ASYMPTaTIC OPTIMALITY AND DISTRIBlITION THEORY OF
NONPARAMETRIC TESfS FOR RESTRICfED ALTERNATIVES
by
Ming-Tan M. Tsai
A dissertation submitted to the faculty of The
University of North Carolina at Chapel Hill in
partial fulfillment of the requirements for the
degree of Doctor of Philosophy in the Department
of Statistics.
Chapel Hill
1987
Reader
A
J
.k·j~
Reader
~
h.
t·. : -
ACKNOWLEDGEMENTS
I would like to express my deepest gratitude to my advisor PRANAB
KUMAR SEN for his effort in guiding this work and preparing me for a
career in academic research.
Also. I would like to thank the other members of my dissertation
committee:
Shrikant Bangdiwala. I.M. Chakravarti. M.R. Leadbetter.
Donald Richards. Gordon Simons for their interest and help.
The manuscript was typed by Ms. Lisa Brooks.
Her assistance and
patience are much appreciated.
My graduate study has been supported in part by a George E.
Nicholson Memorial Fellowship and by a Graduate School On-c8mpus
Dissertation Fellowship.
Finally. I am grateful to my wife and family for their patience
and encouragement.
TABLE OF CONTENTS
CHAPTER I
1.1
INTRODUCfION AND LITERATURE REVIEW
Introduction . . . . . . .
1
1.2 Some testing problems with restricted alternatives
3
1.3 Berry-Esseen bounds of rank statistics . . .
12
1.4 The optimal properties of LR test statistics
17
1.5 Outline of the proposed research
20
CHAPTER I I
.
ASYMPTOfIC OPTIMALITY OF UI-LMP RANK TESTS FOR
RFSfRICfED ALTERNATIVES
.
.
22
2.1
Introduction .
2.2
Preliminary notions
23
2.3
Asymptotic optimality of UI-LMPR tests
28
31
2.4 Multivariate generalizations
2.5 General remarks
.
. . . .
43
CHAPTER I I I ASYMPTOfIC DISTRIBUTION OF UI-LMPR TESTS FOR
RFSfRICfED ALTERNATIVES
3.1
48
Introduction
3.2 UI-LMPR test statistics:
asymptotic distribution theory
3.3
Rates of convergence for distribution of UI-LMPR
test statistics
. . . . . . . . .
.
34
Asymptot i c power comparIson
.
CHAPTER IV
4.1
0
fo..~andR..~
~N
-~ . . . . . . . . .
49
55
60
ASYMPTOTICALLY EFFICIENT RANK TESTS FOR RFSfRICfED
ALTERNATIVES IN RANOOMIZED BLOCKS
Introduction
.
4.2 The UI-LMPR tests based on the intrablock rankings
68
70
4
we have
~
(9)
cP '"
~
a. V 9
'"
~
8 .
0
Let s = {(x .xq • • • • ) . x.
'" 1 ~
'" 1
Then the LR test for the hypothesis 9
hypothesis HI :
~ ~
~
8
0
~ ~
. i=l •.... }.
against the alternative
8 *1 . where 8*1 C 8 1 , is defined by
cP (s)
a
__ { 1
(1.2.1)
o
otherwise
N
rr
i=l
constant
is determined by sup
~~80
~
f(x.;9). i
",1 '"
~
function on
I;
Here the density
cP (s) = a.
a
1. are smooth enough to ensure that cp (s) is a measurable
a
(s.~).
where
~
is the product a-field of subsets of
Let the set of all a-tests be denoted by
"
~.
a
.
,.)','.
~.
A test of the form
cp(s) = 1 (s) is called a non-randomized test and A is the critical
A
region. An a-test CPo ~~afor te~ting 8 0 against 87 is called a
uniformly most powerful (liMP) a test in
~
if
a
(1.2.2)
An a-test cp is called unbiased when
called similar if
~cp(~)
= a. V
~ ~
90 .
~
(9)
cp '"
'"
~
~
9* , and is
1
The envelope power function
~+(9) of an a-test cp is defined by ~+(9) = sup
shortcoming
a. V 9
'"
~
~ (9). and the
cp~
a
cp '"
(9) of an a-test 'P is defined by
cp '"
~ (9) = ~+(9) - ~ (9)
cp '"
'"
(1.2.3)
cp '"
An a-test 'PO is called a most stringent (MS) a-test if
sup*
9~81
~
(9) = min
'P~~
'PO '"
sup*
9~9
a '" 1
~
(9)
(1.2.4)
'P '"
and is called a MSSMP-test if (1.2.4) holds and
~
(9) = 0 for some
'PO '"
9
~
8* .
1
5
Secondly. we present a brief survey of the main results of testing
problems with restricted alternatives for normal d.f.·s which playa
role of limiting testing problems for other non-normal d.f. ·s.
Bartholomew (1959) derived the LR test to test the equality of ordered
means under the assumption of normality.
He also claimed that the power
of his ordered test is greater than that of the ordinary one based on
2
the X distribution.
Bartholomew's test was extended by Kudo (1963) who derived the LR
test for the testing that the mean vector of a p-variate normal d.f. is
zero against an orthant restriction when the covariance matrix is known.
Let
~1""'~- be
~_
independent random vectors with d.f.
~
~ (·;9.~).
P
~
where
f'V"""
f'V
is a known positive definite (p.d.) p x p matrix. For testing the hyPO-
-2
A
-1 A
A
derive the LR test statistic X , = N .9 'J~,:.a._:wheFe 9
N
'" '" '"
is the maximum
likelihood estimator (MLE) of 9 under the alternative hypothesis.
be any subset of P
J. ~
= {1.2 ..... p}
S J S P. we partition X and
_
~=
[
~
and J' be its complement, and for each
~ as
]k(J)
X
(1.2.5)
k(J')
:.:.J
where k(J) is the cardinality of the set J.
-2
XN
= ¢S~sP
-,
-1
-
Let J
-
Thus
-1
N{~:J' ~J:J'~:J,}1{~:J' > ~}1{~'J' ~, ~ O}
(1.2.6)
where
~:J' = ~ - ~J' :J~J' ~J'
and
-1
~J:J' = ~J - ~J' ~'J' ~'J (1.2.7)
and 1(B) represents the indicator function of the set B.
d.f. of
~~
is distributed as
Under H ' the
O
6
-2
P{X
~
P
~ x} =.~
w{p.k{J»
k{J)=l
2
P{Xk{J) ~ x}
(1.2.8)
where X~{J) is a chi-square random variable with k{J) degrees of
freedom. for k{J)
= 1..... p.
*
=k{J)
~
w{p.k{J»
and
-
P{~.J'
.
-1
> ~}P{~'J'
-
~. ~ ~}
(1.2.9)
* represents the sum over all sets J such that the cardinality
where !k{J)
of J is k{J).
For p
~
= 1 •...• p.
4. the weights w{p.k{J». V k{J)
must
be determined numerically.
NUesch (1966) used a concise quadratic programming method to find
A
the MLE
e
and extended the Kudo testing problem to the case where
'V
~ = 0 2 A with known A and unknown 0 2 .
of LR statistic
-2
-2
Xo = XN
/
N
-1
~ X: A
i=1
'VI
P ~ ., . . ~'~t.),
-2
P{XO
~
x} =
~
:-;C
'V
For this case he derived the d.f.
~i
under HO' which is
;,
w{p.k{J»
k{J)=-b";:'.; .i'
k{J) ~Np-k{J)
P B{-2---'
{.
2
)
~
x}
(1.2.1O)
where B{a.b) denotes a random variable having the Beta distribution with
parameters a and b.
The case where
(1969).
~
is completely unknown was considered by Perlman
He derived the LR test statistic for more general problem of
two positively homogeneous sets in the sense that
1
~ ~
f
i
implies
1
2
c ~ ~ f i for all c > O. V i=O.1. Let Y = N2x. ~ = N e and
S
=
HI :
N
~ (!i-~){!i-~)·.
i=l
~ ~
Then the testing problem HO : ~ ~ f O vs.
fl' is the same as
(1.2.1l)
In this general formulation. he obtained the following results:
e'
7
(i)
The LR test for (1.2.11) reject oRO if
= {II~
U(fO.f 1)
22'
.>
2 -1
- l1(~;fO)IIS - 1I~-l1(~;rl)IISL[l+II!-l1(~;fl)IIS]
~ x
(1.2.12)
where 11
(~;fi)'
with respect to
(ii)
i=O.I. denote the orthogonal projection of Y onto f i
<-. ->S
and II-II denotes the Euclidean norm.
The power of the L.R. statistic approaches to one uniformly
However. since the d.f. of U(f .f ) depends on the
O 1
is not possible to calculate a critical value.
unknown~,
it
For some cases, he
obtained upper and lower bouncls for the d ..f. of U(fO,f ) which prOVided
1
a way to get a
conservativ~
test"
For the special case f O = {~} and f
1
is the nonnegative orthant, the 4.f. which deperidson ~ only through
w(p,k(J»
is given by
(1.2.13)
An alternative approach to the problem of testing against
restricted alternatives was studied by Schaafsma and Smid (1966).
A
""
=
°
[~*],
where AO and A* are (p-q) x p and
'Y
Let
x p matrices respect-
A"""""
ively. each of full rank.
Given that a lies in a q-dimensional subspace
q ~ p. defined by AOa = 0, they considered the testing HO : ~*! =
°
vs. HI : A*a ~ 0 both with ~ known and with ~ = a A (A known and a
2
2
unknown).
First they used the Gram-Schmidt orthogonalization process
to construct an orthogonal matrix G. then transform the problem
8
into the special cases of testing problem H
:. TJ
O
j
•
= 0,
j=p-q+ 1, ... ,p-q+r, vs. HI : ,·DTJ· ~i 0'" .where D = A'\C' )-1 and
.
2
'.~.
If we wri te Y = C'X; • then Y _ <P (. :TJ,o I).
......
...... '"'"
......
p
......
......
TJ = C'S.
they restricted attention to the class of SMP tests.
In the case
0
2
=1,
The Neyman-Pearson
Lemma can be used to show that for testing H against the simple alterO
native HI
o
TJ = TJ , the SMP (equals to MP) size a test rejects H
O
p-q+r
p-q+r
L
-a
(1.2.14)
zl_a
L
{ j=p-q+1
j=p-q+1
where zl
for
is the size a critical value for the standard normal d.f.
Let the vector
ec
r
E of length one corresponding to the half-line
(kelk>O}, and for any two such vectors
e and m, let
~(e,m) = cos
-I
e'm
be the smaller angle between i'iila:\lt":Thi;y:showed that if !O c K
.
,
satisfies
~O
~(eO,m)
= sup
mcK
--
.....";'1:-
~
'
= inf sup ~(e,m)
ecK mcK
<~
.-
(1,2.15)
where
K = {Y c E
r
ID
Y ~ 0, IIYII = I),
then the test which rejects H for
O
p-q+r
L
(1.2.16)
j=p-q+1
is MSSMP a test.
In the case when
0
2
is unknown, they focused attention
on the class of SMP-similar size-a tests, for details see Section 6 of
Schaafsma and Smid (1966).
Finally, under the non-parametric set-ups, Chacko (1963) used
ranks to get a nonparametric version of Bartholomew's LR test statistic.
•
9
•
Sen (1968) considered a class·of aligned... rank order test statistics in
the two-way layout.
Barlow et
al~
(1972):p~esented an
extensive review
of these and related testing problem in._,t.
univariate treatment.
Under the multiparameter setup. the tool of LR is no longer
available.
Bhattacharyya and Johnson (1970) proposed a layer rank test
for the bivariate two-sample location problem against ordered
al ternatives.
Let ~1""'~1 be the bivariate random vectors with
continuous distribution F(X) and
x..
J\+
1 ... · .x..
random vectors with continuous d.f. F(X-S).
against HI :
~ ~ ~.
N be the bivariate
~1+ 2
For testing H :
O
~~
although the test based on layer ranks has an
asymptotic normal d.f .. however. its asymptotic power is substantially
•
Chatterjee and De (1972) used the union-intersection (UI)
principle of Roy (1953). based on Puri and Sen's rank statistic (1971.
Chapter 5. c=p=2). to formulate a bivariate linear rank statistic by
maximizing Bahadur efficiency for the problem of Bhattacharyya and
Johnson.
They also proved that the test would be asymptotically more
powerful than the corresponding unrestricted
alternative parameter space.
~
2
test over the entire
De (1976) extended the method of Sen
(1968) by using the UI-principle to obtain an asymptotic aligned UI-rank
statistic for randomized blocks against an ordered alternative.
Chinchilli and Sen (1981a.b) extended the UI-principle to derive
asymptotic distribution-free tests by maximizing Pitman efficiency for
non-negative orthant alternative problem in the multivariate linear
•
models.
For the multivariate linear models
X.
c ..
_1 = _So + _S(C."-~N) + _1
~~l
~,
V i=l ..... N
(1.2.17)
10
where 9
is a p-vector parameter. 9 = (9 ..... 9 ) is a pxq parameter.
1
""q
0
'V
~i
eN
"",
'"
= (c
···· .C
Ni1
N
=
Niq
'"
), are the regression constants. 1 ~ i ~ N.
N.
~ ~
c
and c. are independent identically distributed (iid)
i=1 "",1
~1
random vectors having a continuous p-variate d.f. F.
They consider the
problem of testing
~(1) ~ O.
(1.2.18)
Let
(1.2.19)
and
!N
(1.2.20)
= «TNjk »j=I ..... p;k=I ..... q
where
N
T . = ~ c . ~.(R .. )
NJ k
i=l N1k
J IJ·'
for j=I ..... p; k=I ..... q
with R.. being the rank of X.. among X
IJ
IJ
1J
~j(i)
Xn .• V j=I
nNJ
(1.2.21)
p. and
is a set of scores defined as either
(1.2.22)
or
~j(i)
=
E{~j(U~i»}
(1.2.23)
(i).IS t h e 1. th or d er statIstIc
. . .In t h e sequence F j (X I j) '
h
UN
were
.... F.(~-.)
J -""NJ
VN
"",
and~.
J
are suitable score functions. V j=I ..... p.
. . , »., ., 1
= « vNJJ
= [ [N~ 1
J J = ..... p
~
.
(a..•. ( i )
1=1
NJ
Define
-a...)(
a..•. , ( i ) -a... ,»))
NJ
NJ
NJ
(1.2.24)
with ~j =
1 N
N
~
i=1
~j(i). for j=I ..... p.
Let
e-
11
~ = «b jk ))j=l ..... p;k=l ..... q and
X(B) =
~i - ~(~i - ~). 1 ~ i ~ N
(1.2.25)
and denote R.. (B) the rank of X.. (B) among X .(B) •...• Xu.(B). V j=l,
1J ~
IJ ~
IJ ~
-TIJ ~
Also let
...• p.
TN(~)
"-'" ._
be defined as in (1.2.20) with R .. (B) instead of
IJ
R..• and write TN into a pq-vector.
IJ
"-'"
*
A -1
~ = (~ ~~)
~
~
Further let
*
A
A_I
and ~ = (~ ~ ~ ~ ~ )
(1.2.26)
A
where
~
stands for the Kronecker product and v N =
"-'"
«~N···))·
JJ
.'-1
J.J -
•...• p
where
1
q
-
2
q k=l
~Njj'
E.
~J
k
{
(TN·k(O) - TN.k(E. k ))
J ~
J ~J
o
i f j=j'
(1.2.27)
otherwise
are the p by q matrix having one in the cell (j.k) and zero
elsewhere.
For convenience we let P1 q =a. and partition
~=
~
(l)]aXl
~
and
~
as
(1.2.28)
.:N(2) (pq-a)x1
and
~ = [~(11) ~(12)]
(1.2.29)
~(21) ~(22)
Also define
-1
~(2:1) = ~(2) - ~(21)~(11) ~(1)
(1.2.30)
and
-1
~(22:1) = ~(22) - ~(21) ~(11) ~(12)
(1.2.31)
They used UI-principle and Kuhn-Tucker-Lagrange (KTL) point formula to
construct an asymptotically distribution-free UI-rank test statistic for
(1.2.18) reject H if
O
12
Q~ = ¢S~sP {~(2:1) ~~22:1) ~(2:1) + ~(J:J')
-1
-1
~(JJ:J')~(J:J.)}1{~(J:J')> ~}1{~(J'J')~(J') ~ ~} ~ x
(1.2.32)
2
Under H ' the distribution of Q is
O
N
where ~ = ~(1) and ~ = ?:N(ll)"
asymptotically equal to a weighted sum of chi-square distributions.
(The weights are the sums of multinormal non-negative orthant
probabili ties).
2
When a=pq, Q reduces to have a similar structure to
N
that of the Kudo's LR statistic.
powers of
The comparison of asymptotic local
Q~ and ~ = ~ ~1~ which is the UI-rank test statistic for
global alternative is also studied.
