elements of T are assumed to be real and finite.
We are concerned with the situation when
~v
is not observable.
We have q-dimensiona1 rectangles in the real q-space~q defined by
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125
q
1\ I.
s=l Js's
(5.1. 5)
j
s
= 0
'
1, ... , JI,
s
s
••• < a JI"s < a JI, +l,s
s
=
1, 2, ••• ,q ,
=
00
s
if
o,s
< a
Let
x.€r,
v~
J,s
s
otherwise
(js = 0,1, ••• , Jl,s'
s = 1, ••• , q; i = 1, ••• , v, v> 1).
The observable stochastic matrix is given by
v >
(5.1.7)
JI,
(5.1. 8)
L:
s
1.J ,s Zv~'j ,s
s
s
i = 1, .2, ••• , v, v
(s = 1,2, ... , q;
1 ,
s
j =0
~
1).
Under the model (5.1.1) it follows immediately that
p
(5.1. 9)
F '
(x) = F (x -
s
V ~,s
S
-
So
L: S
c
)
k=l sk kVi
and
p
(5.1.10)
FVi,S,S' (x, y)
= Fss'(X
-
SSo -
L
k=l
Ss k
ck ' ,
v~
p
y-
l<s=f s '-.2q
i
=
1,S
,...~. re. or.dered points on the (extended)
rea11ine [_00,00], s = 1, 2, ••• , q).
(5.1.6)
(where ...00 = a
.S s '0
1, 2, .• ~, v, v > 1 .
-
L: Ss 'k ~Vi) ,
k=l
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126
Let
(5.1.11)
Fv i ,J' ,s
s
= F (a
s
(5.1.12)
F", "
,
vJ.,J ,J "s s
s s
SSo
-
j,s
s
=Pr(X(~)
. VJ.
<
-
aj
*
p
+
v,
p
jJ
1, 0 < j , < J/, ,
s s
i = 1,2, ••• ,
(s') _< a "
s,IR,,)
s' XVi
v
J s"
SsO - k=L: 1 'T sk ck"i'
a.
'Q
"
*)
v
J' , s to- k=l
'T, kCk
s
s
s
V~
= Fss,(a j
s s'
(0 < j
< J/,
s- s
s'
+
1, 1 <
::f: S'
S
L,
.l'
,
_< q,
V ~l).
Define
(5.1.13)
R
=
v,j ,s
s
V
(5.1.14)
R
=
v,j,j "s,s'
s s
V
(0 < j
< J/,
s- s
+
-1
-1
1, 0 < j , < J/, ,
s s
V
L: F . ,
vJ.,J ,s
s
i=l
V
L:
i=l
+
F
"
V J.,J
,
,
_< q,
V
,
s ,J s "s,s
1, 1 <s ::f:
S'
> 1) .
Also let,
=
(5.1.15)
F (a,
- SSO) = pr(x(s) < a
IR),
s J,s
, " vi - j , s 0
s
s'
for all i = 1,2, ••• , V;
(5.1.16)
F,
j
Js, s "s,s
, = Fss ,(aDJ
s'
(s')
< a.
J ,s
s
, X,
VJ.
- (3
s
< a,
-
So
,a
Js"s
o
J , ,s
s
,-
(3 , )
s 0
,IR ), for all i
0
= 1,2, ••• , V;
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127
(0 < j < ~ + 1, 0 < j , < ~ , +1; 1 < s # s' _< q).
- s- s
s s
Define now,
(5.1.17)
. . +1 ,s - FV1,J
. " ,s
P
v i,J' s ,s = FV1,J
s
s
(5.1.18)
P"..
.
,
v1,J s ,s, J s " s
= FVi,j +l,j ,+l,s,s' - FVi,j
s
s
s
,j ,+l,s,s'
s
- F . . . +1
,+ F . . .
, ,
v1,3 , ] ,
,s,s
',)1,] ,J "s s
s s
s s
(0 < j <
- s-
~
s
~
, 0 < j , <
s -
s
,,1 < s # S' _< q, 1 _< i _<
v,
V _> 1),
and let,
(5.1.19)
P
(5.1.20)
j
s
=
,s
F
- F.
j +l,s
J,s
s
s
p..
,
J s ,] s "s,s
= F.J +1 , J
. , +1 ,s ,s , - F.J
s
s
s
. +1 ,s,s '
,J,
s
- Fj +1 .
, + F..
,
J s ,J s "s ,s
s ,J s "s s
(O'::js~~s' O'::js' '::~s"
1 <
#
S
s' '::q).
The above notations will be used repeatedly in the subsequent sections.
The "scale" parameters
0"
s have all been assumed
to be 1 throughout this chapter.
5.2.
A Class of Parametric Tests.
We introduce firs t the "score"-functions
(5.2.1)
~ (u) = -f'(F- 1 (u))/f (F- 1 (u)), 0 < u < 1, s = 1,2, •.• , q.
s
s
s
s
s
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128
Let
?s+1,s
(5.2.2)
6.
J ,s
s
•e
I
/ ?S+1,S
</J(u)du
=
When p.
s
,s :/=
°
J ,s
s
(s = 1,2, ••• , q).
0, otherwise.
:/= 0, we can rewrite 6.
as
J ,s
s
J ,s
s
(5.2.3)
du, when p.
J
F.
F.
J ,s
s
6,
=
J ,s
S
p:J 1
S
,S
[f (a,
J
S
S
B ) -
-
,S
sO
f
S
(a, +1
J
S
- B )].
,S
So
Let
(5.2.4)
J
s
= {js1js E[O,
= {(js' j
(5.2.5)
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I
I
=
1, ••• , 1 s ]' P j ,s > oJ, s
s
,)Ij
s
s
1
(5.2.6)
(5.2.7)
A (F
{r, }) = A
=
ss
s'
Js
s
l:
6
js =0
2
j ,s
s
.:s.
P,
J ,s
s
=
q
2
l:
6 ..
J
EJ
'
s
js s
P
,S
{r. }, {r,
} = Asst
Js
Js '
A(F , F "
s
1, 2, .•. , q.
E[O, 1, ... ,1 ], j , E[O, 1, ... ,1 I],
s
s
s
p.,
, > O}, 1 < s :/= s'
J s ' Js"s,s
2
=
S
11,
S
=
l:
j =0 j
S
=
s
l:
=0
6.
6,
,P"
J s ,s J s ' ,s
J s ' Js '
s'
l: l:
(j S ,j s ,) EJss '
We may note that
(5.2.8)
(similarly as (3.2.8)) •
,
,S ,S
6.
6.
,P"
"
Js's Js"s
J S ,J s "s,s
j
,S
S
;
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Let
(5.2.9)
A
=
«A
ss
,)) •
qXq
We introduce the statistics
R,
V
(5.2.10)
T
=
v,sk
L:
i=l
c*
kVi
s
L:
js =0
Z ••
vJ.J , s
s
t:..
(s
1, ... ,q, k = l, ... ,p).
Js's
We define a test procedure based on the statistics T
's
v,sk
(s = 1, ••• , q, k
=
1, ••. , p).
These statistics are co-ordinate wise
extensions of the statistics used in Chapter III.
Moreover, unlike
the univariate case, the test procedure we are going to propose here
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may not be asymptotically optimal (in Wald's sense) even under the
regularity conditions given in Chapter III.
Nevertheless, it seems
to be the most natural one to consider because of its simplicity and
mathematical tractability.
We may remark also that our statistics
are generalizations in case of grouped data for analogous statistics
considered in case of ungrouped data by Puri and Sen [31].
It is,
however, worth mentioning that the restriction on the "score function"
<p is much less stringent in our case than that in the case of Puri
and Sen who had to impose the conditions of Hajek [18] and Hoeffding
[23] on <p.
Let P ' Q be respective distributions of X* tinder
v
v
-v
We get the following expressions after some algebraic
simplifications:
(5.2.11)
E(T
v,s kip)
v =0
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*
v
*
= (~ ckVJ.' ck'VJ.') A
i=l
ss
(k, k' = 1, •.• , p; 1 < s
~
(5.2.12)
~V = «Tv,sk»
s'
~q).
Write,
•
qXp
Then,
E(!v!P v )
(5.2.13)
=
,
0
~
11.
var(!v1 p v) =
~V
8
A
qXq
pX p
qxp
+
8
11.
as
A
pxp
V+oo
.
qXq
We assume that
(5.2.14)
A isp.d.