Some Monte Carlo studies (a=3)
Q~ to ~.
indicate the asymptotic power superiority of
..
. power superIorIty
proo ff or t h e asymptotIc
available.
0
The analytical
fQ2
N to R~
-~ is still not
However, a partial result is given as follows:
Theorem 1.2.1.
Let A = ((o' .. '-)j j'-1
'"
JJ
'
- , ... ,a
= lim &(ANIHO). If o. =0, V
N-llXl
""-'
Ja
pq
j=1, .... a-1 and is of the form {~c E ; 8 1=8 2 =... =8a _ 1=O,8a >O, ~(2) is
unspecified}, then
1.3
Q~ is more powerful than R~.
Berry-Esseen bounds of rank statistics
The central limit theorem (CLT) provides the asymptotic normality,
it may not provide the rate of convergence.
As such attempt has been
made to relate the rate of convergence to the sample sizes and the first
result of this type is due to Liapounov (1901) who proved
1
<P(x)
I
-2
~ c P3 N
log N (N
~
2), where F is the d.f. of
N
normalized sum of N independent random variable with mean zero, variance
one and the absolute third moment P .
3
Berry- Esseen deleted the
logarithmic factor under the Liapounov conditions.
been shaped by Zolotarev and Beck respectively.
The constant c had
e-
13
Theorem 1.3.1. (Berry-Esseen)
Let Xl, ..
~,~be
N independent random
variables each with mean zero and a finite absolute third moment, let s~
1
= N
N
~
i=1
2
gX i , then
sup IFN(x) - cP(x)
XE-E
1
2
where FN(x) is the d. f. of (N sN)
2
I
~ 0.7975 i 3 ,N
N
~
i=1
(1.3.1)
x.1 , and i 3 ,N = N
1
2
P3
(3]·
sN
If, in
addition to the above conditions, X. are i.i.d. random variables, then
1
(1.3.2)
It follows that the Berry-Esseen bound for sums of i.i.d. r.v. is of the
1
order O(N
2
).
The following theorem is a version of multidimensional
Berry-Esseen theorem.
Theorem 1.3.2.
Let ~1'.'.'~ be i. i.q. ,random vectors in EP having zero
mean vector, the nonsingular covariance matrix
third moment P3j' 1
~
j
~
p.
~
and the finite absolute
Then
1
2
3
P
~
(a
j=1
jj )2
P3j
(1.3.3)
where ~O denotes the class of measurable convex set in EP , FN(e) is the
_! N
d. f. of N 2
~ X. and cP is the multinormal distribution with mean
",,1
i=1
vector 0 and covariance matrix
~, where ~-1 = «a ij » and c(p) is a
finite positive constant depending only on p.
Statisticians are often interested in the general statistics that
are not sums of independent random vectors.
Fortunately many of these
statistics can be approximated quite well by sums of independent random
vectors to ensure the asymptotic normality. To find Berry-Esseen bounds
16
_!+O
~
sup 1PH (QN
0
x~E
x) - P( ~2<_ x )1
~ K3 ( p,o ) N 2
+ K4.(P,~)NO
u
N
~
~
P
~ Ic .. 13 +O
~
i=l j=l
1J
(1.3.5)
•
-1
where QN = ~ ~1
~.
Under contiguous alternative hypothesis, she also had the
following theorem.
Under conditions (a1) and (b1)-(b3), and eO
* such that
Theorem 1.3.4.
max
max
l~i~N l~j~p
le .. 1
1J
< e~,
-
there exist constants K5 (p,o) and K6 (p,o) such
that
1
-1
sup IPH (QN ~ x) - PH1(!'~
x~E
1
w~
x)
1
~ K5 (p·0)N
-2+ 0
+
N L
P ( Ic. . 13+0 + 1e .. 13+0 )N.
0
+ K6 (p.0) }:
. 1 J=
. 1
1J
1J
1=
N
> O.
for any 0
(1.3.6)
where W '" ep (j.l.L<)
'"
p '" ~
with W. = L c ..
J
i=l 1J
~.(F .(X ..• 0».
J
J
1J
j=l. .... p.
When p=l. Hu§kova (1977a) also obtained the following results:
(I) Under (b4) and (b5).
sup Ip{SN
<x
x~E
(
I
I
1
2
2
~ (u)du) } - cf>(x) 1
(1.3.7)
0
(II) Under (b1) through (b5),
1
2
2
sup IPH1{(~ - ~) < x ( 0 ~ (u)du) } - cf>(x) 1
I
I
x~E
(1.3.8)
where
~
=
~
i=l
c Ni
J~(F(xi;O»dF(xi;eNi)
and
K7.
K8 are constants.
Further. for a given measure ~ on the Borel a-field on EP . define
(1.3.9)
where
17
(1.3.10)
f (x) = f(x + y)
y '"
and
(1.3.11)
Hu~kova
(1980) extended Theorem 1.3.3 to the following more general
resul t.
Theorem 1.3.5.
EP .
Let h be a bounded Borel measurable function defined on
Under assumptions (al)-(a3), then there exist constants K9,KI0,Kll
such that
N
P
3
+ K10 Wf (K 11 ~
~ IcN··1 ' ~(., gH~' Var 8 M »
i=1 j=1
IJ
~'
",",'
(1.3.12)
where ~(~) is the d.f. of ~.
1.4 The optimal properties of LR test statistics
For testing the null hypothesis HO:
~
=
~O
against HI:
~ ~ ~O'
Wald (1943) proved the following optimal properties of LR tests.
I
= {~; (~-~O) ~(~-~O) = ~}, N=I, ... , where ~ = O(N
Fisher information matrix at
) and I is the
Then the LR test has asymptotic best
~O.
average power over the surface
-1
~
Let
~
of contiguous alternatives.
Further,
the LR test is asymptotically most stringent for any sequence
{~}.
In
the past two decades, the optimal properties of LR test have been discussed for the testing problem with fixed parameter 9, power
~(9)
'"
vanishing level of significance
~,
as
N~.
and
'"
Bahadur (1960) showed that
the LR test statistic is an optimal sequence in terms of exact
stochastic comparison
(Bahadur efficiency).
an extended real-valued
~
For each N, let TN(s) be
measurable function depending on s only
18
(~1'
through
.... ~) reject HO if TN(s) is large.
Let the tail
probability be defined by
(1.4.1)
where
Po
stands for the product measure on
(s.~).
Also define the
Kullback-Leibler information number
(1.4.2)
Thus the maximal slope of a class of tests is 2
I(~.80)
slope of a sequence of test statistic {TN} then
c(~)
~
2
I(~.80)
=2
inf{I(~.~O);
and (ii) the exact slope of the sequence of LR test
LR
statistic {TN } exists and equals 2
I(~.80)
for V
~
Eo
8 .
1
These
results have been generalized and refined in certain directions by
Bahadur and Raghavachari (1971) who used the LR method to find the exact
slope of a given sequence.
Since Bahadur efficiency is largely influenced by the extreme
tails of the distribution of the test statistics. the probability of
large deviation plays an important role in it.
Hoeffding (1965) had
investigated the rate of convergence of error probabilities in the
multinomial case.
He had the following optimality result of LR test:
If a given test of size
~
is sufficiently different from a LR test.
then there is a LR test of size less than
~.
which is considerably more
powerful than the given test at most parameter points in the set of
alternatives when N is large enough provided that
rate.
~ ~
0 at a suitable
The meaning of considerably more powerful is that the ratio of
e-
19
the error probabilities of two tests tends to zero more rapidly than any
power of N.
Herr (1967) extended the partial results obtained by Hoeffding to
non-singular multivariate normal distribution. Koziol (1978) used the LR
method which was formulated by Bahadur and Raghavachari (1971) to find
the exact slopes of some sequences of test statistics in multivariate
analysis under normal assumptions.
However the application is not
straightforward, since the test statistics may have complicated
distributions.
Hsieh (1979b) used the large deviation probabilities to compute
the Bahadur exact slopes for the test statistics as a function of
noncentrality parameters and showed that LR test for a general multivariate hypothesis is asymptotic optimum in the sense that its slope
attains the maximal Kullback-Leibler infor.mation number for all
alternatives while the other five tests (Roy's largest root,
Lawley-Hotelling trace, Pillais trace, Wilk's U, Olson's statistics) are
not.
Hsieh (1979a) was also devoted to clarifying the asymptotic
optimality of a class of test statistics in multivariate analysis under
normality assumptions.
Although the LR test is asymptotically optimal in the sense of the
Bahadur exact slope in the cases discussed above.
It may not be the
uniformly most powerful test for all sequences of alternatives.
It is
better than other tests only for most sequence of alternatives.
So it
is quite interesting to know how much worse it can be for the remaining
sequences of alternatives.
The shortcoming of LR test plays an
important role in this context.
Oosterhoff and Van Zwet (1970) studied
the behavior of shortcoming of LR test statistic for testing a simple
20
hypothesis against a composite alternative.
•
They proved that under a
condition on the exponential rate of convergence to zero of
~
as
N~,
the shortcoming of LR test statistic converges uniformly to zero.
result can be extended to the more general cases.
This
For multivariate
normal distributions with known covariance matrix, Kallenberg (1980) had
the following strong result.
Theorem 1.4.2.
Let X. be i.i.d. random vectors with
...... 1
is a known p.d. matrix.
~
p
(o;8,!), where!
............
"OJ
For testing H
O
9 * C 9 , the shortcoming of LR test tends to zero uniformly on 9 * ,
1
1
1
1.5
Outline of the proposed research
The purpose of this thesis is to develop statistical theory in the
area of nonparametric tests against.various restricted alternatives.
Chapter 2 deals with the_construction and characterization of
•
~
~ ••
J •• _-:-_
:_
asymptotic optimality of union-intersection locally most powerful rank
(UI-LMPR) tests for restricted alternative problems.
The optimality of
UI-LMPR tests is characterized in the light of the asymptotic most
stringency and somewhere most powerful character.
The univariate case
is studied under general regularity conditions and extended it to the
multivariate cases under some additional assumptions.
Some results of
these UI-LMPR tests which are asymptotically UHF for some specific
problems such as univariate k-sample location or scale alternative
problem, bivariate two-sample ordered location problem, profile analysis
problem and linearly constrained problem are also inferred.
Along with the construction and characterization of the asymptotic
optimality of UI-LMPR tests for a wild class restricted alternative
problems, in Chapter 3, we develop the asymptotic distribution theory of
21
•
the UI-LMPR tests and compare the asymptotic local powers of the UI-LMPR
tests with respect to the unrestricted versions over the restricted
parameter space.
This optimum nonparametric testing procedure against restricted
alternatives can be Widely applied to the multivariate k-sample location
(scale) problems. one-way layout. two-way layouts multivariate analysis
of variance (MANOYA) problems. progressively censoring problems.
growth-curve models and combination tests problems and others.
In
Chapter 4. two classes of the optimal rank tests in the light of
asymptotically most stringency and somewhere most powerful character are
considered for restricted alternatives in a randomized block design with
p treatments and n blocks:
•
on rankings after alignment.
Tests based on the intrablock rankings and
We establish "that the corresponding
UI-LMPR tests are asymptoticaHy' 00; agliinst-1an .ordered al ternatives
wi thin the class of rank tests over"' the"' "eht'ire' parameter space of
alternative hypothesis.
From the comparison of asymptotic power
functions. tests based on ranking after alignment are preferred.
In Chapter 5. the optimal test procedure against restricted
alternatives is extended to the more general case where one has to face
mixed models.
We are successful in constructing another conditionally
distribution-free UI-LMP type pure-rank tests which by invoking the use
of the covariate-adjusted rank statistics by fitting a linear regression
of linear rank matrix of the primary vectors on the linear rank matrix
of the covariate vectors. This testing procedure is still asymptotically
optimal in the light of most stringency and somewhere most powerful
•
character. In the last chapter we discuss some topics for future
research.
CHAPTER II
ASYMPTOTIC OPTIMALITY OF UI-LMP RANK TESTS FOR RESTRICfED ALTERNATIVES
2.1
Introduction
Host of nonparametric tests against ordered. orthant and other
forms of restricted alternatives have mostly been considered on an ad
hoc basis.
The union-intersection (UI-) principle of Roy (1953) has
also been effectively employed by Chatterjee and De (1972. 1974) and
Chinchilli and Sen (1981a.b). among others.
These authors have studied
the (asymptotic) power superiority of the UI-rank tests to their global
versions.
However. these were done mostly under rather restrictive
conditions on the associated covariance matrix and noncentrality vector.
leaving open the possibility for relaxation of some of these conditions.
Sen (1982) incorporated the theory of locally most powerful (LMP-) rank
tests in the UI-setup.
However. the resulting tests may not have the
asymptotic optimality in a conventional sense.
This naturally raises
the question on the asymptotic optimality of UI-LMP (and other related
UI-) rank tests in a well defined manner.
Our main contention is to
provide an affirmative answer in the light of asymptotic most stringency
and somewhere most powerful character of these tests.
Section 2.2 deals with the basic regularity conditions.
preliminary notions and a general formulation of such restricted
alternatives where UI-rank tests have been considered.
Asymptotically
optimal rank tests for restricted alternatives in the univariate case
e-
23
are derived in Section 2.3.
This characterization is extended to the
multivariate case in Section 2.4.
Some further remarks on the
asymptotic optimality of DI-rank tests are appended in the concluding
section.
2.2
Preliminary notions
Let X1 •.... ~ be independent random variables (r.v.) with
continuous distribution functions (d.f.) F ..... F . respectively. all
1
N
defined on E
= (-00.00).
Suppose that F. has an absolutely continuous
1
probability density function (pdf) f .. where
1
(2.2.1)
the ~i
=
equal. and
(c Ni1 .· ... c Niq )' are q-vectors of known constants. not all
~
~
= (~1' .... ~q )'
is a q-vector of unknown parameters; q
~
1.
[To be more precise. we should have started with a triangular scheme of
r.v.·s and d.f. ·s.
However. for the sake of notational simplicity. this
refinement will be suppressed.]
In a nonparametric testing problem. one is usually interested in
the null hypothesis (H ) of the homogeneity of the F . i.e ..
i
O
H : F
O
1
= ... = FN = F
In testing for H against a global alternative. ,we
O
arbitrary.
On the other hand. in testing for H
O
=0
~
(unknown) or. in (2.2.1).
(2.2.2)
.
a~low ~
to be
against a restricted
alternative. we need to pose the parameter space (of
~)
in a more
~
structured way.
We assume that there is a positively homogeneous set
r.
such that under the alternative
H* : ~ ~
r
= {~
~ Eq : ~*
=A ~
and is of full rank a (l
O
2 O. where A is of order a O x q
~
a
O
~
q)}.
(2.2.3)
24
The simplest form of
r
= E+ q .
r
relates to the orthant alternative:
q
the positive orthant in E .
the ordered alternative:
~1 ~
...
~ ~
Other notable forms of
~ ~q'
r
0 i.e ..
relate to
with at least one strict
inequality. umbrella alternatives. tree alternatives. loop alternatives
etc.
These alternatives are all special cases of
r.
defined by (2.2.3).
and often. in practice. they appear to be more realistic than the global
alternative that
~
# O.
Hence. we confine ourselves to a general form
of restricted alternatives
r.
and study the asymptotic optimality of
UI-LMPR tests for this problem.
Generally. for the nonparametric hypothesis testing problem. rank
tests are distribution-free (and hence. valid for a broad class of
d.f. ·s).
However. they are based on the use of some scores. suitably
chosen to maintain the high power property for an important subclass of
such d.f. ·s.
In testing for H against a global alternative. a
O
convenient way to choose such scores is to maximize the local
(asymptotic) power of the test for a specific family of d.f. ·s.
In the
literature. this is referred to as the locally most powerful rank
(LMPR-) test.
A very elegant treatment of LMPR tests is given in Hajek
and Sidak (1967).
We shall find it convenient to adapt their notations
for the restricted alterative case too.
First. note that for the orthant alternative problem. we have
r
=
{~
:
~ ~
O}.
For any given
~
~
= (~1' .... ~ q )'
(with nonnegative
elements). we denote by
~
Then. we have
= o~.
for some arbitrary and positive
o.
(2.2.4)
e-
25
H*=
Also, for every e
> 0,
U
"Yef
(2.2.5)
we let
H~e = {H~ ; 0 < 0 ~ e}
(2.2.6)
The basic idea is to incorporate the LMPR test theory for a given "Y, and
'"
then to use the UI-principle to have the test constructed for the entire
class in (2.2.5).
Towards the formulation of such UI-LMPR tests, we
make the following assumptions:
-
[AI]
-LN
For every N, we define c N = N
..",1'
y:1= 1cN'
..",1,1
and let
(2.2.7)
We assume that there exists a positive definite (p.d.) matrix C, such
-1
that as N increases, N
li"'N_ {
[A2]
~
converges to C, and further
l~N (Su - ~)' S;l(:tIi - ~)} = O.
Let us define f.{x;o"Y) as in (2.2.1) with
1
(i) for every
I"V
i{~l), fi{x;o~)
~
= o"Y, i
'"
'"
(2.2.8)
~
1.
Then,
is absolutely continuous for almost all x
and "Y e f, so that if we let f.S{x;S) = (a/aS)f.{x;S)
",,1
""
I"V
1
and
I"V
f. (x;o"Y) = (a/ao) f.{x;o"Y), then, for S = o"Y, f. (x;o"Y) = "Y'f.S{x;O"Y),
1"Y
'"
1
'"
1"Y
'"
'" ",1
'"
"Y e f, x e E.
o
-1
For almost all x and "Y, the limit f. (x;O) =
1"Y
'"
(ii)
[f.{x;O"Y) - f.{x;O)] exists.
1
'"
1
(iii)
For every "Y e f,
'"
(iV)
largest characteristic root of
The
26
~(f)
=
Be[(a/a~)
log
f(X;~)(a/a~')
f(X;~)]
log
is finite.
Consider now the score generating function (vector):
-1 fe(x;O)
•
f * (x) = {f(x;O)}
= (f *1 (x) ..... f *q (x»
,....,,....,,....,
I
,....,
~(.)
and let
=
(~1(·)"
~j(i)
where U* .... ,U*
N1
NN
N from the d.f.
... ~q(.»1
. x
t
(2.2.9)
E.
with
* *Ni )}. i = 1..... N.
= B{fj(U
(2.2.10)
j = 1..... q .
are the ordered random variables of a sample of size
F (for which the p.d.f. is f(x;O».
be the rank of Xi among Xl' ...
,~.
for i = 1..... N.
Finally. let
~i
Then. we define the
vector of linear rank statistics
1
TN. = N-2
J
l!1= 1 cNIJ..
8...•. (Rn.) =
NJ
-~1
l!1= 1 c*NIJ.. 8...•.
(R__ . ).
NJ
-~1
j=l, .... q;
(2.2.11)
For every (fixed)
~ t
r.
on letting
TN(~)
=
~'~.
and proceeding
as in Hajek and Sidak (1967) and Sen (1982). it follows that the test
with the critical region
suitable critical value
WN(~)
~(a»
the level of significance a (0
g[TN(~)IHO] = 0
TN(~) ~ ~(a)}
is the LMPR test for H . against
O
< a < 1).
Note that for every
(with
*
H~.
~ t
at
r.
and Var[TN(~)IHO] = ~'~~. where
M... = «v .. ,N -1 ('~- ..
NJJ
:-:N
-NJJ
v ... = (N-1)-1
NJJ
= {(Xl" ... ~) :
,»
*
= «v .. ,('~~,
NJJ -NJJ
l!1= 1 8...•.
(i)8...•. ,(i).
NJ
NJ
.,».wIth Yf\J = «vNJJ.. ,»
-vl'
for j.j' = 1, ...• q.
defined by
(2.2.12)
27
Thus. if we let
1
T;(:)
= {:'~ :}-2
~
TN(:). for
t
f
(2.2.13)
.
then. in accordance with the UI-principle of Roy (1953). as in Sen
(1982). we may consider the following UI-LMP rank statistic
QN = sup{T*
N(:) : :
t
(2.2.14)
f}.
As has been mentioned earlier. we consider the case of f of the
form in (2.2.3). so that for the computation of Q in (2.2.14). we need
N
=~ :
to maximize :'~ subject to ~
~ ~
and
:'~:
= 1.
If we let
Kuhn-Tucker-Lagrange (K.T.L.) point formula theorem. we arrive at the
following solution.
Let
(2.2.15)
= {1 ..... a O}
and also let J be any subset of P
a
complement.
For each of the 2
O
set
J.
and J' be its
we partition (following
reorganization. if necessary) ~ and ~ as
U
-
:N -
[~(J)
k(J)
~(J')
and
]
k(J' )
A
-
;:N-
[~(JJ)
~(JJ')
]
(2.2.16)
~(J' J) ~(J' J')
Also. for each J : ~ ~ J ~ P. we let
~(J:J')
= ~(J)
-1
- ~(JJ')~(J'J') ~(J') .
-1
~(JJ:J') = ~(JJ) - ~(JJ') ~(J'J') ~(J'J)'
(2.2.17)
(2.2.18)
Then [viz .. Sen (1982)]. we have
2
QN
,-1
-1, -1
-1
,-1
= ~{~ - ~ ~ ~ ~ ~ } ~ +~;9' {~N(J:J')~ (JJ:J')~(J:J')}
1(~(J:J')
-1
> ~)1(~(J'J')~(J')
~ 0).
where 1(B) stands for the indicator function of the set B.
(2.2.19)
Our main
28
interest is to study the asymptotic optimality of this UI-LMPR test.
2.3
Asymptotic optimality of UI-LMPR tests
As a first step. we shall study the asymptotic power equivalence
of the UI-LMPR test and the UI-LR (likelihood ratio) test for local
alternatives.
Towards this. we define
SN'J
_N
*
*
= X.
1 c Ni J· f J.(X.).
1=
1
~
= (SNl'"
for j
"SNq)' with
= 1, .... q.
(2.3.1)
Note that the summands in (2.3.1) are independent, so that the central
limit theorem (in the multivariate case) can easily be invoked to
conclude that for every
~ ~
f.
I
S;(~) = (~,~~)-2 ~'~ ~ N(O.I)
where
with I(f)
'"
(2.3.2)
[under HO]'
= ct ... ».
JJ
If we let
*
= sup{SN(~)
: ~ ~ f}.
(2.3.3)
then. by virtually repeating the proof of Sen (1982). we obtain that as
N
~
00.
QN - Q~ ~ O. under H as well as a sequence of contiguous
O
al ternatives
I
.
{KN .
Now.
Q~
~
=N
2
~; ~ ~ f}.
(2.3.4)
corresponds to the UI-version of the efficient scores
statistic. and is asymptotically (power-) equivalent to the UI-LR test
statistic (for restricted alternatives). under H as well as
O
{KN}.
As
such. we conclude that the UI-LMPR test based on the specific scores
~(.)
in (2.2.10) is asymptotically power-equivalent (under
{KN}) to the
UI-LR test. and thus the two tests share the same asymptotic optimality
properties.
Note that
5i*
~ ~
(p.d.) as N
P
~
P
00.
matrices M and A, such tha t ~ ~ ~ and ~ ~ ~,
so that there exist p.d.
29
as N ~
00.
As such. defining ~J:J' and ~JJ:J' as in (2.2.18) and
~ ~
(2.2.19). we may conclude that for every
J : ~~ J~
p. such that ~J:J'
f. there is only one
-1
>~
and ~J'J'~J'
~
O.
Then the UI-LMPR
test has asymptotically the best constant power over the ellipsoid in
-1
~'{M
the parameter space specified by
-1
~i:J'~J:J'~J:J'
= constant.
-M
-1
A'A
-1
AM
-1
}~
+
and is also the asymptotically most
stringent test against the restricted domain f.
However. it is
generally not the asymptotically most power test against the entire
domain of alternatives in f; in fact. there may not be any such optimal
test for restricted alternatives.
Towards our main objective of
locating a subspace of f for which the UI-LMPR test has asymptotically
the best power. we consider the following.
Theorem 2.3.1.
O} and f
space.
o
= n
For
~Jg'
f
each J
J
~
(~~
J
~
Pl.
let f
J
+a
= {~= ~ ~ E o. ~J:J'
>
. where E+ao is ao-dimensional positive orthant
For testing (2.2.2) against (2.2.3) (under local alternatives in
(2.3.4». the UI-LMPR test in (2.2.19) is asymptotically most stringent
for f and is asymptotically most powerful for fO' at the respective
significance level a.
Proof.
vector
Consider a r.v.
W haVing a q-variate normal d.f. with mean
M~ and a (known p.d.) dispersion matrix M.
Let Z
= AM- 1W.
and
define
Q*2 =
!.(~-1_~-1~,~-1~-1)!
+
~~g' {~i:J' ~J}:J'~J:J'} 1(~J:J' > 0).
-1
1(~J'J' ~J' ~ 0).
(2.3.5)
where the partitioned matrices and vectors are defined as in (2.2.16)
30
-(2.2.19).
Then. by virtue of (2.3.2)-{2.3.4). it suffices to show that
:
for testing Ha
region Q*2
fa'
f
where
k
~
=~
~
against HI :
~ ~ ~.
the test with the critical
is the most stringent one for f and most powerful for
a
fa
and
are defined in the theorem.
a
show that the shortcoming of Q*2 is equal to
For this. it suffices to
for some ~ inside the
positive orthant f (excluding the null point).
Towards this. we show
that Q*2 is power equivalent to the most stringent test for testing H
a
=~
~
against HI* :
=D
such that B A B'
~ J ~
B
'V
where
~
is a given q x q matrix of real
Since A is p.d .. there exists a upper triangular matrix B
constants.
($
~ ~ ~.
Pl.
(say) is a diagonal matrix.
For each
we take
= BJ = --_I.
so that ~
[a
'V
= ~J
[~JJ
=
~J],
]
~]']' =
~]' J
~J]:J'
[
Then we have
~JJ:J' = ~JJ = ~JJ:J'
We also let Y
!'
=
= BZ
(!j . !j,).
(2.3.6)
and partition Y as in (2.2.I5). so that
Then we may rewrite (2.3.5) as
Q*2 = W'{M-I-M-1A'A-1AM-I)W
(2.3.7)
a :
Then the UI-efficient score test for H
given by (2.3.7).
=~
against H
J
:
~~
>~
is
Thus. for each J{~JsP). the most powerful property
holds for Q*2 whenever
for the region
~
n~JsP
~
lies in f
{f n f J}
n
f
J
. so that Q*2 is most powerful
{which is. by definition.
f o }'
and hence,
31
the desired result follows.
Q.E.D.
Note that in the characterization of the asymptotic optimality of
the UI-LMPR test. the UI-efficient score statistic and (2.3.4) play the
basic role.
We shall see in the next section that this feature remains
true in the multivariate case also.
2.4
Multivariate generalizations
Suppose that
~1'
.... ~ are independent r.v. s with continuous
(p-variate) d.f.·s F ..... F . respectively. where p
1
N
L1
and. as in
(2.2.1). we assume that
F.(x)
= F(X;(M')'
X
~ EP ; ~,
(MI' =
I ~
~ ~,I
~
«~j2CNij2»1~j~p;1~2~q' i=1 ..... N.
where
~i
~
=
«~j2»
(2.4.1)
is a p x q matrix of unknown parameters and
= «cNij2 »· i = 1..... N are p x q matrices of known constants.
The
general multivariate linear model. treated in detail in Chinchilli and
Sen (1981a.b). is a special case of (2.4.1) where fNi =
are q-vectors and F(X;(M') = F(x - (M')'
~
~,I
~
~,I
~i'
the
~i
For the general model in
(2.4.1). the null hypothesis H relates to the homogeneity of the F •
O
i
i.e .•
H : F = ... = F = F
O
1
N
(or
~
= 0).
(2.4.2)
and we consider a restricted alternative of the form
H* : ~ ~
where a
o
r = {~ ~ Epq : ~* = A(vec ~) L O}
is a positive integer
(~
pq) and A is of order a
denotes the pq-vector obtained by stacking the rows of
other.
~
(2.4.3)
0
x pq; vec
under each
~
32
For this generalized orthant alternative problem. the test
procedure is similar to the one considered in the previous sections.
First. we consider a simple null hypothesis H against a simple (local)
O
alternative for which for a fixed
~ ~
r. p
=
o~.
where 0
> O. To obtain
LMPR test for this specific case and to incorporate the same in the
formulation of UI-LMPR tests. we make the following assumptions.
[B1] F is absolutely continuous with continuous density function
~
f(x} = f(x;S}. S = (a 1 ..... a )'
""',....,
"",....,
"",p
(a)
i(l~i~N}. f.(x;o~}
For every
1
r.
all x and ~ ~
'"
r.
which satisfies the conditions:
is absolutely continuous for almost
'"
so that if we let ,....,1
£.S(x;S}
= (a/ae}f.(x;S)
,....,,....,
,....,
1
,....,
and
£i (x;o~) = (a/ao}f.(x;o~).
then for S = o~. £.1,. (x;o~) =(vec ~)' vec
1
,.
f"V
I'V
,....,
- . p
f.S(x;o~}.
,....,1
,....,
"'"
Vx
,....,
~
~(f)
(c)
~}
= &O[(a/a vec
t'V
~ ~
and
E
of Section 2 hold.
,....,
r.
(b)
t'V
Conditions (ii) and (iii) in [A2]
Let
log
f(~;~}(a/a(vec~}'}
(2.4.4)
log f(X;S}]
where &0 denotes the expectation under the null hypothesis.
Then. I(f}
is p.d.
[B2]
For the p-variate pdf
f(x;S}. we denote the conditional pdf of
the jth coordinate. given the others. by f .(x.;slx).
J
J",,,,
and let
g.(x.; Six} = (a/aa.) log f(x;S} = (a/aa.) log f .(x.;slx).
",J J '" ~
",J
'" ~
",J
J J ~ ~
for j = 1..... p.
(2.4.5)
Also. let f[j] denote the marginal pdf of the jth
coordinate. and
*J
f[.](xj;a.}
= (a/aa.) log f[.](x.;a.}. j = 1•...• p.
~J
~J
J
J ~J
~
Note that both the
~j
* are q-vectors.
and :[j]
We denote the
(2.4.6)
33
corresponding pq-vectors by g(x;8) and f * (x;8). respectively. and assume
'"
that there exists a p.d. matrix L* . such that
g(x;O) = L* f * (x;O). for almost all x
~
Ep .
(2.4.7)
For multivariate linear models. (2.4.7) holds for elliptically sYmmetric
d.£. ·s.
For the
[B3]
~i'
[AI] in Section 2 holds (where we use the vec
~i'
if
needed) and assume
sup
sup
l~~~N l~j~p l~i~q
I
c
I
Nij2
= 0
[J
log log N ]
N
(2.4.8)
*
For each j (j=l ..... p) and 2 (2=1 ..... q). f[j]2(x;£)
is
[B4]
differentiable with respect to x on (a .. b.). where
J
-00
~
a j = -sup{x; F[j](x;£) = O}.
~
00
J
b j = inf{x; F[j](x;£) = I}. and
(i) f[.](x;O) ) 0 on its support (a .. b.). (ii)
J
'"
J
let
J
*'
a
f[j]2(x;£) = ax
*
1
f[j]2(x;£).
then assume for some r € (0'2)
r
sup
sup
sup If['·]2(X.,0)I a.s. o(N ). as N -+
l~i~N l~j~p 1~2~q
J
J '"
00
(2.4.9)
Further. let
~ {f*(X.;9)(f*(X.;9»'} = 1* = ((1* ,»
o
",1 '"
'"
",1 '"
'"
mm
(2.4.10)
1(
_m;m ' (_pq
and define the bolck diagonal matrix of 1* by
(2.4.11)
Theorem 2.4.1
For the family of distributions f(x;9). under the
* *-1 .
assumptions [B1] and [B2]. then ,...,
L = ,...,,.....,,.....,
B(l)l
proof.
"'0
"'0
Let :N
9 -- ((9Nji » j=1 ..... p;2=1 ..... q
be the maximum likelihood
34
estimator (M.L.E.) of B
"'*0
"'*0
(parameter of multivariate density function f)
"'*0'.
and!Nj = (B Nj1 .··· .BNjq ) . (J=I ..... p) be the MLE of
the jth coordinatewise density function f[j])'
~j
(parameter of
Under some regularity
conditions. then ~
and ~*O may be solved by the likelihood equations
_N
_N
*
.
"'*0
Yi=1 ~(~i;~) = ~ and Xi=1 : (~i;~) = ~ respectIvely.
~'
~'
iff
=
~
Since L is assumed to be nonsingular. Xi=1 ~(~i;~) =
(!Nl ..... !Np)'
~
Note that!N
~=1 :*(~i'~) =~.