Next we prove two lemmas:
Under P ' the model (5.1.1); the assumptions (3.1.2)-
LEMMA 5.2.1.
v
(3.1.5) and (5.1.3), the joint
asymptotically N . (0, 11.
pq ~ ~
Proof.
e
d~8tribution
of the elements
A).
~
Consider any linear combination
q
p
~
~
e
8=1 k=l
where e
v,s
constants.
of~v ~
k (s = 1, ... , q, k = 1, ... , p) are
v,sk
T
v,sk'
fixed and real
We can write,
(5.2.15)
q
p
~
~
s=l k=l
q
p
where W = ~
~
e k
Vi
s=l k=l v ,s
v
e
v,8k
T
v,sk
~
. 1
J.=
W,
VJ.
R,
*
c
kVi
=
8
~
j =0
s
1, 2, ... ,
v,v
2,1).
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var(wvilp,,) =
v
*
*
q
p
E
E
eke
'k' c kVi ck'''i A ,<
s,s'=1 k,k'=1 v,s
v,s
v
ss
00
(i=1,2, ... ,v).
Again, Iw
Q,
s
E ,E
s=1 J =0
s
q
,I
v~
16.
Js '
s
Iz
'j
v~
s'
s
Hence, writing m
v,pq
Q,
(5.2.16)
Iwvi 1
3 < 3
p
_ mv,pq 4
*
P I
E c '
k=1 k v~
s
q
(E , E 0
s=1 J =
13
S
*
P
< m3
4P+q E I C ,I
- v,pq
k=1 kV~
=
p+q
3
mv,pq 4
*
16,J
3
S
Iz ,
,s V i J
S
,S
)
3
q
E
s=1
Q,
p
3 q
S
Elck"i
E
E
v
k=1
s=1 j =0
l
16,J
S
s'
S
I
3
zv~J
..
s'
s
Hence,
(5.2.17)
1
3
max
(max
< m
- v,pq
1<s<q js EJ
s
; ,2 )
J ,s
s
q
3
4P+q
1
,1
E (
k=1 kv~
s=1
f c*
1
::i:
m3
max
(max
v,pq
1<s<q j e:J
- s s
;.2 )
J ,s
s
4p+q
3
J
1¢2(u)ldu)2
s
0
q
1 3 ( E A3 (F
'
k \)~
s
s=1
PI*
E c
k=1
1
».
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Thus,
1
v
(5.2.18)
3
r E(IWViI3IPv) -< mv,pq
(max
max
i=l
l<s<q J' EJ
s
,e
I
s
s
*
IckVi
I ( rq
p
[r
max
k=l l<i<v
s=l
max
max
l<ks> l<i<v
= 0(1),
* I
I~Vi
q
(r
.
3
A (F))
s=l
s
from (3.1.4).
v
r Var(W
(5.2.19)
i=l
,Ip )
V1
v
=
Ie
I
I
I·
I
I
I
I
;'-J2,s )
= 0(1) ,
using (3.1.5), (5.2.8) and the fact that eV,sk are finite and real
elements, and also the summations involve a finite number of terms.
(5.2.17)-(5.2.19) validate Liapounov's criteria under which
the classical C.L.T.holds.
Now, using (5.2.13), the lemma is proved.
Q.E.D.
Let
(5.2.20)
h
vsi
LEMMA 5.2.2.
=
p
r
k=l
T
sk
*
c
( s = 1,2, ... , q; i
kVi
Under Qv, the model (5.1.1) and the same assumptions
as in lemma 5.2.1., -=:.;:;....oL_==.-::.=-==.::..:=..::..;;..:.:.....::.=...~~-=-===--=.;;:;.
the joint distribut10n of the elements of -V
T -is
aSymptotically
N (D (A) T A , A3 A), where,
pq - -
1) •
I
I
I·
I
I
I
I
133
(5.2.21)
D (A) =
All
0
o
Using Taylor expansion, we get from (5.1.11), (5.1.15),
Proof.
(5.1.17) and (5.1.19),
(5.2.22)
for j EJ , where, 0 <
s S
I
I
(5.2.23)
Ie
Now,
+ 6. j ,s h Vs i + h VS1. e.Js's g j ,8. (h"s-f)]'
v ...
s
s
P ..
= p.
[1
V1,J ,S
J ,s
s
s
gj
(h
s
,8
vsi
e.J
,s
s
< 1, and
) = p- 1
[f' (a.
j ,s
s
s
J ,s
s
- f' (a. +1
s
v~
=
I
t
*
c
s
So
-
ej
s
,s
- e
j
,s - Sso
y)
s
,s
y)] dy
s
E
kVi j =0
s
i=l
J
S
-
6..
P.
(1 + h .6..
Js's Js's
VS1 Js's
I
I
I
I,
=
v
p
*
*
E T k'( E ckVi ck'Vi) Ass + 0(1), s =
k'=l s
i=l
1, •• , q, k = 1, ••• , p.
Hence,
I
I
e
I
I
...............
*
*
V
LT k( E c .c .)A
k=l q i=l 1 V1 k V1 qq
p
V *
* .)A
C
•C
k=l qk i=l pV1 kV1 qq
p
E
T
(E
I
I
.e
I
I
I
134
~1 (1)
+
(where by ~1 (1) we mean a qxp matrix each element of which tends
to zero as V + 00)
An
·.. LIp
..·...
1"q1 ·.. 1" qp
,
1"n
=
I,
I
... 0
0
A
qq
A
+
~V
= D(A) 1"
0(1) + D(A) 1"
~
V
*
L: c .
i=l 1 V1
•
V
*
*
L: c . c .
i=l 1 Vl. pVl.
A as
V
...
2
* *
L: c . c .
i=l 1 V1 pVl.
·..
V
•
•
6
+
*
L: c .
i=l pVl.
~(1).
2
V + 00
Also,
I
Ie
I
I
I
I
I
I
I
.e
I
Ji,
*
*
s
Ji,
s
V
IQ )
• L: 11.
Z •. s' L: c.., . · L: 0 fj j ,s z.
Vl.j ,s V
i=l
K Vl. J =
i=l KVl. j =0 Js's Vl.J s '
s
s
s
s
V
Cov. ( L: c..
=
Ji,
s
V
(
L:
= L: 1k'Vi ck'Vi Var
j =0
i=l
s
*
Ji,
*
s
But, Var [( L:
j
Ji,
s
=0
s
= E[( L:
j =0
s
Ji,
s
Z ..
- [{E( L:
j
s
=0
Vl.J ,s
s
6.
J ,s
s
)}2IQ.v J
I
I
135
.I
I
I
I
t
=
j
I
I
,I
2
6j ,s P,J,S (1+ hVS1,6,J,s + hVS1,e,J ,s gJ' (hvsi »
=0
S
s
s
s
s
s
L:
j =0
s
= ASS + O(hV1,) •
Hence, from (5.2.25),
(5.2.26)
I
Cov (Tv,s k' Tv,s k' Qv )
v
*
*
= (i~l
c kVi ~'Vi) Ass + 0(1), s = 1,2, ... , q; k,k' = 1, ... ,p.
= (L:v
c
Similarly,
Ie
I
I
I
I
I
s
s
t
I
I
L:
i=l
* *
kVi ck'Vi) Ass, + 0(1),1 < s # s'
~
q; k,k'
= 1, ••• ,
Using (5.2.26) and (5.2.27) we get,
Var (T-v IQ v )
(5.2.28)
(where by
~2(1)
= -v
A
3 A
_+
_2 (1) ,
O
we mean a square matrix of order pq each element of
which tends to zero as V
+ 00).
Thus,
(5.2.29)
Let W , (i
V1
= 1,
2, ••• , v) be the same as in lemma 5.2.1.
inequality (5.2.16),
Using the
p.
I
136
I
J
'-
I
I
I
I
I
I
Ie
3
< m
-
4P+q
v,pq
R,
* 3
[c,1
p
q
~
k=l kV1
s
3
I I1
1
J"s' s
P ViJ'
. s'
s
R,
P
* 3 ~q ~ s 111 • 1 3 (P . +
q ~ I~.I
J ,s
J ,s
k=l
V1
s=l j =0
s
s
P+
3
4
= m
v,pq
0
(1) )
s
Using now (5.2.18) and (5.2.30), we get,
(5.2.30(a) )
q
R,
'*
p
s
Also, E (W • 1Q ) = ~
~ e
c '. ~
11
(P •
+ 0 (1) )
V1 v
s=l k=l v,sk kV1 j =0 js'S Js's
s
Hence,
I
I
I
I
Using the inequality IwV1" - E(W 1.)1
v
I
Similarly,
I
(5.2.32)
I.