Hence
(4.4.12)
Under the regularity conditions of Wald (1943). ~ is a consistent
_N
estimator. so that the usual expansion of the scores
_N
Xi=1 :
* (~i;!N
"'*0
)
around
~
Xi=1
"'0
~(~i;!N)
and
[under HO] and the use of the weak laws of
large numbers lead to
1
1
2
N (Sn-O)(I+o (1»
~,
p
~
1
= 1- (f)
1
2
N (S:
~,
2
[N- ~-1
1-
g(x.;O)]
~
~1
~
(2.4.13)
1
- 0)(1 +
~
0
p
(1»
= B-
1
~
2
(1*) [N- ~-1
1~
f*(X.;O)]
~
~l
~
(2.4.14)
and these two equations lead us to
LB(1*) = 1(f) = L1*L'
(2.4.15)
This completes the proof of the theorem.
Let Rij be the rank of Xij among the set X1j .X2j ...
i=I •.... N. and j=I ..... p. and let
matrix and
!N* be
columns of
~
~
"~j'
for
be the p x N rank collection
the p x N matrix which is framed by permuting the
in such a way that the top row is in the natural order; it
35
is termed the reduced rank collection matrix.
.
* let
For any glven!N'
S(~) be the set of N! possible rank-collection matrices which can be
reduced to ~ by column permutations.
Then. by virture of the
Chatterjee-Sen (1964) rank permutation principle. we have
P{~
We denote by
~N
=:
*
I S(!N)'
HO} = (N!)
-1
.
v:
€
S(~)
(2.4.16)
the permutational probability measure generated by
*
Also. for every j (=I ..... p). working with the f[.](-:O)
(2.4.16).
"" J
define the scores
~(i).
we
""
i=I .... ,N. as in (2.2.10) (With F being
replaced by F[j])' note that
~j(k)
=
(~jl(k)
~jq(k»I.
.....
Consider then the p x q matrix
T~ji
1
= N-
2
~=1
c Niji
for k=I ..... N and j=I ..... p.
(2.4.17)
~ = «T~ji» of linear rank statistics.
~ji(Rij) = ~=1
C;iji
~ji(Rij)
j=l, .... p; i=I .... . q.
(2.4.18)
* , » . '-1
= N-1 c..
= «c".
~
rmun m.m - ..... pq .
(2.4.19)
For later use. we write
:!:N
= vec
o
:!:N
and
*
c:.~
~
(2.4.20)
for every i =1 ....• N. and let
-1 _N
V
Pol = (N-l)
x:1= 1 lL.
R.'.. = « v N I) )
~,
~1 ~1
mm
m.m '1
= •...• pq
I
We assume that
~
is p.d .. in probability.
(2.4.21)
Then. as in Puri and Sen
(1969). we have
(2.4.22)
where
(2.4.23)
36
Further. under the Assumptions [B1]-[B4].
and
TM
-vi'
where
~
~
{~ }
~
(·;z~.z).
pq
A = vec
~.
(2.4.24)
""" '"
is a sequence of contiguous alternatives. defined as in
{~}
Section 2.3.
Theorem 2.4.2.
For the family of distributions in (2.4.1). under the
assumptions [B1] through [B4]. the test with critical region
(vec
~)'~ ~ ~ ~(a)
is locally most powerful rank (LMPR) for testing HO
in (2.4.2) against H*
'Y
~ = 6 ~. ~ E r and fixed. at the significance
level a.
Proof: Note that
P{~
=
~IH~}
N
=
fR····f .rr f(~i;fNi) ~1 ~
...
1= 1
~=~
'"
~
N
= f .... f
~=~
+
~
f{~i;~)~1 ... ~
i=1
f~~~f
[':1
f(~i;fNi) -
':1
f(~i;£)]~1 ... ~
(2.4.25)
Then. virtually repeating the elegant proof (for the uniparameter case)
in Hajek and Sidak (1967. pp. 71-72). and extending it to the
multiparameter case. we obtain that under [B1] through [B41~ as 6
! o.
N
f .... f
~=~
~
i=1
f(x.;O)dx
",1 '"
'"
N
+6
~.
1
.~
p
N
q
Z Z Z cN .. n~.nf.... fg.n(x .. ;olx) rr f(x.;O)
i=1 j=1 2=1
1J~ J~ ~=~
J~ 1J '" '" i=1
",1 '"
~1 ... ~ + 0(6)
(2.4.26)
Noting that the gradients in (2.4.25) have all null expectations. we
obtain on summing over all
~ E S(~) that as 6 ! o.
37
(2.4.27)
From (2.4.26) and (2.4.27). we have
P{~.
;;.:N
=r I
'"
H*} / [ }; *
'"r
:ES(!N)
N
p
P{~
(2.4.28)
= rIH*}]
'" 2.
N
q
1I
}; f ~.=r
.... f cN"'e
.. ;olx)
IT dF(:<:I';9.)r
]
IJ '"r'J(; g·lI(x
J(; ",IJ
'" '" 1. _=____
1
._-. };_1_J_=_(;
.}; 1 11=1
= _1 1+0 _1=
_ _....r!....c.
0(0)
{
N!
N
f . ... ~i IT dF (x. ; 0)
~=!N i=l
",1 '"
where (N!)
-1
represents the probability ratio under H '
O
Hence. using
the Neyman-Pearson Lemma on this conditional probability set-up. we
.
obtain that a LMPR test statistic for testIng
H vs. H'*"r
o
is given by
the coefficient in the right hand side of (2.4.28); using (2.4.7).
(2.4.16) and (2.4.17). we may further simplify this as
*
N
N
(vec '"r)' L }; f~.=~f [~ (:<:i;?') x c M.] IT dF(x.;O)
'" i=l ;;.:N __ . _ . _ . ~,1
i=l
",1 '"
(2.4.29)
N!
f .... f
N
IT
~=~* i=l
dF (x. ;0)
",1 '"
where for every k(=l ..... p) and i(=l ..... N). f * (x.;O) x CM1' =
",,1 ""
~,
'V
«f*[1](X. 1 ;0) x c M. 1 )' ..... (f*[ ](x. ;0) x CM' )');
'"
1 '"
~,1
'" P
Ip '"
~,IP
*
*
*
![k](Xik;~) x ~ik = (f[k]1(xik;~)cNik1· .... f[k]q(xik;~)cNikq)
(2.4.30)
Now for every k (=1.2 ..... p). let U . i=l ..... N be i.i.d. r.v.·s
ik
having the uniform (0.1) d.f .. and define
-1
F[k](t) = inf{x : F[k](x;~) ~ t}.
Also let
for t
€
(0,1).
(2.4.31)
38
1
~(t) = N-
1 _N
Yi=1 I(U ik ~ t)
2
-1
~(t) = N [GNk(t) - t], t € (0,1),
and
(2.4.32)
where the quantile function ~(.) is defined as in (2.4.31).
(N+l)
-1
Rik < t
-1
-1
F[k](~(t»
~
(N+l)
-1
Then for
(R ik+l), a heuristic Taylor series expansion of
yields that
f[k]e(xik;~)
=
1
f[k]e(F[~](t);~)
+ [N-
2
~(t)] [f[~]e(F[~](f);~)]
(2.4.33)
1
for some f such that It-fl
(25 log2N)IN.
C
~ N-21~(t)l.
Let c N = (25 log log N)IN =
1
Then, by [B4], we have for some r € (0'2)'
~~~1-c If[~]e(F[~](f);~)/f[k](F[~](f);~)1 a~s·o(Nr).
N-!o-
(2.4.33a)
N
Further, by Theorem 4.5.5 of Csorgo and
R~vesz
(1981), the first factor
of the second term on the right hand side of (2.4.33) satisfies
1 1 1
lim
sup
N-21~(t)I/{t(1-t)}2/ (N- 1 log2N)2 ~ 4 a.s.. (2.4.33b)
N-llX) ~~t~1-~
Thus, combining (2.4.33) through (2.4.33b) and letting
*
*-1
f[k]e(Xik;~) - f[k]e(F[k](Rik/(N+l»;~),
e:
1
~
e
~
q, as N
~ 00
Wke(R ik ) =
we obtain that for every
,
1
sup
/ (N+l)~I-cN
~~Rik
max
IWke(R
l~k~p
Next we claim that for every k(1
ik
~
)I a~s. O«(log2N)INI-r}2}.
k
~
p) and
e
r-1
N2
(1
~
e
~
(2.4.34)
q), as N ~
00,
1
(N+l)
-1 sup
IWke(R ik ) I = o«N
Rikc (cN,I-cN)
We verify (2.4.35) only for Rik
the upper tail.
Let
°< t
~
cN.
~
log2 »
a.s.
(2.4.35)
(N+1)cN, as a similar proof holds for
For N sufficiently large, if
e-
39
U~ik) ~ t. then
IWk~(t)1 = If[k]~(Xik;~)
~
-
f[k]~(F[~](t);~) I
log2N
r
o(N ) t
~ o(Nr)~ = o[ N1- r ]
(2.4.35a)
a.s.
ik l t. then
If U
IWk~(t) I
~
r
o(N )U ik
= o(N
r
-1
)GNk(t)
~ o(Nr )( I~("II) = 0(log2N
This proves (2.4.35).
sup
sup
1~i~N Hk~p
'N I
+ "II)
1-r
/ N )
~ o(Nr )
H
!O;N
';'N(1-"II)
a.s.
+"11]
(2.4.35b)
By (2.4.34) and (2.4.35). we obtain that
rR ];~] I a.s.
l~~~q If[k]e(Xik;~) - f[k]e(F[~]LN:~
0
N
[IOg2 ]
N 1 r
(2.4.36)
Using (2.4.8) and (2.4.36). we may reduce (2.4.29) (a.s.) to
(vec
~)'~
. and this completes the proof of Theorem 2.4.2.
We may remark that for the coordinatewise independence case. we do not
need (2.4.7) as the Hajek-Sidak (1967) arguments directly apply to
(2.4.27)-(2.4.29).
~)'~
In general. the LMPR test statistic based on (vec
depends not only on
~
but also on the unknown matrix L.
For
testing (2.4.2) against (2.4.3). according to Roy's (1953) UI-principle.
the overall test statistic is given by
40
QN
= sup{~'~ EN /
~'~ ~'~
. ~ = vee 2.' 2.
c f}
(2.4.37)
for this non-linear programming problem. the K.T.L. point formula
theorem can be used again.
the point
* * *
(~ '~1.t2)
For the lagrange function
is a K.T.L. point if it satisfies the system
(a)
(b)
(c)
~*'~ X'~*
(d)
*'
*
~1 ~ ~
-
1
=0
(2.4.38)
=0
=0
From (2.4.38)-(e). we have
2
t; ~* = (~ ~ ~,)-1 ~ EN
+
(~ ~ ~,)-1 ~'~7
(2.4.39)
After some manipulations. consequently we arrive
(2.4.40)
Define
(2.4.41 )
and
(2.4.42)
where
~(~)
is defined as in (2.4.11). then by Theorem 2.4.1 and
(2.4.23) we have
~
p
-
~~~
as
N
~
(X)
(2.4.43)
41
Further let
(2.4.44)
Then from (2.4.40) and (2.4.43) we have
*'
*
~1 (~ + ~ ~1)
If we denote
=0
in probability as N ~
(2.4.45)
00
J be any subset of AO = {1.2, .... a O} and J' be its
complement. thus (2.4.45) implies
*
~l(J) =
o.
*
-1
~l(J') = -~(J'J') ~(J') ~ 0
(2.4.46)
and
(2.4.47)
~(J:J') > ~
a
Moreover. Kudo (1963) showed that the collection of all 2
O
sets
-1
; ~(J:J') > ~. ~(J'J') ~(J') ~ ~}
aO
~(J) = {~ € E
(2.4.48)
a
is a disjoint and exhaustive partition of E 0
*'
,-1
Therefore we have
,-1
!
~ ~ = {!N:N !N - ~(J') ~(J'J') ~(J,)}2
(2.4.49)
where J is the set such that conditions (2.4.46) and (2.4.47) hold.
Therefore the UI-LMPR statistic in (2.4.37) is given by
~ = !fi[~1 - ~-l(:N)~,~l~ ~-1(:N)]!N + ~~CA {~(J:J')
- - 0
-1
-1
~(JJ:J')~(J:J')} l{~(J:J') > ~} l{~(J'J') ~(J') ~~}
(2.4.50)
in probability. where the partitioned matrices and vectors are defined
as in (2.2.16)-(2.2.19).
Theorem 2.4.3.
For testing (2.4.2) against (2.4.3) (under the local
alternative {KN}). [B1]-[B4] hold. then the UI-LMPR test statistic ~
and the corresponding UI-LR test statistic are asymptotically
power-equivalent.
42
Proof.
Define
o (( N
}; c *NIJi
.. g. (X .. :oIX) ]] .
~ = i=1
Ji IJ ~ ~
J=I, ... ,p:i=I, ... ,q
(2.4.51)
*0
~ =
(2.4.52)
};
[[ N
i=1
c * ..
NIJi
f* .
[J]i
(X .. : 0) ]] .
IJ ~
J=I, ... ,p:i=I, ...
,q
and write
~
= vec
*
o and
~
~
= vec
*0
~
then it is easy to show that
(2.4.53)
where
~
and
=
((.1':m c~mml ]]m:m'=l. ... ,pq
A_I
*
-1 A*
Also, let ~ = ~ ~ ~ (~ = ~ ~ (~)~)
-1
~
A* A*
-1
(~) ~ ~
A*
(~)~').
(2.4.54)
A
A1
A*
and ~ = ~ ~~. (~ = ~
Then the Ur-score statistic for testing (2.4.2)
vs. (2.4.3) turns out to be of form
~O ~ ~[~;/ !:;l~.~lSlh ~S4o {~(n') ~lJJ:J') ~(J:J')}
+
-
1{~(J:J')
> ~}
A_I
1{~(J'J') ~(J') ~ ~}
(2.4.55)
~
O}
(2.4.56)
Note that
A
~
=
A*
~ ~ ~',
(as
~
=
*
~), and
the implications of contiguity, we have
hence, by Theorem 4.1 and
e-
43
P
~ - ~(~) ~-1~(~)
-+
0
as
N
-+
00
under HO as well as under
{~}
(2.4.57)
Thus. by (2.4.57) and (2.4.12)-(2.4.14).
~o and 0;2 are asymptotically
{~}.
equivalent in distribution under H as well as
O
Furthermore. under
some regularity conditions. the UI-score statistic 0;0 and the
corresponding UI-LR statistic are asymptotically equivalent in
distribution under
{~}.
For this we may refer to Chapter 4 of Puri and
o
Sen (1985) where the asymptotic quadratic mean equivalence of T
and
Niki
_N
*
*
ri=1
~iki f[k]i(Xik;~) (for k=1 •.... p; i=1 ... .. q) has been established.
Thus. the test statistic 0; and the corresponding UI-LR statistic both
have the same asymptotic power function under
{~}.
This completes the
proof of the theorem.
Let ry = A B-1 (~)~ and A =
Theorem 2.4.4.
rv
J ($ s J
and f
O
=
rv
rv
rv
rv
'VI'V
lim&(~
IH
N~
For each
)'
O
+a
SAO)' define f
J
= {ry
€ E
O
~J:J' = ~ - ~J'
-1
~J'J'
~. ~ O}
'"
n
f . Then for testing (2.4.2) vs. (2.4.3) (under
J
~JSAo
the UI-LMPR test statistic with critical region
WN(~)
=
{~;
0;
{~}).
~ ~(a)}
is asymptotically most stringent for f and is asymptotically most
powerful for f
Proof.
O
at the respective level of
signifi~ce
a.
It is easily followed via Theorem 2.3.1 and Theorem 2.4.2. and
hence is omitted.
2.5.
General remarks
We consider here some specific problems and append some useful
discussions.
(I)
Univariate k-sample location and scale alternative problems.
X • j=1 •.... n. i=1 ..... k
ij
Let
be the independent r.v.·s with continuous
44
Fi(x). x c E. where
d.f.·s
F.j(x)
= Fi(x) =
1
F«x-~.)/o.).
1
1
(2.5.1)
J=1.2 ..... n; i=l ..... k.
In the conventional case. we take the scale parameters 01' .... ok to be
all equal (and equal to 1. without any loss of generality). so that
(2.5.1) corresponds to a special case of (2.2.1) where
~1
= ... =
~n
=
(1.0 ..... 0)'. ~n+1 - ... = ~2n = (0.1.0 ..... 0)' ..... ~[(k-1)n+1] =
... =
= (0 ..... 0.1)'. and N = kn.
~
Here. we are interested in
testing for
(l).
Ho .
~1
=
... =
~k
(2.5.2)
(i.e .. the homogeneity of the F i ).
against the simple ordered alternative
0
~
€
r =
{~
€
Ek ; A ~ l O. where A =
0
-1
:]
(2.5.3)
Let R.. be the rank X.. among the N observations of the combined
IJ
IJ
sample. for J=1.2 ..... n. i=l ..... k.
order statistics by
~(1)
We also denote the corresponding
< ... < ~(N)'
respectively.
Thus. as in
(2.2.7). we have the scores
~(r) = ~{-f(~(r»/f(~(r»}' for r=l ..... N.
(2.5.4)
and the corresponding linear rank statistics are given by
(2.5.5)
and let
(2.5.6)
Then the dispersion matrix of
~ = a~(k!k
-
!N
(under H ) is given by
~'). ~ = (1 ..... 1)'
O
.
(2.5.7)
e"
45
1
2
-1
Also. note that under ~. ~ = N :. ~ = ~ and UN = ~~, so that
~ = ~ ~ 1 ~' = oN-2 ~* where
*IJ
o..
Partitioning
~
*
~* = «oij»
with
if
i=j
if
li-jl=l
if
li-jl~2
(2.5.8)
and A* as in (2.2.16)-(2.2.18). it is easy to verify that
here
~J:J' ~~. V J : <P S J ~ P
= {I, .... k-I}.
if ~ ~ ~.
(2.5.9)
and hence, by Theorem 2.3.1 we may conclude that for testing (2.5.2)
against (2.5.3) (under local contiguous alternatives). the UI-LMPR test
based on the score function in (2.5.4) is asymptotically uniformly most
powerful for all alternatives in the order-restricted parameter space.
Consider the next scale problem (with homogeneous locations).
where
-1
F.(x)
= F(o.1 x).
1
for
i=l ....• k. x c E.
(2.5.10)
Here the null hypothesis relates to the homogeneity of the o. while we
1
have an ordered alternative as in (2.5.4) with the vector
~
replaced by
For this k-sample scale ordered alternative problem. using the scores
(2.5.11)
it can be shown that the corresponding UI-LMPR test is asymptotically
UMP for the entire (restricted) parameter space under the ordered
al ternative.
(II)
Bivariate two-sample ordered location problem.
Chatterjee and De
(1972. 1974) developed an UI-rank statistic for this problem and showed
that their test statistic is better than the unrestricted one.
A
46
generalization of this result to the general multivariate case is still
an open problem.
However, instead of the maximization of the
Bahadur-efficiency [as has been considered in Chatterjee and De 1974)],
we may maximize the Pitman-efficiency to determine the UI-LMPR test for
the general restricted alternative problem considered by Chatterjee and
De (1972, 1974), our UI-LMPR test is asymptotically uniformly most
powerful for the restricted parameter space under the alternative
hypotheses.
(III) Other models.
The asymptotic UMP property of the UI-LMPR test
holds for other problems as well:
(i)
a positive orthant alternative
when the limiting dispersion matrix is diagonal, (ii) an ordered
alternative problem when the limiting covariance matrix is of form
(a~ ~
+
a~ ~'),
and this is typically the case with randomized block
designs and with multi-sample problems,
(iii)
profile analysis and (iv)
in alternatives which put constraints
on the parameters in the form of lower dimensional hyperspaces.
Chinchilli and Sen (1981 b) have considered the power superiority of the
UI-rank test in the general multivariate problem under certain
restraints on the covariance matrix.
Under the same condition on the
covariance matrix, by our Theorem 4.4, we are able to ,obtain the
asymptotic UMP character of the UI-LMPR test for a wider class of
alternatives.
In passing, we may remark that the UI-LMPR test is
invariant under any permutation of the p-coordinates, while the
corresponding rank test based on the step-down procedure is not so, and
the latter may not be asymptotically UMP.
(IV)
A nonparametric version of Schaafsma and Smid (1966) procedure.
For simplicity, we consider only the case of the positive orthant
e-
47
alternative where A = ",pq
I
and
r
+pq
= E
.
Under assumptions [B1] and [B2]
in Section 2.4, the use of the Neyman-Pearson Lemma (on the distribution
of the ranks) leads us to the LMPR test of size
a
which rejects H for
O
(2.5.12)
where
~
and
~
are defined as in (2.4.19) and (2.4.22) respectively.
Assume i is the half-line of points 0 A, 0
>0
and for any two
such vectors i and m, let
(2.5.16)
where
<.,.>~
denotes the inner product with respect to
the Euclidean norm with respect
to~.
~
and
II.II~
is
Following Schaafsma and Smid
(1966), the test in (2.5.12) is asymptotically most stringent somewhere
most powerful if
r
satisfies
= sup
~(iO,m)
!O E
~o
mEr
'"
= inf
sup
iEr
mEr
'"
~(i,m)
(2.5.14)
where the desired half-line !O can be determined by applying the method
of Abelson and Tukey (1963).
o
~(!'~1)
Namely
0
o
= ~(!,~) = ... = ~(i,e
",,,,pq )
(2.5.15)
where the edges e~ (j=1, ... ,pq) is defined by
",J
o .
e . . A.>O
",J
J
and
Ak=O
i f k #j
v
j ,k=!' ... ,pq
(2.5.16)
Since the maximum shortcoming of this test is an increasing function of
~(!O'~)'
hence it is unsatisfactory for large values of
~O.
aIAPTER III
ASYMPTOTIC DISTRIBlITION OF UI-LMPR TFSfS FOR RFSfRICfED ALTERNATIVES
3.1
Introduction
Since the classical LR test for H : p = 0 vs. HI :
O
~ ~ ~,
has an
exponential rate of convergence (to 0) for the error probability (under
nonlocal alternatives) which does not improve for a restricted
alternative, Brown (1971) advocated the use of the classical LR test for
testing H vs. H* (defined in (2.4.3».
O
However, some Monte Carlo
studies [viz., Barlow et al. (1972)] exhibit the power superiority of
the restricted LR tests to their unrestricted counterparts.
In the
nonparametric setup under consideration, we have incorporated the
notation of LMPR tests along with the UI-principle in the construction
of some UI-LMPR tests which are asymptotically power-equivalent to the
corresponding LR tests under restricted (contiguous) alternatives.
Like
in the parametric case, the asymptotic distribution theory of such
restricted alternative tests becomes more involved.
In fact, the simple
chi square (central under H and non-central under the global
O
alternative) distributional approximations are generally not tenable for
such restricted cases.
Our main contention is to study the asymptotic
distribution theory of the UI-LMPR test statistics and to incorporate
the same in the study of the asymptotic power properties of these tests
and their classical versions.
49
The asymptotic null and non-null (under contiguous alternatives)
distributions are then derived in Section 3.2.
The rates of convergence
of the actual distributions to their asymptotic forms are investigated
in Section 3.3.
Some asymptotic relative efficiency and asymptotic
optimality results are presented in the concluding section.
3.2
UI-LMPR test statistics:
asymptotic distribution theory
The task of studying the exact distribution theory (even under H )
a
becomes prohibitively laborious as N increases.
For this reason. we
take recourse to the asymptotic distribution theory.
Let
Q*2 = W'(2-1-B-1(2)A'A-1AB-1(2»W +
""I"'Vt"V
~
f"V
I"V
(3.2.1)
where W is the random vector haVing a pq-variate normal d.f. with mean
vector 2A and covariance matrix 2 (i.e .. W ~
I"V
~ (·;2A.2». Z = AB- 1 (2)W.
J>Cl
~'"
t"V
tf'VJ;'V
t"V1"V
and the partitioned vectors and matrices are defined as in
(2.2.16)-(2.2.18).
Theorem 3.2.1.
Under the Assumptions [Bl]-[B4].
(3.2.2)
where
~-aa+k(J)
represents a Chi-square random variable with
pq-aa+k(J) degrees of freedom. and ek(J)
~
a} with
2*
= 2*
k(J)
PH {~.J'
a
.
> ~.
-1
~'J'~'
denoting the sum over all set J(~JSAa) such that the
k(J)
cardinality of J is k(J).
Proof.
Since the first term and the second term of r.h.s. of (3.2.1)
are independent. so
50
(3.2.3)
where
*
Q (J)
. -1
= ~:J'~JJ:J'~:J'
*
and a (J)
-1
= 1{~:J' > ~'~'J'~
~ ~} (3.2.4)
Note that the matrix ~(~-1_B-l(~)A'A-1AB-l(~}) is idempotent. and thus
rank[~(~-1_B-1(~)A'A-1AB-1(~»]= tr[~(~-1_B-1(~)A'A-1AB-1(~»]
,...,
",,"'.-v
f"V"V
""
,....,
,...,,...,,...,
#"'I"III"V
,....,
(3.2.5)
Hence we have
gHO{elt!.(:-I_~-I(:)~.~-I~-I(:»!.}= (1-21t)
(3.2.6)
If we wri te
R(J)
=
{~
-1
aO
t
E
; ~J:J' > ~. ~'J'~J' ~ O}
(3.2.7)
a
then the collection of all 2 O set R(J) is a disjoint and exhaustive
a
partitioning of E O.
it
gH {e
O
~
Thus we get
Q*(J)a*(J)
<t9S\O
}
=
=
f~;,J e
~
~JSAo
~
it
Q*(J)a*(J)
<t9S\o
I··· JeitQ*(J)d~
R(J)
(Z;O.A)
ao ~ ~ ~
(3.2.8)
Kudo (1963) showed that. for each J(~JSAo)' Q*(J) and R(J) are
independent under HO'
Thus (3.2.8) can be rewritten as
¢C~cA [f~klJ)f eltQ*(J)d~k(J)(~'J';~'~J'J')][f~(~)f d~ao(:;~'~)]
- - 0
e"
51
k(J)
2
=
(3.2.9)
so
aO
.
(3.2.10)
2
e k (J)(1-21t)
o
k(J)=O
Therefore. using the Fourier inversion formula on (3.2.10). we obtain
BH e
{
itQ*2}
=
that
a
~~ P{Q~ ~
xIHO} =
O
k~J)
ek(J)
P{~-ao+k(J) ~
(3.2.11)
x}.
for every real x. and this completes the proof of (3.2.2).
Next. we consider the non-null case and restrict ourselves to a
sequence of restricted (contiguous) alternatives [as in (2.3.4)].
Note
that under {KN} in (2.3.4). for any J(~JSAo)' Q*(J) and R(J) are no
longer stochastically independent. and this introduces complications in
the form of the asymptotic distributions under consideration.
We have
the following.
Under {KN} and the Assumptions [B1] through [B4].
Theorem 3.2.2.
00
2 ~ x IKN} = 2 d* P{X___
2
lim P{QN
h ~ x}
N-lOO
h=O h
1Xl a O+
(3.2.12)
where
if h
= O.
(3.2.13)
1
e
e
--f
2
-~f
2
j+k(J)=h
{ e*
d(k(J).j;~.A)
(~)(f/2)
aO ~
h/2 [(h/ 2)!] -1 +
if h is odd. k(J)=l ..... a O.
2
k(J)+j=h
.
d(k(J).J;~.A)
}
~ ~
if h is even. k(J)=l ..... a O.
with f = A'[2-2B- 1 (2)A'A- 1AB- 1 (2)2]A. ~ = AB-1(2)~A.
(3.2.14)
52
(3.2.15)
and
d(k(J).j;~.A)
m
~
=
(f/2) (m!)
-1
C(k(J).e;~.A)
(3.2.16)
e.m~O
e+m=j
where
e
--1
C(l.e;~.A) = f((e+1)/2)22
C(k(J).e:~.A)
(e!~)-l
(3.2.17)
=
(3.2.18)
for all k(J) = 2 ..... a O'
*
~(J)
denotes the sum over all subsets
J(~JSAo) such that the cardinality of J is k(J). and for each
J(~JSAo)' the vector ~ is defined as
sin B
J1
cos B sinB
J1
~
J2
•
= •
-~2- ~ B . ~ ~2-' V j=1 ....• k(J)-2
JJ
~os BJ1 ••••• cosBJk(J)_lsinBJk(J)_l -~ ~ B (J)-l ~ ~
Jk
cos BJ1 ·····cos BJk (J)-l
Proof.
(3.2.19)
Note that the characteristic function of Q*2 under {KN} is
53
g~{ e
itQ*2}
=
(3.2.20)
where
(3.2.21)
and
d4>
(Z;O,A)
a O '" '" '"
(3.2.22)
A
Let QJ(t) be the integral of the r.h.s. of (3.2.22).
J
~
(2~)
~J:J'~:J'>O
-k(J)
2
e
1
*
--(1-2it)Q (J)
2
I
1
-2
Then
i
(~:J'~J:J'~:J') d~:J'
(3.2.23)
Therefore we have the following:
A
(a)
If k(J)=O. then QJ(t)=1.
QPl
=
;
i=O
(b)
(t!l-lrr:lJ2~-1
If k(J)=1. then
x
54
(3.2.24)
And (c) if k(J)=2 ..... a O' then make a transformation to the polar
coordinates (p J .9 J1 .··· .9 Jk (J)-1)'
Let
(3.2.25)
~J:J' = PJ~; PJ ~ O.
where -~ ~ 9Ji ~ ~ and -~ ~ 9 (J)-1 ~ ~ V i=1.2 ..... k(J)-2. Then the
Jk
k(J)-1 k(J)-2
k(J)-i-1
Jacobian of this polar transformation is P
.rr
cos
9 .
Ji
J
1=1
We also let R* (J) be a set of restrictions on P and 9 's of the form
J
J
1
2
~JJ:J' ~J ~ ~.
Then. for every k(J) : 2
1
A
Qlt)
=};
(n l.)-1
c;
~
k(J)
~
and PJ ~ 0
(3.2.26)
a O'
-1
I
e-2~J:JI~JJ:J'~J:J'
e'
2=0
k(J)-2
rr
i=1
where
d~
cos
k(J)-i-1
9 . d9 dp
J 1 ~J J
denotes d9 J1 d9 J2 ... d9 (J)-1'
Jk
(3.2.27)
After some simplications. we
have
00
QJ(t) = }; C(k(J).2;~.A)(1-2it)-(2+k(J»/2.
2=0
Using (3.21) through (3.29). we conclude that
~{eitQ*2} = [e-~ m~ 2-mCml)-lfmCl-2it)
aO
-pq-a +2m
2°
]
(2+k(J»
00
[e* (~) +};
};
aO ~
k(J)=1 2=0
C(k(J).2;~.A)(1-2it)
2
]
(3.2.28)
55
(k(J)+j)
ao
CIO
+!
!
k(J)=1 j=O
d(k(J).j;~.A)(1-2it)
2
J
(3.2.29)
Thus. we obtain that
2
lim P{QN
N-llXl
CIO
~ xIK__ } = !
h=O
-""N
~ x}
d*h P{X:__
h
lXJ. a O+
(3.2.30)
This completes the proof of Theorem 3.2.2.
Note that in the unrestricted alternative case. we have a central
2
2
X distribution under the null hypothesis and non-central X under the
(local) alternative.
However. the picture is different here.
It is a
mixture of central chi square d.f.·s under the null hypothesis. but the
non-null case may not be reducible to the same mixture of non-central
chi square d.f. ·s.
3.3
Let
Rates of convergence for distribution of UI-LMPR test statistics
~
be the distribution function
~
defined in (2.4.19).
We suppose
pq
~ is a measure on the Borel a-field on E . and denote B(w.t) the open
ball with center wand radius t. i.e .• t
pq
B(w.t) = {y € E ;lIw-yll
where II-II denotes the Euclidean norm.
> O.
pq
Vw€ E .
< t}
(3.3.1)
For any J(~JSAo) and any
real-valued Borel measurable function gJ on R(J) defined in (3.2.7). we
define, V t
> 0, the oscillation function S
g
J(B(~,:))
as
56
S J(.;C) = S J(B(~,c))
g
g
=
sup
{Ig(z)-g(y)
z€R(J)
-
;z,y € B(w,c), y € R(J)}
-
-
(3.3.2)
and also define S J as the supremum of the average modulus of
g
oscillation of gJ with respect to a finite measure A over all translate
of gJ by
(3.3.3)
where
(3.3.4)
For simplicity, we assume that
[C1]
N
*
N
Lemma 3.3.1.
pq
E
*2
}; cN".f! = 0 and }; cN'.f! = 1, V j=l, ...
i=l
IJ
i=l
IJ
,p; f!=1, ...
,g.
(3.3.5)
Let h be a bounded Borel measurable function defined on
, then under the assumptions [B1]-[B4] and [C1], there exist
*
constants d 1 ,d and d (not depending on N. cNijf!,h)
such that
2
3
N
I
f
h(w)d(~(w)~ (w;O'~'))1 ~ d 1 suppq Ih(w) I };
Epq
pq - - ~
wcE
i=l
+ d2 ~ [d3
Proof.
3 0
p
q I *
};
};
};
cNijf! 1 + N0 '~pq(.;~'~) ]
i=l j=l f!=1
N
(3.3.6)
It follows along the lines of the Theorem 1.3.5, and hence is
omitted.
Theorem 3.3.2.
Let
~i'
1
~
i
~
N, be N independent p-dimensional
stochastic vectors having continuous c.d.f. F(x;C
CM. defined
M.), where --.l,l
_ --.l,l
in (2.4.1).