I
I
s
~
~
s=l j =0
get now from (5.2.30(a»
3
-< 4(IWV1,1
~
i=l
3
+ IE(WVi)1 ), we
and (5.2.30(b»,
v
(5.2.31)
3
E [ 1W , - E (W ")
V1
V1
1
3
]
= 0 (1)
•
v
~ Var (wviIQv)
i=l
=
=
q
~
p
~
v
*
eke
'k' A ,( ~ ckV1'
v,s
ss i=l
s , s'=l k ,k'=l v,s
0(1) .
*
C
K
'''1" + 0(1»
v
I
I
,.
I
I
I
I
I
I
1:_
I
I
I
I
137
Using (5.2.24) and (5.2.29) and arguing as in the last part
Q.E.D.
of lemma 5.2.1, we get the result.
T being a stochastic matrix, for actual test construction,
-v
it is more convenient to use a real valued function of T
-v
As in chapters II, III and IV, a suitable statistic
statistic.
will be a non-negative definite quadratic form in the pq elements
of
!v
where the discriminant of the quadratic
form is the inverse
of the matrix E = A 8 Av = «e
~v
~
v,s k ,s 'k')) (say).
[We may note
. that according to our convention, Var (!~Ip) = ~ 8 ~).
Consider
the statistic
(5.2.33)
S
V
-1
where ~v
=
q
q
p
L:
L:
L:
P
L:
sk s'k'
Tv , s'k' ev '
v,sk
s=l s'=l k=l k'=l
T
l
l
ss' ;\kk'
= «e~k,s 'k' )) = A- 8 A= «a
~v
-
»
*
A size-E test procedure based on the critical function "l/Jl(~V)
is as follows:
i f Sv > S
V,E
(5.2.34)
*
"l/J l (~)
if S
= SV,E
if S
< S
V
0
,
V
V,E
I
where SV,E and 0lVE are so chosen that E["l/Jl (~~)
I
we get a similar size-E test.
I.
I
I
as a test
IP v ]
= E.
Thus,
By virtue of lemmas 5.2.1 and 5.2.2 Sv is asymptotically
2
~q
under Pv '
Then S
V,E
+
2
2
X
(upper lOOE% point of X ),
""Pq,E
""p q
I
I
tt-
138
o.lVE
+
0 as V -+
·2
> X
-
I
I
I
I
I
I
<
The large-sample test procedure will be as follows:
00.
pq,E
, reject H in (5.1.2)
2
Xpq,E
' accept H o
We may also observe that writing!v* = (Tv,ll'···' Tv,lp'···'
T 1' ••• ' Tv
)f, we get,
v,q
,qp
,I
S
V
= T*f(A
-v
~
e
A
~~
)-1 T* = T*f(A- I
~v
~V
~
e A-I)
~V
From (5.2.24), we get,
*
E(!vIQv) =
(5.2.35)
-p
V
*
*
Tlk ( L: c lVi c kVi ) All
k=l
i=l
L:
P
L:
V
P
L:
V
P
L:
V
*
*
Tlk ( . L: cpVi ckVi ) All
1.=1
k=1
Ie
I
I
I
I
I
I
I
0
*
*
T k ( L: c lVi ckVi ) Aqq
k:=l q i=l
*
*
T k( L: cpVi c kVi ) Aqq
k=l q i=l
+a column vector with pq elements each of which
+
0 as V
+
00.
We can write the first vector in (5.2.35) as
~V
=
~
--' *
D(A) A
T
~v ~
pqxl
*
where T = (TIl' ••• ' TIp'.'.'
~
.-.J
L
ql , ••• ,
Thus, D(A) A
= A
~v
~
as V
+00
,
where E*
e
L
,......, A e I and
p
qp )', D(A) =
*
* *
T+ET
A := E
i.e. B=E
~v
~v
-v·~
~~
* v)
Thus, E(TvIQ
~
+
E*
T
*
as
V +
00.
T*.
~v
I
I
tr
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
,e
I
139
*
Similarly, from (5.2.29), Var (~vIQv)
Using lemma 502.2., we get now, under H ' Sv is asymptotically
v
2
~ (n),
where
(5.2.36)
5.3.
n = (E
* T *) ' E*-1
(E
* T *)
* * T * = T *'
= T ' E
(A 8 II.)
T
*
A Class of Nonparametric Tests.
The unconditional distribution of Sv depends on the unknown
~.
J ,s
(j
s
s
=
0, 1, ••• ,
~
s
, s
=
under a permutation model, S
V
1, 2, ••• , p) even under H.
0
provides a distribution-free test.
Since, the vectors X 1' ••. ' X
~v
However,
~vv
are i.i.d. under H , they have a
0
joint distribution which remains invariant under vi possible permutationsof the V vectors.
Thus, in the V dimensional real space
v , we have a set of vi permutationally equiprobable points.
~
Let
~v denote the permutational probability measure defined on this set.
Under ~v' all the vi equally likely realizations have common probability (vi) -1.
We introduce the following notations:
v
(5.3.1)
~
i=l
v
(5.3.2)
~
i=l
zVl.J
..
zVl.J
..
S
S
,s
,s
=
v.
J ,s
s
zvij
s
,s'
=
v..
J ,J "s,s
s
s
,
I
I
~
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
140
= 0, F ,
\i,D,s
\! j +l,s
s
(5.3.3)
=
(5.3.4)
\!
=
F
-1
F\!,Js
. +1,s
JF\!,Js's
.
= 0,
du if \!.
¢(u)du
Js's
~
0 ,
otherwise.
A
We may note that \!.
,\! . ,
"f::,.
Js's
Js,Js"s,s
\!,Js's
(0
~
j
s
~
£ , 0
s
~
~
j ,
s
£ "
s
Permutation of the vectors
1
~
s
~
s'
~ q)
x* 1 ' ••• , x*.
~\!
remain unaffected by the
It follows now immediately
~\!\!
that
E(Z..
I(?) =
\!J.J ,s \!
(5.3.5)
s
.
E(Z .,
\! J.J ,s Z\!ij' s' s!tY\!) = 0 ,
s
\! .
(\! , ,
- 8. .,)
J ,s
Js's
JsJ s
f3
E(Z ..
\! J.J ,s Z\!J..'01
J ,8 ICY)
\! =
\!(\! - 1)
s
s
(5.3.6)
(5.3.7)
(where 0
~
~ j~ ~
js
£s' 1
are Kronecker deltas).
(5.3.8)
E (Z i'
\! J s ' S
~
s
~
q; 1
~
i
~
i'
~
\!, \!
~
1; 8 . .
JsJ s '
Also, we get after some simplifications,
Z ..
\! J.J s' ,
s,I~)
(5.3.9)
=
\!
-1
(\! - 1)
-1
\! .
\! .
J s ,s J s "s
,
\!
-1
(\! - 1)
-1
\ ! j,J' "s,s
.'
ss
I
I
~
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
141
(0
~ j
~
s
t
s
0
t
~ j
s
, ~ t
s
" 1
~
i ~ i' ~ v, v ~ 1).
Define the permutation test statistics,
(5.3.10)
(5.3.11)
(1
~
~
s
=
~
s'
q).
Now, after some lengthy algebraic manipulations t we get,
v,s k Ie?)
v
(5.3.13)
E (U
(5.3.14)
~
-1
E(U".skU",s'k' I V-,)
= v(v -1)
(L: c k .c k ' .)A
,
v,
v
v
i=l V1
V1 V,SS
= 0;
v
(1
~
k t k'
$
*
*
p, 1 ~ s, s' ~ q).
Writing
(5.3.15)
U
"'v
«Uv , sk»
qxp
(5.3.16)
where A
"'v
The following test statistic (which is a quadratic form in
U
v,sk
's) is proposed:
I
I
Ie
I
I
I
I
I
I
Ie
142
(5.3.17)
q
L
M
Kg = A-I
-v
p
L
P
L
U
Ksk,s'k'
s=1 s'=1 k=1 k'=l Uv,sk v,s'k' v
v
e -v
A
where ~
K v=-!I.v
~v
q
L
e -v
Ag
and
= «Ksk,s'k'»
v
denotes a' generalized inverse of K
-v
(b~ is a generalized inverse of
hv )'
We may also remark that A
-v (as we
shall show later on) is p.d. in probability.