Consider the UI-LMPR test statistic Q~ given in (2.4.50),
then under the assumption [B1]-[B4] and [C1], there exists a constant d
4
57
such that
(3.3.7)
Proof.
For every J(~JSAo)' defining Q*(J) as in (3.5). we let
Q(J) = W·(~-1 - B-1(~)A'AAB-1(~»W + Q*(J).
t1"V
Then. we have for every x
=
~
~JSAo
IIEpq 1{Q(J)
~
~
t1"V
t1"V
(3.2.8)
""""""'"
o.
x} a*(J)d[PH {Tl'J
0 "",
N
~
p
~
i=1 j=1
~
w} - 4> (W;O.2..._)]
'"
pq '" '" ~
I
3 +0 0
*
I
c Nije
N +
e=1
q
1
~
(3.3.9)
where
J
g (~) = 1{Q(J) ~ x} 1{~:J'
-1
> ~'~'J'
By noting that (under HO as well as {KN})
~. ~ ~}.
(3.3.10)
58
1{~:J'
> O.
-1
~'J' ~, ~ O} = 1{~J:J'
> O}
-1
1{~J'J' ~. ~ O}.
(3.3.11)
we have
(3.3.12)
and
'"
S J(c.~) = ~
sup
f pq S J(w+y;c)d~(w;O.~)
<ffJSAo g
<ffJSAo ~cR(J) E
g '" '"
'" '"
~
(ii)
~
=
<ffJSAo
f sup
f
~cR(J)
pq
E
sup
~cR(J)
[lgJcW+Y) - gJ (v+y)
'" '"
'" '"
I.
"~-:::" < ~. ::: ~ R(J)]d4>(~;~.~)}
=
~
~O~.J (8A/w»
sup
<ffJSAo ~cR(J)
",';:N
c
(3.3.13)
+ y)
'"
'"
where
pq
AJ(~) = {~ c E ; Q(J) ~ x. ~:J'
and
of
(8AJ(~»
AJ(~)
pq
of E
c
> ~.
-1
~'J'~J' ~ O}
(3.3.14)
is the set of all points whose distances from the boundary
less than c.
Let
~O
be the class of all convex Borel subsets
• then for every J(<ffJSAo)' AJ(~) c ~O.
Hence. using Corollary
3.2 of Bhattacharya and Rao (1976) and some manipulations. we get
~
<ffJSAo
N
S J(c,~)
g
p
q
Taking c = d3 ~
~
~
i=1 j=1 i=1
!
~
~ 1~1222
pq+l
f(--2--)
(f(~»-1
c .
I c*NIJ· ·il 3+0N0 and combining (3.3.9).
and (3.3.15). we obtain that (3.3.7) holds.
Q.E.D.
(3.3.15)
(3.3.12)
59
Next we investigate the rates of convergence of distribution
Q~ to its limiting mixture central X2 distribution under a
function of
sequence of restricted contiguous alternative
{KN}·
Under
distribution function depends on unknown parameter fNi =
«8
Nije
».
{KN}. the
«~je
c*
Nije » =
so we make some extra conditions on the distribution function
and the unknown parameters 8
V i=l ..... N. j=l ..... p and e=l ..... q.
Nije
[C2] For every j=l •.... p and e=l ..... q
~1= 1
8NIJ~
· ·n = 0
and
~1=1
2
8NIJ~
. ·n = 1.
(3.3.16)
[C3] For the jth marginal density function f[.](x .. ;8.). we assume that
J
there exist constants dS and d 6 such that for
IJ ",J
max
max
max
18Nijel
l~j~p l~e~q l~i~N
(3.3.17)
Lemma 3.3.3.
Let h be a bounded Borel measurable function defined on
pq
E . then under the assumptions [B1]-[B4] and
constants
If
Epq
~.
d
S
and d
9
[C2]-[C3]. there exist
such that
h(w)d(PK__{Tn ~ w} - ~ (W;L.A.L.»I
'"
-""N -vi'
'"
pq '" ;::N", ;::N
(3.3.1S)
Proof.
It follows from Lemma 3.3.1 and (7) in remarks of Hu~kova
(1980).
Theorem 3.3.4.
Let X.• 1
",1
~
i
~
N. be N independent p-dimensional
60
stochastic vectors having continuous c.d.f. F(X:CM')'
'" ,...1,1
((~Ji
c*
Niji
»·
CM' =
",HI
Then under the assumptions [Bl]-[B4] and [C2]-[C3].
there exists constant d
10
(not depending on N) such that for every real
x.
sup
XE.E
N
p
q
~ d 10 .2
2
2
1=1 j=1 i=1
I c*Niji 13+0N0 (1+ I~ji 13+0 ).
The proof follows directly from Theorem 3.3.2 and Lemma 3.3.3. and hence
is omitted.
3.4
2
Asymptotic power comparison of Q and
N
EN2
While testing against a restricted alternative. in the parametric
case. one may compare the classical LR test with its restricted
alternative version. so as to gather information on the gain in the
sensitivity of the test.
In the nonparametric case. we intend to
compare the asymptotic power functions of
for H vs. HI) with the same objective.
O
Q; and ~ (the UI-LMPR test
The power superiority of the
restricted tests over the unrestricted ones in some specific
nonparametric problems has been studied by Chatterjee and De (1974) and
Chinchilli and Sen (1981). among others.
randomized block design problem. this
De (1976) has
Q; and
the
s~own
that for a
traditio~l ~
both
have the same approximate Bahadur-slope for any alternative in the
restricted parameter space.
We may observe that this feature is
generally true for the LR test.
Towards this. we assume that X.•
i=I •...• N. are i.i.d.r.v. with the d.f.
to be given.
",1
~ (.:~.20). where 20 is assumed
pq
"''''
Consider the usual LR test and define
-1
0
0
KN(~:~'~) = N {log[L(~:~.: )IL(~:~.: )]}: ~ = (~1"" .~).
(3.4.1 )
61
where
L{X:~.!O) stands for the likelihood function.
By the classical
Law of Large Numbers. we obtain that
(3.4.2)
where
I{~:O)
stands for the Kullback-Leibler Information: it is defined
as
For testing HO:
I{~:O)
~
=~
= ~1{!O)-1~.
vs. K:
~ #~.
it is easy to check that
for every ~ #
'"
'"
o.
(3.4.4)
'"
Thus. the unrestricted LRT achieves the Kullback-Leibler Information in
(3.4.4).
For testing HO:
~
=~
against K* :
~ ~ ~.
where
~
is defined
by (2.2.3). it is easy to show that the restricted LR test also achieves
the same information.
For this restricted alternative problem. the LR
test statistic is given by
(3.4.5)
where the partitioned vectors and matrices are defined as in
(2.2.16)-{2.2.18) and
1
and
1
o 2-2_N
Z = N Me. = N A{Y:_1
~
~
~
1-
X.).
~l
(3.4.6)
Thus
stands for the sums of the multinormal orthant
probabilities corresponding to the
~:J"
for which the k{J) have the
62
common value k{=O •...• a ).
O
that for every
-1
li~~{-N
Therefore. by some routine steps. we obtain
t{~O).
log PH {N
-1 *2
o
QN
~ t}}
= t/2
a.s.
~
o
pq (.;~.~).
.....,.....,
(3.4.7)
Moreover. under the same setup.
(3.4.8)
-1
It is easy to show that (3.4.8) equals to ~.~O
~. so that the exact
Bahadur slopes for both the unrestricted and restricted LR tests are the
same.
Thus. we need a finer asymptotic comparison to discriminate the
two LR tests.
In this context. the concept of approximate
Bahadur-Gochran deficiency. developed by Chandra and Ghosh (1978). may
be adapted to force this distinction.
Let
~(~) and u 1N{u2N ) be the critical value and size of the
.
.
,
unrestrIcted (restrIcted) LR1 sequence
~
is
*
~.
*2
{~
*2
}. ({QN }) when the power at
Then. using the results in Section 2 of Chandra and Ghosh
(1982). we obtain that
~
= ~':O
-1
~+~-1{1_~*)
j ~':O
-1-!
~ N 2+[{~-I{I_~*»2+pq_l]N-l + 0{N- 1 ).
(3.4.9)
(3.4.10)
63
and further that
kNo - ~
= o(N-1 ).
so that we have
pq-aO+k(J)
2
_!~
e 2
pq-aO+k(J)
e~(J)[r(
2
)]
1
-1
(1+[(pq-aO) +
k(J»/2-1](~)-1
1
+ 0(N- ».
(3.4.11)
so that
log a
2N
= log
a
1N
+ log e
o + 0(1).
(3.4.12)
aO
Consequently. by using Theorem 2.3.1 of Chandra and Ghosh (1978) we
conclude that the approximate Bahadur-COchran deficiency of the
unrestricted LR test with respect to the restricted one is
_2(~,(~0)-1~)-1 log eO
a
Q~2. for HO: ~
=
When
o
is not known. the LR tests statistic.
O
~ against K*: ~ ~ O. has the same form as in (3.4.5)
with ~O being replaced by S
'"
~
'"
= (N-1)-1
~_1(X.-~.)(X.-~.)I. In the case.
1-
",1:..::N
",1:..::N
following Perlman (1969). we obtain that under HO' for every x
02
aO
0
2
2
P{QN ~ x}} = ~(J)=O ek(J) P{~-ao+k(J)/XN-pq ~ x}.
~
0,
(3.4.13)
By very similar steps. it follows that in this case. both the
unrestricted and restricted LRT's have the same Bahadur slope. and the
approximate Bahadur-COchran deficiency of the former with respect to the
64
latter is still positive.
We may note that in the above discussion, we have allowed both a
and a
lN
to converge to 0, and for a fixed power p* , we have drawn the
2N
relative performance picture.
This conclusion may not necessarily apply
to the conventional case where the level of significance is held fixed,
and for contiguous alternatives [such as in (2.3.4)], the relative power
pictures are studied.
In situations where the Pitman-efficiency measure
is adoptable, the two sequence of competing statistics have the same
type of distributions [viz., non-central chi square or normal] differing
in some noncentrality parameters, and the relative picture of these
noncentralities convey the efficiency picture.
and
However, in our case,
Q;
~ have different asymptotic distributions and the Pitman-measure is
not adoptable.
Another plausible approach for this local comparison is
to go through a second order local scheme.
i.e., to compare the slopes
of the asymptotic local power functions at the null point.
{~}
note that under
in (2.3.4)
pR2(a;~} = exp(-(A'~}/2} -hY~=o(A,~}h
'V
'"
For this,
'V
I'Uo'V
2-
h
(h!}-lp{~_ 2h ~ x'}, (3.4.14)
llQ+
'V'V
a
2
where x' stands for the upper lOOa% point of the central X d.f. with pq
a
2
Also, by Theorem 3.2, for the UI-LMPR test Q , under
DF.
N
{~}
in
(2.3.4), the asymptotic power is given by
PQ*2(a;~}
where x
co
*
= ~=o
dh
is the critical level for Q*2 .
a
2
P{'1xl-a +h ~ xa },
o
(3.4.15)
.
SInce
both (3.4.14) and
(3.4.15) are smooth functions, by the Taylor expansion, we have for
small
(A'~)
1
R
fJ
)
2(
a;~
R
'"
=a
+
O({(~ ,~~}2}2}.
I\.
"-I\.
(3.4.16)
65
~ *2(a;~) = a
Q
1
1
+ b*(A'2A)2 + O({(A'2A)2}2)
(3.4.17)
'"
where
b*
=
1
~A*' 2 A*
.
*
wIth ~
{f
= c -1
1 ~
a -1
11*' A- 1Zd<I> (Z;O.A)]P{ 2
'" '" '" a O '" '" '"
~-aO
-1
A Z~O
and
*-1
= c 1 ~. for c 1
~
~x}+ ~
r
r(J)]
__2
_
k(J)-1
- k(J)
2(~)
a
If A = I. then (3.4.18)
> O.
reduces to
2
- P{~-ao+k(J) ~ xa }] }
> O.
where
-*
11
aO
= aO
In general. b* is not very simple in form.
this further as
-1
*
i:111i
(3.4.19)
However. we may simplify
66
a
o
L
+
*L
k(J)=1 k(J)
a
*' A-1 Z
~
R(J) ~
l I
O
*L
L
+
I
k(J)=1
~
2
d~
P{x_~
(Z;O.A»
aO ~ ~ ~
x }] +
a
~
~-aO+k(J)+1
*' -1
2
~J,AT'J'
ZJ' d~ (Z;O.A» ] [P{x_~_
k(J) ~ x }
~
~
ao ~ ~ ~
~ a o+
a
R(J) ~
(J)
P(~-aO+k(J)+l ~ xa)l}·
-
~
(3.4.20)
where R(J) defined in (3.2.7).
By noting that
U
-1
A
*' A-1 Z
~
Z~O
~
(Z;O.A) ]
aO ~ ~ ~
d~
2
P{x_~
~-aO
~
x }
a
(3.4.21)
we conclude that the slope of the asymptotic power function at the null
point for the restricted UI-LMPR test is larger than that of the
unrestricted one.
This result gives us the asymptotic local power
superiority of the restricted test
~.
Q;
to that of the unrestricted one
in the restricted alternative space A =
O
{~ ~
pq
E :
~ ~
O}.
We may finally remark that for testing the homogeneity against
alternatives which put constraints on the parameters in the linear form
of lower dimensional hyperspaces i.e .• HO:
pq
E :
~
vec
~
=
~}.
~
=
~
vs. H*1 :
~ ~
r* =
under local (contiguous) alternatives. both
~
{~~
and
~
have non-centrality chi square d.f.·s with pq-a and pq DF respectively.
67
and with the same non-centrality parameter.
Since, the power function
of a non-central chi square variable, for a fixed non-centrality
parameter, is a nonincreasing function of its DF. we conclude that for
all
~
*
cr.
Q2
N is asymptotically locally more powerful that
RN.2
CHAPTER IV
ASYMPTOTICALLY EFFICIENT RANK TESTS
FOR RFSfRICfED ALTERNATIVES IN RANDOMIZED BLOCKS
4.1. Introduction
We consider the usual randomized block design model consisting of n
blocks and p treatments. Each block contains p comparable experimental
units and the p treatments are assigned at random to them. Let X.. =
",lJ
b e t h e vector
.. , ... , X(q»,
..
(X (I)
IJ
i
th
IJ
block, V i
= 1, ... ,n;
j
0f
response
= 1, ... ,p,
0
f t h e J.th treatment
0
f t he
and assume that the random vector
X.. has a q-variate continuous c.d.f. F IJ
.. , 1 -< i -< n, 1 < j -< p, where
",lJ
F .. (x) = F.(x-{3.)
1
J '"
1
'"
",J
q
x E E , V i
= 1, ... ,n;
j
= I, ... ,p,
(4.1.1)
,
(1)
(q)
{3. = ({3. , ... ,(3. ), j = 1, ... ,p, are the vector of treatment effects.
",J
J
J
Here, the fundamental assumption is that blocks are mutually
independent. In the literature, the most common testing problem against
a restricted alternative pro-blem in a randomized block design relates
to testing the null hypothesis
(or {3
= 0,
where {3
= ({31'···'{3
»
'"
",p
(4.1.2)
against the ordered alternatives
H2 :
~1 ~ ~ ~
...
~ ~p
with at least one strict inequality.
(4.1.3)
69
For q = 1. De (1976) extended the method of Sen (1968) by effectively
employing the UI-principle to form an aligned rank test for testing H
O
vs. H when X.. are assumed to be identically independent distributed.
2
IJ
However. no attempt was made to establish possible optimal properties.
Boyd and Sen (1984) used the concepts of LMPR test and UI-principle to
construct the UI-LMPR tests based on intrablock rankings and on rankings
after alignment respectively. They did some Monte-carlo studies which
indicate that the ranking-after-alignment procedure does a better job.
although an analytical proof of the above assertion is still not given.
This provides scope for extending their result to a more general
setting. The main purposes of this chapter are attempts to fill in these
gaps by (i) giving an analytical proof of the asymptotic power
superiority of the ranking-after-alignment procedure over the intrablock
ranking procedure for restricted alternative problems. (ii) establishing
that the UI-LMPR tests are asymptotically UMP against the ordered
alternatives within the respective rank class over the entire
alternative parameter space. and (iii) indicating that the UI-LMPR tests
have the property of asymptotically most stringency and somewhere most
powerful character within the corresponding class for testing H against
O
H*: ~ €
r = {~€Epq; A vec ~ ~ O. A is defined as i~ (2.4.3)}
(4.1.4)
Throughout this chapter. results of Chapter 2 will be used
extensively. The UI-LMPR tests based on intrablock rankings and on
rankings after alignment for testing (4.1.2) vs. (4.1.4) are studied in
Sections 4.2 and 4.3 respectively. The asymptotic (local) power
comparison of these two procedures is investigated in the last section.