Now a (permutation) test procedure based on a critical
function ~2(~~) can be formulated as follows:
1
if M
2VE'
if M
v
a
if M
(
(5.3.18)
V
> M
V,E
\
~2 (~~)
I
=l
0
V
M
V,E
<
M
V,E
where O2
and M
are chosen in such a way that E[~2(~:) 10:,] = E.
VE
V,E
v
v
Then, E[~2(X*)lp ] = E, so that we get a similar size E test.
-v
v
The limitations of the permutation test procedure for moderately
I
I
I
I
I
-I
large sample sizes have already been discussed in chapter II and
reemphasized in chapter III.
We next want to find the asymptotic distribution of M.
v
this end in view, we first prove the following lemma:
LEMMA 5.3.1.
Under the model (5.1.1), the assumptions (3.1.2)-(3.1.5)
and (5.1.3), A
-v
~
A in P - (or in Q -) probability as v
v
v
-
--
~
00.
A in P - (or in a -) probability as
ss
v
'v
Proof.
Lemma 3.3.1 gives A
V,ss
v
(We can work with univariate marginal distributions in this
~
00.
case) •
I.
I
I
With
~
So, only we need to show that the elements A
v,ss'
P - or in Q -probability, where 1 s s
\!
\!
~
s' S q) as v
~
00.
~
A , (in
ss
We can write
I
I
.e
143
(5.3.19)
I
I
I
I
I
I
..I
I
-
A88 ,
R,
R,
~8
~
,
S.A
[ii
, -0
j =0 J ,-
I
I
I
I
I
I
Ie
Av ,88'
V
8
8
R,
=
.
J
A
.
,J 8 ,8
ii.
V, J
8" 8
,(V..
,Iv)
J8, J 8 " 8 , 8
,
(t:.
I: 8
=0
V,.J' ,8
8
8 '
R,
~
8
,
A
(ii,
' =0
J 8'
R,
~
V
,
,J 8
' ,8
,
8
j 8' =0
ii.
ii,
,Iv)
,[(V.,
J 8,8 J 8' ,8
J 8 ,J 8 ' ,8,8
Applying Cauchy-Schwarz inequality to the 8quare of the fir8t
term on the right hand 8ide of (5.3.19), we get,
A
- ii.
(5.3.20)
J
- ii,
J
8
,8
8
)ii.
,8 V,J
)2(v.
J
8
,(v..
,Iv)]
,,8
J ,J ,,8,8
8
2
8.
Iv)]
8
,8
R,
8'A2
ii,
,(V.
,Iv)]
,=0 v, J 8' ,8
J 8' ,8
~
[
j
8
A
Now, a8 in lemma 3.3.2, we can 8how here that ii V,J. ,8
8
(either under P - or under Q -) for each J'
v
v
8
0,1, ... ,
ii.
J8
R,.
8
=
,8
0
P
Since
(1)
I
I
I·
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I.
I
I
144
A2 (F ,) <
s
and the summation of the right hand side of (5.3.20)
00
involves only a finite number of terms, we get the square of the first
term (and hence the first term) on the right hand side of (5.3.19)
is
0
p
(1).
The same technique (i.e. Schwarz inequality) enables us to
write
(5.3.21)
[2
nd
term on the right hand side of (5.3.19)]2
As before, the first factor on the right hand side of (5.3.21) is
(either under P - or Qv-probabi1ity).
v
0
p
(1)
The second factor can be written
as
- P . s .].
J s' ..
The first term in the above is uniformly bounded in j
s
_1
by A2 (F ), while
s
we can show as in chapter III, v.
Iv - p.
= 0 (v 2) either in P
Js's
Js's
P
v
(or in Q -probability).
v
(5.3.19) is
0
p
(1).
So, the second term on the right hand side of
Write the third term as
Use the fact for each pair'
(js' js')
E
3ss"~'
Js'
s'~'
bounded valued, while (v..
,Iv) - p . .
,
J s ,J s "s,s
J,]
s s "s,s
P - or in Q -probability as v
v
v
~
00.
Js "
+
s' are
0 either in
The last statement can be proved
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
145
by writing
(F
,
v,J
+ ' + 1,8,8 '
81,J,
8-
F,J
8
+1,J' +
, 1,8,8 ,)
8
- (Fv,J. +1 ,J' ,,8,8 , - F,J +1 ,J' ,,8,8 ,)
s
8
8
8
- (Fv,J. ,J,
. +1 ,8,8 , - F.J J
'+1
,)
8 8
8, 8, ,8,8
where
FV,j +l,j ,+1,S,8'
8
v
-1
8
(j S = 0, 1, ... ,
]1,8;
j 8' = 0, 1, ... ,
]1,8',
1 :,; 8
;t
8' :,; q), and then
8hwoing each term in the above sum i8 0 (1) by u8ing the Cheby8hev
p
inequality and the fact that under Q", H . .
8' - F. J'
8 s' ='0(1),
v
V,J 8 ,J 8 ,,8,
J s ' 8" ,
(j 8 = 0, 1, .•• , ]1,8+1 ; j 8' = 0, 1, ..• ,
]I,
8
,+1;
1:,; 8
-
;t
8' :,; q).
the third term on the right hand 8ide of (5.3.19) i8 0 (1), the
p
8ummation involving only a finite number of terms;
COROLLARY 5.3.1.
A
"'\!
e "'v
A
~
A
'"
e '"A -in
Under the condition8 of the above lemma
P - or in Q,,-probabi1ity a8 v
v
v
-
~
00
The elements of A are all bounded above by 1, while
v
Hence
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I
q
11
A
S=l
5.3.1 and tbe
ss
::;
q
11
<
00
We get the result by using lemma
•
s=l
assump~ion
that
~v
+
~
as v +
00.
Our next theorem gives the permutation limit distribution of
THEOREM 5.3.1.
Under the model (5.1.1), if the assumptions (3.1.2)-
(3.1.5), (5.1.3) and (5.2.14) hold, then the joint distribution of the
elements of U is under (j)v asymptotically
N (0, A eA).
--pq
.~v
Proof.
~
~
The proof proceeds along the lines of Puri and Sen [31].
any given
m=
For
(m 1 , ••• , m )', define,
p
U
~voo
We can write,
v
P
L ( L
i=l k=l
mkc k* .) (
=Uvsoo(say),
Vl.
Q,
Ls
•
J =
a
z..
V l.J
s
s
,S
(j
0
v,J , s
s
)
s=1,2, ... ,q.
We have to show that Qvoo is asymptotically Nq(Q,
(m'~)~).
To prove
q
this, we consider any linear combination ~e'U
~voo
LeU 00' where,
s=l s vs
e'U
=
~voo
~
It is easy to see, in view of (3.a.4) and (3.1.5) that
p
*
v
p
*
max ( L mkc 0)2 = 0(1), and L ( L mkc 0)
k VJ.
k VJ.
l::;i::;v k=l
i=l k=l
2
m+m'Am as v +
= ro'A
f"tJ""'Vi""W
r"V~
00,
Qv .
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147
v
p
L: ( L:
Le.
i=1 k=1
mkc~
.f =
p
V1
satisfies the Noether condition.
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I
I
v
.e
I
q
L: { L: e
i=1 S=1
q
L:
zv 1J
..
s
s
,S
Also,
l:.v,J.
s
,s
}2
q
L:ee,
s=1 s,'=1 s s
in PV - (or in QV -) probability as v
~ ~'A~
.-
~
00 •
Also, we can show as in (2.3.19)-(2.3.21) that
lim k
v
Iv
=
O=> lim v -1
v~
r
l~il<
q
where, W (i)
v
Ie
I
I
I
I
I
I
I
v
-1
*
Hence, the (double) sequence { L: mkc .}
k\>1
k=1
0(1) •
max
•.. <ik ~v
v
Z..
L:
f::,.
vJ s ' s
V 1J S ' S
s=1
v
q
so that W = v-I L: W (i) = L: e
v
s=1 s
i=1 v
II
~ . (u)du =
0 s
o.
Now, use Theorem 4.2 of [16J along with (5.3.16) to get the result.
Q.E.D.
The next lemma will establish the "asymptotic equivalence" of
the permutation test statistics to the parametric test statistics.
LEMMA 5.3.2.
U - T
~
~
Under (5.1.1), (3.1.2)-(3.1.5), (5.1.3) and (5.2.14),
~
Q
qXp
-in
P - (or in Q -) probability.
V
\!
V
.Q,s
L: (f::,
•
T
= L: c * .
v,sk
i=1 kv 1 js =0 v,J s,s
A
Proof.
Write, U
v,sk
-
f::,
j , s) Z i'
s
v Js's
(s = 1,2, •.. , q, k = 1,2, .•. , p), and proceed as in lemma 3.3.2 to
,
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148
show U k - T
v,s
v,sk
+
0 in P - or in Q - probability for each pair
v
v
(s, k), where s = 1,2, ... , q and k = 1,2, ... , p.
Bonferroni
inequality, we get the result,
Then using
Q.E.D.
As an immediate consequence of lemmas 5.3.1 and 5.3.2. we get
the permutation test statistic M defined in (5.3.17) is "asymptotically
v
equivalent" to the parametric test statistic S defined in (5.2.33).
v
Thus M and S
v have asymptotically the same distribution either under
v
HO or under H.
v
Hence the test procedure based on M given in (5.3.18)
v
is asymptotically power-equivalent to the one based on S
v
given in
(5.2.34) .
5.4.
A Multivariate Generalization of Tests in Chapter IV.
We have the model (5.1.1) and we want to test in this section
(5.4.1)
Q*
HO:
qx(p+l)
Q
qx(p+l)
where,
(310
(5.4.2)
e ••
(:3
qp
against the set of alternatives Q*
;I!
Q.
The assumptions (3.1.2)-(3.1.5)
hold in this case also, although we may remark as in chapter IV that the
assumption (3.1.2) is unnecessary here once we assume that the matrices
v
A
"'\i 0
« i=1~
ck*, .)) and
Vl.
(p+l) x(p+l)
c~.
Vl.
/::'0
«A kk ,))
(p+l)x(p+l)
are p.d. where /::'vo
+
/::'0
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149
as v
+
We may note that as in chapter IV, c Ovi = 1,
00.
v
cOv =v,
-1..
2
In the present situation, we assume in addition that the
q-variate d.f. F is diagonally symmetric about the zero vector, that is
if £ = (El"'"
E
p )' has d.f. F, so has -£.
This implies that any E.
J.
has marginally the same distribution as -E. (i = 1, 2, ••. , p) and any
J.
pair (E.
J.,
1 :::; i
;t
E.) has the same joint distribution as (-E., -E.),
J
J
J.
j :::; p.
We keep the notations F .(x'), F.
VJ. ~
Vl,S
(x.·), F
(x y) F(~'),
vi,ss"'.-
F (x), F ,(x,y), H (x), H
(x), H
,(x,y) (1 :::; s
s
ss
v ~
v,s
v,ss
as in section 5.1.
;t
s' :::; q) same
However, the sequence of alternatives to be con-
sidered here is as follows:
(5.4.3)
v
where ~vo
D =
rO -
1/2
,
«T sk »
qX(P+l)
For notational convenience, we consider here q-dimensional
rectangles in the' real q-space ~q defined as
q
(5.4.4)
(\
s=l
-~
s
q
= ('\
s= 1
{x la. 1
< x :::; a.
1 }
s J s -2"' s
s
J s +2"' s
, ... , -1, 0, 1, ... ,
~
s
,
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I
I
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I
I
I
I
I.
I
I
150
_00
=
a
denote a finite set of
1 < ••• <
-Q, - -
s 2
ordered points on the real line with a . s = -a . ,. let a a
-J s '
Js's
,s
s = 1, 2, ••. , q).
The observable stochastic matrix is
* ..
X* = (X1,
~v
(5.4.5)
~v
*
·,!,,),v;;::1,
vv
where X* . = «X(l»*
vi
, •.• , (X(q.»*)I
V1.
'
~V1.
(X(~»*
(5.4.6)
V1.
0;
with
Q,
=
L;s
js=-Q,
(s = 1, 2 , ••• , q; i =
I.
J s ,s
s
1,.~tI,
v,
V
Z
vij ,s
s
;;:: 1) ,
and
1
zV1.J
.. S ,s
(5.4.7)
=
o
otherwise ,
-Q, , ••• , -1, 0, 1, •.. , Q, , s = 1, ... , q, i
s
s
1 , 2, ••• , v, v ;;:: 1).
For the testing problem considered here, a class of permutationally distribution-free tests has been proposed which is a multivariate extension of the permutation argument in chapter IV.
We shall
discuss more about it later.
First, we consider a class of parametric tests.
we need introduce the following notations:
(5.4.8)
F ..
V1.,J ,s
s
= pr'(x(s)
vi
$
a.
IH)
Js's v
For this,
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I
II
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151
(5.4.9)
Fv i "
,J ,J
8
v ;::: 1).
8
,,8,8 ,
(8')
Pro (X(8)
vi
X.
V1
:::;
Let
v
(5.4.10)
-1
v
L:
F ,
8
v
-1
v
(5.4.11)
,
V1,J ,8
i=1
L:
F ,.,
V1,J
i=1
8
,J
8
,,8,8
,
(£
+1'' -£ 8'
- 8 -< J' 8 :::; £8
A18o, let
Pro (X(8) :::;
vi
=
(5.4.12)
(5.4.13)
=
(-£ 8
:::;
j8
:::;
£ +1' -£
8 '
8'
:::;
Pro (X(~)
V1
:::;
j8'
:::;
a.
J
8 ,8
£ ,+1; 1
8
:::;
(8')
, X\)1.
8
;t
8'
:::;
a,
J
8 ,,8
,IH a);
:::; q) •
Define
(5.4.14)
= FVi ,j +1,8
8
The notation8 (5.1.17)-(5.1.20) are the 8ame except that
range8 from -£
8
to £
8
for all 8 = 1, 2, ••• , q.
J'
s.
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152
The "score" functions
~.
Js's
<jJ
s
(u) are the same as in (5.2.1).
are the same as in (5.2.2).
We introduce another class of
"score" functions 1jJ (u) defined by
s
,°
1jJ (u)
(5.4.15 )
s
u
<
1, 1
<
~
~
s
q.
Further, let
(5.4.16)
(5.4.17)
F*
j
s
p*
j
s
F.
- F-j +l,s
Js's
s
,s
,s
= F*j +l,s
S
= 2F.J s ,s -1 (j s = 1,
F':
Js's
(
2P.
Js's
J.s
2 , ••• ,
1, 2 , ••• ,
9,
s
+1),
9,8) ,
/
F*
(5.4.18)
~*
j ,s
=
J;s+I'8 1jJ(u)du
s
l+F.J* +1
s
r
du
fF*s
j
Js's
=
;*
F,J + 1, s
s
,s
,s
2
du
) l+F~
Js's
2
=
~o
,s
=
0, for all s
(5.4.19)
=
1, ••. , q.
A
Oss
=
Let
p.
-Js's
(j
s
=
1, ... ,
9,
s
),
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I
I·
I
I
I
I
I
I
Ie
t
I
I
I
I
I
153
(5.4.20)
A (F , F
O S
s
I .} " {I. } )
Js
JSI
I' {
=
(j
S
fj,j
= 0, 1, ... , 2- ; j
S
S
I
= AOs S
fj,.
s ,s J s "s
0, 1, ... , 2-
I;
IP..
I •
J s ,J S "s,s
1 ::s; s
;t
s
I
::s;
q).
Hence, from (5.4.20), we can rewrite
2IS
= 2
2IS
j =1 j
S
S
I
1=1
(1 ::s; s
(5.4.21)
Assume A =
((AO
o
ss
I'))'
.
is p.d.
q(p+1)xq(p+1)
We introduce the statistics
v
2IS Z
7J.
(5.4.22) T
=
I c* .
v, sko
2- v ij ,s j , s
i=l kV1 •
J s =-
S
S
S
=
v
2-
~ c* . (
L; s U ..
k v 1 . "I v 1J
1= I
J =
S
LJ
,s
S
fj, •
J
,s
S
)sgn x(s.)
where,
1
if
Ix(~) I e:'I.
V1
I.
I
I
s
I
(5.4.23)
U
II.