70
4.2. The UI-LMPR tests based on the intrablock rankings
R~~)
Let
be the rank of
IJ
e = 1....• q.
(e)
X~~).
(e)
IJ
j
= 1 •.... p;
i
= 1..... n.
(e).
among X'1 1 ' X'1 2 . . . . , X.
Ip . SInce F.1 €
continuous d.f.) V i
= 1 ....• n.
~o
(the class of
ties among the observations may be
ignored. We also let
R.
",1
=
V i
= 1 •...• n.
(4.2.1)
and define R~ to be the matrix derived from R. by permuting the columns
",1
",1
*
in such a way that the top row in the natural order. Let S(R.).
",1
i
= 1 •.... n,
be the set of matrices which are permutationally equivalent
* Since under HO' ",1
to ",1
R..
X. 1 , .... ",lp
X. are interchangeable random vectors
whose joint distribution remains invariant under any permutation of
these p-vectors. This implies that under H ' given a particular realO
* the distribution of R. will be uniform over S(R.).
* That
ization of R..
",1
",1
",1
is for any r. € S(R.).
* we have
'" 1
P{R.
",1
",1
= ",1
r. IS(R~).Ho} = (p!)-1
",1
for all F .• i
1
= 1....• n.
(4.2.2)
Moreover. we also note that the ranking in the different blocks are
mutually independent (since the presence of nuisance parameters-block
effects does not allow comparisons of observations arising from distinct
(rl.r~ ....
blocks). so if
P{R.
",1
'"
= ",1
r ..
whatever F i €
i
~O'
~
* i = 1.2 ..... n}. then
,r ) € {S(R.).
-vn
",1
= 1 •.... nIS(R~).Ho'
",1
i
= 1 ..... n.
i
= 1 ....• n} = (p!)-n
(4.2.3)
71
Denoting ~(1) as the probability measure over these {p!)n equally
n
likely rank permutation. Define the matrix of linear rank statistics by
o
~=
where
o
»
«TNjt
=
j=l, .... p; t=l, ...• q
~jt(r).
n
(t)
«-n1 i=1
}; ~·t{R
.. »)
J
IJ
(4.2.4)
r = 1....• n. j = 1..... p. t = 1..... q are defined as in
(2.4.17). It is thus easily seen that
(4.2.5)
Also. it is easy to verify that
o
0
I
(1)
Cov(TNjt • TNj't' ~n
for j.j' = 1 •.... p;
e.e'
) = (D j j , - p
-1
)vnee ,
(4.2.6)
= 1 ..... q. where 0 .. , is the usual Kronecker
JJ
delta and where
(4.2.7)
is defined by
(4.2.8)
Thus. if we write
0'
~1 = vec ~
(4.2.9)
then
(4.2.10)
_.1-
Now we consider a sequence {K .
n'
~
= n
2~.
~
is fixed} of
alternative hypothesis where K specifies that the random vectors
n
72
x.. -
",IJ
a. - n -"i'Y •• j
",J
",1
= 1 ..... p
are interchangeable for i
P
'Y
'" 1
•.... 'Y
",p
are p real q vectors with
~ 'Y.
. I",J
J=
= 1 ..... n
and
= O. Thus under {K }
n
(4.2.11)
( 4.2.12)
Moreover. we assume that the following limits exist
(4.2.13)
(4.2.14)
and let
eThen. under assumption [Bl] and following Lemma 7.3.10 of Puri and Sen
(1971). we have. as n
~
00.
P
~1 ~ ~1
(4.2.16)
=
and
(4.2.17)
where
~1 = .-Y.l
.Furthermore. we let
~
(I
",p
-!p ",p",p
1 1')
(4.2.18)
73
and
(4.2.19)
Based on !N1 and the use of UI-principle. the overall test
. in (4.1.4) is granted as
statistic for H versus H* defIned
O
-1
l{~l(J:J') > ~. ~l(J'J')~l(J') ~ O}
(4.2.20)
where ~1 denotes the generalized inverse of ~1'
Theorem 4.2.1. Let ~1
=~
~
(1)
J ($ ~ J ~ AO)' assume r J
> O} and
- '"
r(l) =
0
n
Av-JCA
~ - 0
-1
(:1):1 ~ and ~1
= {~1
= ~~
I
8(~1 HO)' For each
+aO (1)
€ E
-1
;~:J' = ~lJ-~l(JJ')~l(J'J')~lJ'
r(l). If the family f.(x-~) satisfy the
J
I
'" '"
assumptions [B1]-[B2] and [B4]. then for testing H vs. H* :
O
(under {Kn }). the UI-LMPR test
stringent for
~
€
r
Q~l in (4.2.20) is asymptotically most
r and asymptotically liMP
for
ra 1 )
within the class of
intrablock rank tests at the respective level of significance a.
Proof. The proof follows directly from Theorem 2.4.4 and hence is
omi tted.
Corollary 4.2.2. Under the same assumptions as in Theorem 4.2.1. then
the corresponding UI-LMPR test is asymptotically liMP (under {K }) within
n
the class of intrablock rank tests for (4.1.2) against (4.1.3) when q=l
at the respective level of significance a.
74
Outline of Proof. The testing (4.1.2) against (4.1.3) can be written as
A = ,..,q
I
for testing (4.1.2) against (4.1.4). where ,..,
-1
1
-1
0
0
0
0
1
~
L with
,..,
0
0
L =
(4.2.21)
0
0
-1
0
1
Then we have
~1
(4.2.22)
=
e-
with
-1
. j=1,2 ..... p
(Diag VI )V 1 "Y.
'"
'" ",J
where Diag
~1
.
denotes the diagonal matrix of
-1
.
(4.2.23)
~1.
And
*
-1
~1 = (Dlag ~1 ) ~1(Dlag ~1 ) ~ ~
where A* is defined in (2.5.8).
(4.2.24)
Let P = {q.2q ..... (p-l)q} and J be any
subset of P. J' be its complement. then V $ ~ J ~ P
~~1, ~
By noting that if q=l. then
*
"Y.
,..,J
0
=
if
"Y ••
,..,J
~1 ~ ~.
V j=l ..... p. hence the corollary is
proved.
4.3. The UI-LMPR tests based on the rankings after alignment
It is obvious that some information is lost if we consider the
intrablock ranking procedure which is absent from interblock
75
comparisons. The ranking-after-alignment procedure utilizes the possible
information contained in the intrablock comparisons. Here. intuitively.
one feels that it should improve the efficiency using the aligned
rankings procedure. To do this. the observations are aligned. namely.
transformed in such a way that the block effects are eliminated. Subtracting the block average or the block median from each observation of
a given block will produce such a result. Some Monte Carlo works suggest
that the choice of the alignment function is not crucial. so we define
the aligned random vectors as
Y..
~lJ
1
P
=x .. - - };
~lJ
P
X..
j
j=I~lJ
(4.3.1)
= 1, .... P; i = 1, .... n.
For convenience. we assume that Y.. has a q-variate continuous c.d.f
",1 J
F?.
Vi
1J
= 1 ..... n;
(2)
j = 1 ....• P . Le t S.. be the rank of
1J
(2) 'Y (2) ..... Y(2) for
N (= np) observations Y
np
II
I2
j
= 1 ..... p.
y~~) among the
1J
i
= 1 ..... n.
2 = 1 ..... q. Thus. corresponding to the aligned observation Y... we have
~lJ
a rank vector S .. =
~lJ
(S~~) ..... S~~»'. j =
1J
1J
define the rank collection matrix
~
by
1 •...• p. i = 1 ..... n. We also
(~II'
...
'~p).
Note that under
H ' Y. .Y. ..... Y. are interchangeable random vectors. so the joint
O ~1 I ~1 2
~lP
distribution of YN = (Y ' ..... Y· ..... Y· ) remains invariant under the
"",
~I I
~IP
,Jlp
finite group
~n of transformation {gO}
(which maps the sample space onto
n
itself). Thus for any
g~
€
~n'
there exists
tionally equivalent to!N. If we denote
corresponding to!N'
* then
*
~
=
°
g~
*
~
~
=
g~
which is permuta-
the rank collection matrix
and is permutationally equivalent to
~. Thus. under H ' the conditional distribution of ~ over the (p!)n
O
76
realization
{~ = ~; g~
~n} is uniform. each realization having the
€
conditional probability (p!)-n. Let us denote this conditional prob-
~~2) and define the scores ~(k). k = 1 .... N. as in
ability measure by
(2.4.7) with F[j] being replaced by
~2' ~2' ~2
and
respectively with
2
= 1 ..... q.
~2
-~
~
.-
Similarly define
~2' ~2'
Q~I' ~I' ~I' ~I' ~I
the same as in
and
~I
~j2(k) being replaced by ~j2(k) for j = 1 ..... p;
Furthermore. consider a restricted (contiguous) alternative
p
_.1.
{K__ :
F~j]'
= N
2~.
~
= O}.
~~.
€ f.
. I",J
(4.3.2)
J=
Then we have
o
F i [j]2(x)
= F0i [e]2 ( x
-~ (2»
- N ~j
.
(4.3.3)
and
(4.3.4)
Finally. we define
~
and
:2
~I
the same as in
F i [e]22'(x. y ) being replaced by
F~[e]2
and :1 with F i [e]2(x) and
F~[e]22'(x.y) respectively.
and
Then under parallel arguments as in the previous section we have
~
~2 {~}) tpq(e; ~.~).
Theorem 4.3.1. Let ~
(4.3.5)
= ~-1 (~)~
(2)
J ($ ~ J ~ AO)' assume f J
= {~
.
= ~:
and ~
+aO
(2)
€ E
; ~:J'
-1
(2)
~(JJ')~(J'J')~J' ~ ~} and f O =
n
f
(2)
CA J
qg-o
I
B(AN2 HO)'
For each
= ~J . If the family
0
Fi(~-~)
satisfy the assumption [BI]-[B2] and [B4]. then for testing (4.1.2)
against (4.1.4) (under
{~}
defined in (4.3.2». the UI-LMPR
77
test
for
2
~2
is asymptotically most stringent for r and asymptotically UMP
r~2) within the class of aligned rank tests at the respective level
of significance a.
Corollary 4.3.2. Under the same regularity assumptions as in Theorem
4.3.1. the corresponding UI-LMPR test is asymptotically UMP (under {KN})
within the class of aligned rank tests for testing (4.1.2) against
(4.1.3) when q=1 at the same level of significance a.
Note that both the proofs of Theorem 4.3.1 and Corollary 4.3.2 are
very similar to the proof of Theorem 4.2.1 and Corollary 4.2.2. and
hence. are omitted.
2
2
4.4. The asymptotic power comparison of Q and Q
N1
N2
2
The optimal property of ON1 may not generally hold when we extend
the domain to the class of tests based on rankings after alignment.
Since the rankings of
Q~2 bases on the aligned observations which
2
disregard blocks. so Q should have the optimal property within a more
N2
broad class. For testing homogeneity against global alternatives. Hodges
and Lehmann (1962) were successful in establishing the superiority of
the rankings after alignment procedure over the intrablock ranking procedure when the underlying distribution is normally distributed. For a
wider class of distribution. this result was extended by Sen (1968) and
studied in detail by Puri and Sen (1971). The assertion of the
superiority of the aligned rank tests to the intrablock rank tests can
be easily extended to the testing against alternatives which put
constraints on the parameters in the linear form of lower dimensional
hyperspace. However. for a wider class of restricted alternatives. this
assertion may not be true. The following Theorem provides a partial
78
answer when the aligned rank tests are asymptotically power-superior to
the intrablock rank tests in the restricted alternative space.
*
~
Theorem 4.4.1. Let
'"
= Al-~~1'
'"
'"
~1{~'~1}
'" '"
. P{Q2
= 11m
N1
n~
~
x {I} IK} and
a
n
Outline of the proof. Let us define
AI' = Var X... Aq
'" 1
for
j # j'
•
~1
",IJ
{4.4.1}
= Cov{X ... X.. ,}
",IJ ",IJ
= 1..... p; i = 1..... n. and
.
Al = 11m n
'"
n~
-1 n
~
AI" Aq = lim n
i=I'" 1 ~
n~
-1 n
~
Aq
{4.4.2}
•
i=I~1
Then via Theorems 4.2.1 and 4.3.1. we have
-1
A
~1 = ~1 - :2
and
~1 =
,
~ =~.
say.
E:.!. ~
and
-1
A} .
p {A
",1 - :2
~ = E:.!.
{4.4.3}
Namely,
:1 = p
~1 =~~
{4.4.4}
Next, we define
2
.-1
Ok = ~~£Ao{~{J:J'}~{JJ:J'}~{J:J·}}I{~{J:J'}
-1
~{J'J'}~{J'} ~ O}
where
V k = 1,2.
> O.
{4.4.5}
e-
79
It suffices to show that if
P{Q~ ~ x~k)IHO} = a. V k = 1.2. then
P{Q~ ~ x~I)IKn} ~ P{~ ~ x~2)1~}.
assume
=~
~1
Without any loss of generality we
and note that
(4.4.6)
= };
¢f;J~o
[fB (J)d~a
2
(z;
O '"
Jp ~1
11. I) '" '"
fB (J)d~a
1
(z; 11. I)]
O '"
'" '"
where
(4.4.7)
x(l)
a
o
aO
(k)
~(J)
= {:
Since
P{Q~ ~ x~k)IHo} = a. V k = 1.2. therefore it is obvious that
= x(2).
a
> O. For
~
€ E
;~. ~ ~. ~
> ~.
II~II ~ x a
Furthermore. we write x(l) = x and 110
a
a
'"
€ 00' (i) if a = 1. then we have
O
} V k
= 11
'"
= 1,2.
+ 01. where
'"
(4.4.8)
(ii) if a
O
= 2. by regrouping the sums and sYmmetric. arguments. we have
.JX-T}
1
=-
2".
{
a
f
1
rvx -11 -0
a
1
-11-0
2
f
-00
80
+
~-'112
-0-11 1
S
...IX-0-'11
S
a
2
--Q)
+ o{l)
~
o.
(4.4.9)
By mathematical induction. thus p{Q21 ~ xa IKn } is non-decreasing in
11* whenever
~ €
00· Therefore we have.
~ €
00'
and hence the theorem follows.
Remark. For a O = 2. let
11*1
~
O. 11*
2
1vx
r -}
~ 3
a
U
°*
*
{~
€
1r- *
* 2
= {~* € E2 ; 11*
1 ~ 3 vxa ' 112 ~ ~} U {~ € E ;
E2 ; 11*1
~
*
O. 112
~
O} U 00' then under
parallel arguments as in (ii) of Theorem 4.4.1. we have
( 4.4.10)
Generalizing this result to the higher dimensions is still an open
problem so far.
e-
CHAPTER V
ASYMPTOTIC OPTIMALITY OF NONPARAMETRIC TESTS
FOR RESTRICfED ALTERNATIVES IN MULTIVARIATE MIXED MODELS
5.1. Introduction
Let Z.
",1
p.r
~
1]. i
= (Y",1.• ",1
X.)
[where Y.
",1
= 1.2 ..... N.
by F~l)(y) (F~2)(x»
1
'"
1
'"
and X.
",1
= (X.1 1 ..... X.l r )
be N independent random vectors having a
(p+r)-variate c.d.f. F.(z), i
1
= (Y.1 1 ..... Y.Ip )
= 1.2 •.... N.
p+r
z € E
. Also let us denote
'"
the marginal joint c.d.f. of Y. (X.) and let
",,1
",,1
F~O)(ylx) be the condition c.d.f. of Y. given X. for i = 1..... N. x
1
"" ""
and y
€
",,1
€
r
E
",,1
EP . Then we assume
""
i ~ 1,
fx)
where the
~i
= (cNi1 "
(5.1.1)
... c Niq )' are q-vectors of known regression
constants. not all equal,
~
=
(~1'
"',...,
.... ~ ) is a pxq matrix of unknown
""q
parameters pertaining to the design-effects. f is a pxr matrix of
""
unknown parameters pertaining to the effects of the concomitant
variates, and the c.d.f. F is also of an unspecified form. If we let
N
= nq.
and take ~1
= ... = ~n = (1.0 ..... 0)'.
(0.1.0, .... 0)' ..... ~[n(q-1)+1]
~(n+1)
= ... = ~2n =
= ... = ~nq = (0 .... ,0.1)'.
then
(5.1.1) represents a multivariate analysis of covariance (MANOODVA)
model which can be viewed as a special case of (5.1.1). with
~.
(px1
""J
vector) denoting the jth treatment effect for j
= 1 ..... q.