V1J
S
,s
=
o
otherwise
J s' s'
V1
;t Sl
::s;
q) •
I
I
,I
I
I
I
I
I
154
Us = 1,2, •.. ,
(5.4.24)
and,
Q, )
s
sgn x(~)
V1.
if x(~)
J:
II
>
a
i f x<s) =
a
i f x(s) <
a
V1.
vi
vi
(s = 1, 2, ... , q; k = 1, 2, ... , p; v ;: : 1).
respective distributions of ~v
X* under H
O
LEMMA 5.4.1.
(3.1.2)-(3.1.5), (5.1.3) and the diagonal symmetry of the q-variate
d.f. F, the joint distribution of the elements of
I
,I
I,
is asymptotically N(p+l) q (Q, /;:'0 ~ ~o)·
T
~vo
LEMMA 5.4.2.
, kO))
= «Tv,s
qx (p+1)
Under Q , the model (5.1.1) and the same assumptions as
v
in lemma 5.4.1, the joint distribution of the elements of T
~v
-is
asymptotically N(p+1)q (Q(A)Xo/;:'o' /;:'0 @ ho)' Q(A) being defined in
(5.2.21) .
We may remark at this stage that for testing the null hypothesis in (5.4.1), the test statistic we use is a quadratic form in
the elements of T
I.
I
I
The following
Under P , the model (5.1.1), and the assumptions
v
(5.4.25)
I
= Hand
H.
ov
v
two results (similar as lemma 5.2.1 and 5.2.2) are immediate:
Ie
I
I
Let P and Q denote
v
v
~o
(5.4.26)
E
wh ere, ~vO
s vo =
=A
~
0
Ill'
The test statistic is
q
q
p
P
L:
L:
L:
L:
s=l s'=l k=O k'=O
A
~vo
T
T
esk,s'k'
v,sko v,s'k'o vO
'
Sk
= ( ( evO,sk,s'k'·) ') and ._~v-01 = «ev o ' s 'k ' ) )
I
I
.-
I
155
The test procedure based on the statistic 8
vO is specified by the
critical function \)J 3 (X~) , where
1
W3 (~~)
(5.4.27)
I
I
t
I
I
0 3VE
0
if 8
vo
if 8
vo
if 8
vo
> 8
= 8
< 8
VO,E
VO,E
VO,E
where 8 VO ,E' 03VE are chosen in such a way that E[~3(~~) IP v ]
= E,
so that the test is a similar size E test.
From lemmas 5.4.1 and 5.4.2, it is easy to see also that
under HO' 8VD is asymptotically X~(P+l)' while under Hv ' it is
X~(p+l)(nO)' where,
(5.4.28)
Ie
I
I
I
I
I
I
II
I
However the above test is not distribution-free.
To obtain
a (conditionally) distribution-free test, we use once again a permutation argument.
Xes) =
(5.4.29)
viO
where II
j
(j
s
=
S
,s
1, 2, ••• ,
(5.4.30)
First, the following notations are introduced:
fI,
•
s'
s
= 1,
2, ••• , q; i = 1, 2, ... , v, v
*
~O
Xes)
v10
(s)
X
VVO
~
1).
I
.1
.-
I
I
I
I
I
I
Ie
156
*
(5.4.31)
(1)
XCI)
sgn Xv l' ••• sgn vV f
sgn .¥.VO =
sgn X(q) ••• sgn
vI
v
(5.4.32)
L:
-9., , ••• , -1,0,1, ... ,9.,);
S
s
Z"
i=I . VJ.JS'S
v
(5.4.33)
L:
Z"
i=I
I
I
I
I
I·
I
v"
J s ' J s ' ,s , s
,
( -9.,s ~ J's ~ 9.,s·, -9.,s' ~ J' S , ~
(5.4.34)
F
0, FV,j +I,S
v,-9., ,s
s
Fv, -9., , -9., "
s
s
=
s, s'
Fv,j +I,j ,+I,s,s'
s
s .
=v
L:
(5.4.37)
F*
9., )
s
°
js'
js
-1
L:
L:
m =-9., m ,=-9., ,
s
s
s s
(-9.,
(5.4.36)
~ J'
ss
~
9., " -9., ,
s
s
~
FV,j +I,s
s
v,j +I,s
s
*
Fv,j +I,j ,+I,s,s'
s
s
Fv,j
s
+ 1 ,j s ,+I,s,s '
- Fv,J'+1 ,(- J
',
) ,s,s ,
s
s
Fv, (
' ,+ 1 ,s,s ,+Fv, (
',
) ,s,s ,
-J' ),J
-J' ),(
-J
s
s
s
s
(l~J'
s
~9."l<J',~9.,,;
s
-
I),'
m =-9.,
s
s
~
(5.4.35)
n
N
S
5s
= v -1
s·
I
I
I
,=
Z"
v J.J s ,s v J.J s ' ,s
s
s
s;ts');
J' ,
s
~
9., ,)
s
•
I
157
I
.-
1/i(u)du
(5.4.38)
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
(l+F*
v,j +1,S)/2
.J
s
. (l+F* ' ) / 2
v,Js's
=
~(u)du
(l+F*
v,j +1 ,s) /2
J
*
s
du
(1+F,
v,J ,s )/2
s
Consider the statistics,
v
(5.4.39)
(s
U*
v,skO
= 1,2, •.. ,
9.-
l: c* ,(
i=l k.v~ j
=
q, k
s
l:s
=1
= 1,2, ••• ,
p).
The permutation test procedure,
we are going to discuss below, will be based on U*
v,skO
(s = 1, ... , q; k= 1, ... , p).
Under HO in (5.4.1), X 'a (i = 1,2, .•. , v) are v i.i.d.
"'v ~
random vectors.
Hence, the joint distribution of the elements of
~~O rema:ins invariant under v! possible permutations of its columns.
v
Further, under HO, there are 2 possible sign-changes of the columns
of sgn ~~, each such sign change being independent of the permutations
of the columns of X*.
"'va
9.-
(5.4.40)
L:
s
j =1
s
Hence, if we denote
1, ... , v; s = 1, ... , q),
I
I
,-
I
I
I
I
I
I
Ie
t
I
I
158
(5.4.41)
then the distribution of W* remains invariant under any permutation of
"'v
v
its columns, and, further, 2 possible sign changes of the columns
"'Va
If we now. consider a finite group {G } of transformations
v
{gv} which maps the sample space onto itself, where a typical transformation g
v
(5.4.42)
(_l)ffiv
W* =
V"'V
................................
d = 1,2, ... , v and (i , ••• , i ) is any permutav
v
tion of the integers (1, ... , v). Thus there is a set of 2 v!
where, m =
d
0,1.;
transformations in G.
v
Under HO, the conditional distribution over
the set of 2v v! realizations (generated by G ) is uniform, each
v
v
having common conditional probability (2 v!)-l. We denote the
conditional probability measure by
(5.4.43)
I
Let,
I
g
is such that
(_l)ffiv
I
,
I
,-
W(~)
(i = 1, ••• , v; s = 1, ••• , q).
V1
of sgn X*do not affect
(5.4.44)
E(U*
v,skO
A
ov,ss
=
Ie? v')
=
O?'.v
Then,
a
A*2
8..
V,J s ,s
(v.
+v(.) )/v,s=1,2, ... ,q;
J,s
-J s ,s
s
I
I
,e
I
I
I
I
I
I
159
(5.4.45)
A
Ov,ss'
1 :'> s
;t
s'
q •
:'>
Using now the results,
(1 :'> j
E[U ..
V1J
= [v
U. .
'S V1J "S
S
S
(s)
(s')'~
V1
v
, sgn (X . ) sgn (X i
j
, jS, S "s,s
V..
J S ,-J S "S,S
)
s
:'> fI,
s
,
IV v' ]
,
Ie
I
I
I
I
I
I
,I
,-
I
we get,
(5.4.46)
E(U*
U*
v,skO v,s'k'O
(k, k'
1-,0')
~v
0, 1, ... , p; s, s' = 1, 2, ... , q)
Write
(5.4.47)
Q~o = «U~,skO))'
qx(p+l)
1 :'> s :'> q)
,
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.-
We have,
I
(5.4.49)
E(U* I{P') = 0
"'v 0 v
'"
(504.50)
var(u*ol~') = A
"'\)
v
"'\)
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,
I
Ie~
I
I
(5.4.48)
A
"'Ov =
«AOV ,SS,))
qxq
e "'ov
A
= K
(say)
"'OV
Again, assuming A is p.d. so that K is p.d. the test
"'ov
"'OV
statistic we propose is given by
q
~
pE
pE . U*
U*
E
6
, Ksk,s'k'
v,skO
v,sk
0 Ov
s=ls'=1 k=O k'=O
(5.4.51)
where, «Ksk,s'k')) = Kg
a generalized inverse of K
Ov
Ov'
Ov'
We can show,
here, that K is p.d. in probability.
Ov
The following test is proposed based on the critical function
W4(~:0'
~:)
sgn
given by
1
(5.4.52)
1}J
4
ex*
~vO'
sgn X* ) =
-'v
cS
4VE
0
> M*
VO
VO,E
if M* = M*
vO
vO,E:
i f M* < M*
vO
VO,E:.
i f M'~
where M*0 ,6
are so chosen that E [I/J (X*O' sgnX*) l(pl] =
v ,E
4VE
4 "'v
"'v
v
E:.
This implies that E[,i,'1'4 ex*
X*)J'p ] = E, so that the test is a
"'v 0 , sgn "'v
v
similar size
E:
test.
However, as in the earlier cases, in this case also, we need
find the asymptotic distribution of M* •
vo
first prove the following lemma.
With this end in view, we
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assumption of diagonal symmetry of F, we get,
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Under (5.l~1), (3.1.2)-(3.1.5), (5.1.3) along with the
LEMMA 5.4.3.
+
~ov
in Pv - (and in Qv -) probability.
A
~O
We can show Ao
v,ss
Proof.
+
AOss as v
+
00
in P - (and in Q -)
v
v
probability similarly as in lemma 4.3.1.
Now, proceeding as in lemma 5.3.1 we can show A
,
Ov,ss
as v
+
00
~
(1
s
7
s'
~
q).
+
A
Oss
Q.E.D.
We next find the permutation limit distribution of Q~o •
THEOREM 504.1.
Under the model (5.101), if the assumptions (3.1.2)-
(3.1.5), (5.1.3), (5.2.14) along with the assumption of diagonal
symmetry of F hold, then the joint distribution of the elements of
Q~o is under~~ asymptotically Nq(p+l)
Proof.
(Q, ~O
e ho)'
The proof follows by combining Theorem 5.3.1 and Theorem
4.3.2 and hence is omitted.
The next lemma establishes the llasymptotic equivalence"
of the parametric test statistics T
ko to the permutation testv,s
statistics U~~
v,skO'
LEMMA 5.4.4.
U*
~v 0
ProoL
U*
v,skO
Under the same assumptions as in Theorem 5.3.1,
- ~v
T o·~
+ a
qx(p+l)
-in Pv - or in Qv -probability.
-
Write,
T
v,skO
-
*
fj,.
J
s ,s
)U. ,
V 1J S ' S
] sgn. X(s)
. •
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162
Using lemma 4.3.2 we can show that U* k - T
v,s 0
v,skO
Q\) -probability for each s = 1, 2, ••• , q, k
the Bonferroni
= 0,
° in Pv -
1, ••• , p.
and in
Using now
inequality, we get the result.
The above result along with lemmas 5.401 and 5.4.2 shows that
under Ho, the joint distribution of the elements of Q~o is asymptotically Nq(p+l) (Q, ~O @ ~O), while under Hv it is asymptotically
Nq (p+1) (.J2(A).roJl o '
110
9 ~O)·
Thus, M~o is under HO asymptotically
XCp+l)q' while under Hv it is asymptotically
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+
X~p+l)q(nO)' no being
defined in (5,4.28).
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We may note that the above results can be extended to the
case of a countable set of class-intervals by using co-ordinatewise
contiguity.
5,5.
Concluding Remarks and Scope for further Research.
Unlike tests in the univariate case, namely those in chapters
III and IV, we cannot claim here asymptotic optimality from the point
of view of Wald in a quite general sense.
It has been shown by Purl
and Sen [31] that when the derivative of the logarithm of the
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conditional density of one variable given the rest is a linear combination of the derivatives of the logarithms of the marginal densities,
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-
then the tests proposed by them in case of a similar problem for
ungrouped data is asymptotically optimal according to Wald's criterion.
The assumption holds in case of multinormal distribution or for
any coordinate wise independent distribution,
The latter con-
elusion holds true here as a trivial consequence of the results
derived in the univariate case.
But, the former conclusion is not
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apparent.
Studies can be pursued along this line, but at the moment,
we are not aware of any particular statistics which will give
asymptotic optimality in our desired sense.
Also, we comment as in
Sen and Puri [31] that the model (5.1.1) can be made a little more
general by taking
X(s,)= (3. 0 +
s
v~
(s
p
~
k=l
(s)
(3sk c k ' +
v~
= 1, 2, .•• , q;
i
E
(s)
.
v~
1, 2, ••. , v, v
~
1).
The model presents certain complications, since, then, the
variance-covariance matrix of the statistics cannot be expressed as
a Kronecker product of two non-negative definite matrices, thereby,
rendering the analysis more difficult.
However, in many actual
situations, (e.g. the c-samp1e problem), such requirement is met.
A further scope for future research is to extend the findings
of chapters III, IV and V to the situation where the predictors are
stochastic.
We have solved the problem only in the univariate case
with ungrouped data in chapter II.
A multivariate extension of those
results for ungrouped data also seems feasible.
Also, extending the
findings of chapter II to the case of signed-ranked-statistics we can
tackle the analysis of covariance problems in paired comparison
designs.
Finally, we make the comment that the problem of tied ranks
can be tackled very easily with the techniques developed earlier,
if for tied observations, we use "scores" described in earlier
chapters, the class-intervals being properly picked from the ranks of
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these observations, while for untied observations, we take classintervals (depending on the ranks of such observations) with
arbitrarily small probabilities of belonging to them.
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CHAPTER VI
A NUMERICAL EXAMPLE
6.1.
Introduction and the Data.
In this chapter, we consider an example to illustrate some
of the test procedures developed in the earlier chapters.
The
example, we shall be using, will illustrate test procedures developed in Chapter III, although more general examples can be easily
constructed.
The problem is to test the effectiveness of several contraceptive devices.
1500 married women in the age-group 15-25 are
observed for a period of 36 months.
Women are either subject to
no contraceptive device, (hereafter, referred to as contraceptive
1 i.e., the 'control group') contraceptive 2 or contraceptive 3.
There are 500 women in each group.
A 'conception delay' is defined as the exposure months
preceding
but not including the month of conception.
We treat
conception as a chance event and 'conception delay' as a random
variable.
An 'ovulatory cycle' is equated to a calender month.
For simplicity it is assumed that the population is 'homogeneous'
i.e. the monthly probability of conceiving remains constant from
one month to another and for a homogeneous population conception
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166
delay is distributed geometrically.l
The data shown in Table 6.1,
though fictitious, corresponding to this fact, admitting, of course,
sampling fluctuations.
Conception delay is calculated by subtracting
nine months from the interval between marriage and the first livebirth.
There are 12 class intervals within each of which we know
the number of observations belonging to the three groups earlier
mentioned.
Hence, the problem is a 3-samp1e grouped data problem.
Throughout, this chapter the suffix
V
has been dropped,
whenever possible, without ambiguity.
TABLE 6.1
showing distribution of conception delays for women
subject to contraceptives 1, 2, or 3.
Conception
Frequenc of women sub;ect to
ContraDelay
ContraContra(in months) ceptive 1 ceptive 2 ceptive 3
0-1
1-2
2-3
3-4
4-5
5-6
6-7
7-8
8-9
9-10
10-11
11 & above
Total
145
107
76
45
42
28
16
17
6
6
3
9
500
140
102
65
56
43
23
23
18
8
5
3
14
500
128
94
69
60
40
25
18
17
15
12
5
17
500
Total
Class
FreQuency
413
303
210
161
125
76
57
52
29
23
11
40
1500
1see e.g. a paper by Mindel Sheps, "Uses of stochastic
models in the evaluation of population policies. I. Theory
and approaches to data analysis", Proceedings of the 5th
Berkeley Symposium on Mathematical Statistics and Probability
Vol. IV 115-136.
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Let T , T , T denote conception delays for women subject to contra2
l
3
ceptives 1, 2 and 3 respectively.
P2 , P3 > O.
Let PI (t), P2 (t), P3 (t) denote
Then log T , log T + log P and log T + log P are
2
2
l
3
3
identically distributed.
In other words, weare assuming that the
use of a contraceptive is effecting a shift in the scale of the
distribution of conception delay for women who are not subject to
any contraceptive device, so that when we take logarithm of this
variable, there is a shift in location of the transformed variable.
Assume Pr(log T < u) = Fl(u) =F(u-S').
l
a
Hence, Pr(log T. < u)
J -
= F(u - 13' + log P.), j = 1, 2. If now X denotes logarithm of cona
J
ception delay and the 1500 r.v. 's are labelled as Xl' X2 ,···, X1500f
t h e d • f • 0 f X ( concept10n de 1 ay f or 1 th woman ) can b e wr1tten as
0
0
0
0
1
I, if the i
where, c
(k
ki
th
woman is subject to contraceptive k
{ 0, otherwise,
= 1,2,3; i = 1,2, .. ,1500).
where
~i
=
{
(6.1.1) can be rewritten as
Of oth
2/3, 1 1 . woman is subject ot contraceptive k
-1/3, otherwise,
(k = 1, 2, 3·, i = 1, 2, ••• ,1500).
Thus,
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168
1500
~
(601.3)
c
i=l
1500
~
i=l
ki
= 0, k
1, 2, 3.
~i = 500(2/3)2 + 1000(-1/3)2 = 1000/3
1500
~
c
i=l
ki
1000(2/3)(-1/3) + 500(-1/3)(-1/3)
ck'i =
= -500/3, 1
So,
~v =
(6.1.4)
t
-.
J
1
-.5
-.5
1
-.5
-.5
-.5
1
~
k
~
k' < 3.
which is of rank 2.
So, if we work with a principal submatrix of order 2 of A , it will
-v
be of full rank.
for all k
= 1,
2
max
1<i<1500
Also
2, 3.
1k
1500
= 4/9 and
i
~
i=l
c
2
= 1000/3,
ki
Hence,
max
c * 2 = 4/3000 = .001333 •
1<i<1500 kl.
(6.1.5)
0
(6.1.2) is the same as 0.1.1).
Also, from, (6.1.3), (6.1.4) and
the remarks below it,and (6.1.5), we find that (3.1.2)-(3.1.5) are
satisfied.
The underlying distribution F is assumed to be such that
(3.1.6) is satisfied.
(6.1.6)
We want to test
H
g
The usual chi-square statistic for testing the homogenity of
several populations can be used in this case.
In the next section,
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we compute the chi-square statistic, and the permutation test
statistic defined in (3.3.6) using (1)
(2)
Normal scores.
The number of observations being large,
the asymptotic chi-square distribution of the permutation test
statistic seems reasonable.
6.2.
Calculation of actual test statistics and analysis of the
data.
We calculate first the homogeneity chi-square statistic.
Let f
and k
denote the observed frequency for the jth class interval
jk
th
group (j = 0, 1, ... , 11; k = 1, 2, 3) .
Jo
l:
=
k=l
Let
11
3
f.
f
jk
(j = 0, 1, ... , 11) ; f
ok
=
l:
j=O
f
jk
(k
1, 2, 3) •
Then, under H in (6.1.6), the expected frequency for the jth
o
th
= f. f k!1500 (j = 0, 1, ••• 11;
class-interval and k
group is f'
Jk 0
J0 0
k = 1, 2, 3).
The homogeneity chi-square statistic is given by
11
l:
2
f.k!f' k - 1500 ,
J 0
j=O k=l J
(6.2.1)
3
l:
which under H is a central chi-square with llx2 = 22 degrees of
o
freedom.
The observed value of chi-square in this case turns out
to be 18.6506.
significance.
It is insignificant even at upper 25% level of
We can, thus, conclude that if a homogeneity chi-
square test is used, the data bear no evidence to reject the null
hypothesis •.
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Next, we proceed to calculate the test statistic given
in (3.3.6) using Wilcoxon score.
Then
~(u)
=
2u - 1,
A
6 v,.J.
Fv,J. + FV,j'+1 - 1.
=
A
TABLE 6.2.
showing values of 6 , for the Wilcoxon scores test.
VJ
t
i
0-1
1-2
2-3
3-4
4-5
5-6
6-7
7-8
8-9
9-10
10-11
above 11
Total
0
1
2
3
4
5
6
7
8
9
10
11
A
cumulative
frequency frequency
413
303
210
161
125
76
57
52
29
23
11
40
1500
V6
413
716
926
1087
1212
1288
1345
1397
1426
1449
1460
1500
Vi
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-1087
-371
142
513
799
1000
1133
1242
1323
1375
1409
1460
Using the formula (3.3.4), we get
(6.2.2)
(U , U , U )
1
2
3
=
(-1.0619, -.1521,1.2140).
From (6. 1. 4) we find a generalized inverse of
(6.2.3)
A*
-v
= 4/3
1
.5
0
.5
1
0
0
=
4/3
2/3
0
2/3
4/3
0
0
0
From (3. 3 •5) ,
(6.2.4)
~V
2
A (F , {I,})
v
J
=
given by
~
.32202299 •
Hence, using (3.3.6) we get observed value of the chi-square
(1)2
statistic as X
= 5.4331.
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2
Under H ' the statistic is distributed as X2 •
o
upper 10% value
ofax~ is 4.605 and upper 5% value of a X; is 5.991.
So, the observed statistic (using Wilcoxon score) is significant at
10% level of significance, but not at 5% level of significance.
At
any rate, we can see that in this case, it is better than the homogeneity chi-square statistic.
We next consider the normal scores test.
x
ep
¢(u)
-1
(u), where, ep(x)
=
_
A
;;.
(6.2.5)
J
-1/2
(2~)
In this case,
2
exp(-t /2) dt.
Then,
ex>
.
vJ
A
= 0, 1 ..• ,11) being defined in (3.3.1), (3.3.2)
Vj ' FV,j and ;;'Vj (j
and (3.3.3) respectively, and
~(x)
-1/2
2
(2~)
exp(-x /2).
A
TABLE 6.3.
j
0
1
2
3
showing values of ;;'Vj for the Normal Scores Test.
;;.
-1. 21260679
- .31894609
.11955300
.44431388
Then, (U , U , U )
2
1
3
2
j
Vj
A (F ,{I.})
v
J
;;.
.
VJ
4
.72908880
5
.96909789
6 1.16507737
7 1.36861240
j
;;.
.
vJ
8 1.56503017
9 1. 73387609
10 1. 87671818
11 2.31332213
=
(-1.9152, -.2940, 2.2092)
=
.91897472 •
Then, the observed value of the chi-square statistic (defined in
(2) 2
Ie,
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The theoretical
(3.3.6)) is given by X
6.2643.
Thus, the statistic used
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(a
X~) is significant at 5% level of significance, and we may draw
the conclusion, that, in this case the Normal Scores test is better
than the Homogeneity chi-square test or the Wilcoxon test.
The above is an example where our proposed tests are much
better than the classical tests available in the literature.
The
drawback with the homogeneity chi-square test is that too many
degrees of freedom are associated with it, thus making it less
likely to detect any heterogeneity among populations than the tests
we have proposed.
It is expected that in usual situations, the
value of a chi-square per degree of freedom will be much less for
a homogeneity chi-square than the one used by us.
This suggests
the superiority of our proposed tests to the classical tests.
The case of tied ranks for the p (2:. 2) sample problem has
been treated in the null situation by Kruskal
3
Wallis •
2
and Kruskal and
Also, asymptotic power properties of such tests can be
treated very easily by using asymptotic distributions of U-statistics.
4
The treatment of ties in the "normal scores" case is not, however,
available in the literature.
Our proposed methods include that
situation as a particular case.
2
-Kruskal, W. H. (1952)0 A nonparametric test for the several
sample problem. Ann. Math. Statist. 23 525-540
3Kruskal, W. H. and Wallis, W. A. (1952). Use of ranks in
one-criterion variance analysis. J. Amer. Statist. Assoco 47 583-621
4Hoeffding, W. (1948). A class of statistics with asymptotic
normal distribution. Anno Math. Statist. 19 293-325
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We might remark that in this case no attempt has been made
to study the permutation distribution of the test statistics.
The
difficulties associated with their use in practice have already
been mentioned in earlier chapters.
However, attempts are being
made to study the permutation distribution with specific scores
and moderately large sample sizes.
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REFERENCES
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--
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