The usual
MANOODVA is valid only if the treatments do not affect the values of the
82
covariates. This means that
= F(2)(x)
q
(5.1. 2)
""
Assuming this distributional homogeneity of the covariate, against the
null hypothesis of no treatment effect,
F(O)(Ylx) = F(O)(Ylx) =
H :
o
1
"" ""
2
"" ""
~
= O.
or
= F(O)(Ylx)
q
"" ""
(or
= 0)
~
(5.1.3)
we may be interested in the alternatives of the ordering of treatment
effects while utilizing the information contained in the covariates,
Le.
(5.1.4)
with at least one strict inequality holding for some y
(or
~ ~
...
~ ~
""q
€
EP and x
€
Er
).
In testing for H in (5.1.3) against the simple ordered alternative
O
in (5.1.4), Boyd and Sen (1986) effectively used the DI-principle and
the rank principle of Chatterjee and Sen (1964) to build a class of
conditionally distribution-free DI-rank tests for the univariate case.
However their testing procedure was considered on an ad hoc basis and
hence it may not be asymptotically optimal. We incorporate the concept
of LMPR test and DI-principle to construct the DI-LMPR tests which are
asymptotically power-equivalent to the restricted LR tests for testing
H :
O
~
=~
H*: ~ €
r
against restricted alternatives
= {~
pq
€ E
; A vec ~ ~ 0, A is defined in (2.4.3)}
(5.1.5)
83
under a sequence of contiguous alternatives
{KN} for the general multi-
variate models. The main purpose of this chapter is to extend this
optimum nonparametric procedure for testing H in (5.1.3) against H* in
O
(5.1.5) to the general multivariate mixed models. We may remark that for
the multivariate mixed models, usually the parameters f relating to the
effects of the concomitant variates are unknown. so the optimum nonparametric testing procedures for models only invoking nonstochastic variates may not work out here. Sometimes one can eliminate the nuisance
parameters by suitable transformations. however this often results in
the loss of information due to data reduction. To avoid this. we first
consider models as in (5.1.1), then apply the robust aligned rank procedure of Sen and Puri (1977) to eliminate the nuisance parameter f and
incorporate in the optimum construction of the UI-LMPR statistics on
testing H against (5.1.5).
O
Section 5.2 deals with basic assumptions. Section 5.3 with asymptotically optimal aligned rank tests for restricted alternatives. while
asymptotically optimal pure-rank tests are presented in the next
section. The last section deals with the comparison of asymptotic power
functions.
5.2. Preliminary notions
Toward the formulation of the UI-LMP aligned rank test statistics.
we make the following assumptions:
·
d.
[D1] . For every N . we de f Ine
:Ni
= ~.
~~i'
.)
~i
d...
. ~
1 N
= N-
! d ·• and
i=l N1
84
~
N
=
i:1(~i
And assume that
~)(~i
-
~
-
~)I
= «dNkk ·»k.k·=l ..... q+r·
satisfy the conditions: (i) there exists a positive
definite matrix D such that lim n. = D. (ii) for q+r
~
max
max
l~k~q+r l~i~N
integer NO =
i.
N- 2 (d
~i
i.
Nkk
=
)2
N~ ~
~
IdN.kl = 0(~log2NIN). (iii) for each ~
> 1.
> O. there exists an
1
N(~)
>~
(5.2.1)
such that for all N
max
l~i~N
~~)- ~7)·
NO'
Id . - ~- I. V 1 ~ k ~ q+r. (iv) if q+r ~ 1. then
N1 k -Nk
~
i
~
1. where for each k = 1..... q+r. s = 1.2.
~~,
is
non-decreasing in i.
~i'
[D2]. For the
condition [B3] holds.
Let R.. (S .. ) be the rank of Y.. (X .. ) among Y1 ····· .Y .
IJ IJ
IJ IJ
J
NJ
(X 1j ... "~j) V i = 1
N; j = 1..... p(r) and let
~ = «Rij »i=l
e-
N;j=l ..... p and ~ = «Sij»i=l ..... N;j=l ..... r.
(5.2.2)
For every j (=1 ....• p). working with
with F[j] being replaced by
F~~~).
f~~~* (.;~)
(defined in (2.4.6)
we define the scores
i = 1..... N; j = 1..... p. as in (2.4.17) with
Further. consider the p x (q+r) matrix
*
~
F~~~
~~)(i).
replacing F[j]'
= «T*
Nje » of linear rank
statistics. defined by
*
T .n =
NJ ~
for
e
N
(0)
}; d . n a..... (R .. )
i=l Nl~
l'iJ
IJ
(5.2.3)
= 1..... q+r; j = 1..... p. We also write
~
= vec
*
~.
(5.2.4)
85
Moreover. denote by
(5.2.5)
(0) _
Vjj' -
where
00
S-oo
SOO
-00
(0)*
(0)*
(0)
. .,
f[j] (u)f[j,](v)dF[j.j'](u.v). J~J =1 .. ··.P.
F[(~) "] is the bivariate marginal c.d.f. of the (j.j.)th component
J.J
and assume
v(O) =
((v~~~»
is p.d. and finite.
JJ
(5.2.6)
5.3. The proposed aligned rank order tests
Based on the mixed-rank
nuisance parameter
f.
first we estimate the
statistic~.
We denote this rank estimator of
"-
f
by
fN.
Then we
"-
use the ranks of the residuals Y. = Y. - f(dM.-dM). 1 ~ i ~ N. to
",1
",1
'" --..1,1 --..1'
construct suitable aligned rank statistics which are robust and
asymptotically distribution free for testing H against (5.1.5). To
O
explain this. we let
~ = ((b ji »j=l ....• p;i=l ..... q+r
(5.3.1)
and define the new random vectors
Y.(B) = Y. - B(dM.-dM)
",1
'" --..1,1 --..1'
",1 '"
Vi=l. .... N.
(5.3.2)
also let R.. (B) be the rank of Y.. (B) among Y .(B) ..... Y .(B). V
1J '"
NJ '"
IJ '"
IJ '"
i = 1 ....• N; j = 1 •.... p. and partition B as well as TM(B) (defined as
'"
--..1' '"
in (5.2.4) with R.. being replaced by R.. (B»
IJ
IJ '"
into pxq and pxr matrices
respectively. namely
B = [~1 . ~ ] and ~(~)
pxq pxr
= [~(1)(~)'
pxq
~(2)(~)]
pxr
(5.3.3)
86
Also. let
(5.3.4)
and
(5.3.5)
where
(5.3.6)
Furthermore. we define
~N = {Bq
~
fN
pr
€ E
;
p
q+r
2
2
j=1 i=q+l
ITN.n(0.b~(2»1 = minimum}
(5.3.7)
Jc ~ ~J
"'-
and let
of
be the centroid of
Jure~kova
~N.
Then by the argument parallel to those
(1971). it follows that
p
sup II£N - fll
~~N
~
0
as N
~
(5.3.8)
(X)
~
and
(5.3.9)
"'-
where
~2
is lower r by r principal minor of
~.
Then under HO' £N is a
translation invariant. robust. consistent and asymptotically normally
distributed estimator of f. Now we consider the aligned rank order
statistics
(5.3.10)
87
where
*
N
(0)
A
UNJ'(J = L dN'(J a.;•. (R .. )
e;
i=1 Ie; NJ
IJ
V j = 1..... p; 2 = 1..... q
(5.3.11)
and the aligned ranks are defined by
A
A
A
Roo = Roo{O.f PJ )
IJ
IJ
O'V
(5. 3.12)
V i = 1, .... N; j = 1 ..... p.
ovl'
Moreover. we define the permutational aligned rank covariance
matrix by
A
A
Mu = «m.•...
NJJ
-TI
and partition
~=
» J.J
... =1 ..... p
~
1
N
(O)
A
(O)
0 '0»)'
A
= «N-1 L a..... {R .. )a..... (R.
- i=1 NJ
IJ NJ
IJ
(5.3.13)
as
~11
[ ~21
~12
] q
~22
r
(5.3.14)
r
q
If we wri te
*
(5.3.15)
~ = vec ~ .
then it is easy to verify that
(5.3.16)
where
-1
(5.3.17)
~11:2 = ~11 - ~12~22~21'
Lemma 5.3.1. Under the assumption [D1]. lim n'
N~ ~ 12
= O. in probability.
O'V
under H as well as under a sequence of restricted contiguous
O
alternatives
{~}
in (2.3.4).
88
Proof. Note that
(5.3.18)
= «dN12 (ki»)k=1 ..... q;i=1 •.... r
and via Chebyshev inequality. we have
•
V k=1 ....• q; i=1 .. ... r.
then it follows from the assumption [D1] that dN12 (ki)
probability under HO' as N
under H ' as N
O
~ 00.
~
00.
~
O. in
V k = 1 •.... q; i = 1 ..... r. So
P
~12 ~ ~
The second part follows from contiguity and the
first part.
Lemma 5.3.2. Under the assumptions [B1] and [D1]. as N
~
00
(5.3.20)
where
(5.3.21)
Proof. The lemma follows by the similar arguments as in Lemma 7.4.1
through Lemma 7.4.3 of Puri and Sen (1985) and using the results of
Lemma 5.3.1 and the property of contiguity.
Based on
~.
we can derive a suitable test statistics fer testing
89
H vs. H* . Parallel arguments as in Chapter 2. this overall test
O
statistic for H vs. H* is granted as
O
(5.3.22)
•
=
-
~ ~ ~
.
-!N
-1
~
,-1
!N + +
~~~o{!N(J:J')~(JJ:J')!N(J:J')}
-1
1{!N(J:J'»~' ~(J'J')!N(J')~~}
where
(5.3.23)
Theorem 5.3.3. Let ~ = AB-1(~)~ A and A = lim 8(An
rv
rv
t"Vt'V
rv
rv
rv
N~
~.,
IHo ).
-1
+aO
J (~~ J ~ AO)' assume f J = {~ € E
f
O
=
n
f
~J~O J
. If the family
For each
; ~J:J' = ~J-~JJ'~J'J'~' ~ O} and
F~O)(y) satisfy the assumptions [B1]. [B2].
I
'V
. H vs. H* (under
[B4] and [D1]. then for testIng
O
{~}).
the UI-LMP
aligned rank test statistic Q~o is asymptotically most stringent and
somewhere most powerful for f and asymptotically liMP for f
O
at the
respective level of significance a.
Proof. It follows directly from Lemma 5.3.2 and Theorem 2.4.4. and hence
is omitted.
Corollary 5.3.4. Under the same assumptions as in Theorem 5.3.3. the
corresponding UI-LMP aligned rank test is asymptotically
testing H in (5.1.3) vs.
O
~
liMP for
in (5.1.4) when p=1 (under a sequence of
restricted contiguous alternative) in ANOOOVA models (pertaining to
randomized block designs) at the level of significance a.
90
5.4. Conditionally distribution-free rank tests
The testing procedure considered in the previous section possesses
the asymptotically optimal property in the light of most stringent
somewhere most powerful character. However. there are some disadvantages
for this procedure. First. the computation of it is quite cumbersome.
Secondly. it is based on aligned observations which are not independent.
and this may invalidate the invariance structure underlying the scope of
the usual conditionally distribution-free rank tests. To overcome these
drawbacks. we wish to establish. under parallel regularity conditions. a
conditionally distribution-free procedure which is easy to compute and
also shares the same asymptotic optimality as
Q~o. We also aim to
construct the class of optimal tests only by invoking the coordinatewise
ranks of the Z .. i = 1 •.... N. As in Section 2.4. the permutational
",1
invariance structure still holds for
conditional measures by
of scores
~
and
(~.~).
so denote these
~~2) and ~N respectively. First we define sets
~~){i) (~){i».
j = 1..... p; i = 1..... N. (k=1 ..... r.
i=1 •.... N) as in (2.4.18) using
f~~~ (f~~~)
instead of
f~~~. V
j = 1 •.... p (k=1 ..... r). Then we consider the linear rank matrices
0
:!:N1 =
({T~~~»
1 N
(1)
= {{- }; {cN" 2 - c 2)BN· (R .. ) »
vff i=1
1
J
IJ
(5.4.1)
V j=l, .... p; 2=1 ..... q
0
:!:N2=
({T~~»
1 N
(2)
= ({vff i:1{c Ni2 - c 2 )BNk (Sik) »
V k=l, ...• r; 2=1 •.... q
Also. let
(5.4.2)
91
1
V'PJ11 = «N-1
~,
N (1)
(1)
! a.~. (R .. )a.~ . , (R .. ,) ))pxp
i=l NJ
IJ NJ
IJ
(5.4.3)
(5.4.4)
and
(5.4.5)
Thus if we write
~1
~1' ~2
~2
and
~
then. under permutational measure
~N'
we arrive at the following results
= vee
= vee
=
(~1' ~2)'
(5.4.6)
by some standard computations:
(5.4.7)
where
~=
~11
[ ~12
p
~12
] P
~22
r
(5.4.8)
r
Further. we denote
(5.4.9)
~11 = «V 11 (jj'»)' v 11 (jj')
= fro
-00
fro
-00
(1)*
(1)*
(1)
f[j] (u)f[j,](v)dF[j.j'](u.v)
V j.j'=l ..... p
(5.4.10)
ro fro
~2 = «v22 (kk'»)' v 22 (kk') = f -00
-00
(2)*
(2)*
(2)
f[k] (u)f[k,](v)dF[k.k'](u.v)
V k.k·=l, .... r
92
and
(5.4.11 )
~12
= «v 12 (jk»)'
00
v 12 (jk)
00
= f-oo f-oo
(1)*
(2)*
f[j] (u)f[k] (v)dG[j.k](u,v)
V j=l ..... p; k=l ..... r
(1)
•
(2)
where F[j.j'](u.v). F[k.k'](u.v) and G[j.k](u.v) are the bivariate
marginal distribution functions of (Y ... Y.. ,). (X.k.X. k ,) and (Y ... X. k )
IJ
IJ
1
1
IJ
1
respectively. V j.j'
= 1 ....• p;
Sen (1985. p. 320). under
T
M
~,
~
~ ~
H
o
q
(
p+r )(.;O.V
~ ~
~N
~
k.k'
= 1.... ,r.
Then following Puri and
and the assumptions [B1] and [D1], we have
(5.4.12)
D11 )
~
where
v=
~
~12 ]
[~11
V·
~12
(5.4.13)
~2
e-
From (5.4.12). we then can use the covariate-adjusted rank statistics by
fitting a linear regression of
~1
on
~2.
So the residual rank
statistic of this regression is given by
(5.4.14)
And it is easy to show
(5.4.15)
where
(5.4.16)
Moreover. for the conditional random vector
~.
from (5.4.12). we have
93
(5.4.17)
where
(5.4.18)
Thus. along with (5.3.20). (5.4.18) and the fact that ~
we arrive. as N
(0)
= ~11:2'
~ 00.
(5.4.19)
~
- ~ ~ 0
Base on
~.
in ~N probability under HO as well as under {~}.
and parallel to the argument of previous section for
testing H vs. H* . the proposed permutationally distribution-free pure
O
2
rank test statistic. say QN1' can be easily obtained with
replaced by
~
~
being
under less stringent conditions (here. use [D2] instead
2
2
of [D1]). The result of (5.4.19) tells us that both Q and QNO have the
N1
same asymptotically local power function. hence share the same
asymptotic optimality. Moreover. one of the most important advantages of
2
2
Q over QNO is its robustness against departure from linearity in
N1
(5.1.1). This ensures that residual rank procedure has generally broader
and more flexible structures of the model.
5.5. Asymptotic power comparison of MANOOOVA rank test statistics
and MANOVA rank test statistics
If we totally ignore the covariates and proceed the same as in
Chapter 2 to obtain the corresponding UI-LMPR test. say Q;2' for testing
HO vs. H* (defined in (5.1.5». Then the
~2
is derived solely from
hence it limiting covariance matrix in (5.4.17) would have involved
~1'
94
~ = ~11 8 ~11 instead of :1 = ~11:2 8 ~11· For testing HO against
~ #~,
global alternative H :
1
the asymptotic relative efficiency of the
covariate-adjusted rank test relative to the MANOVA rank test is bounded
below by one. This is due to the usual MANOOOVA models incorporated the
information contained in the concomitant variates in some plausible way.
Naturally, the same question arises here whether the covariate-adjusted
rank test asymptotically dominates the one which ignores this
information for testing H against restricted alternative
O
H*
in mixed
models. The following theorem provides the answer to this and also
clearly reveals the greater power for the covariate adjusted test than
the unadjusted one over most of the restricted parameter space when we
have an ordered alternative for a MANOOOVA model pertaining to
completely randomized block designs.
1
Theorem 5.5.1. Let
~
(k)
in Theorem 5.3.3 with
x~k)IKN}' V k=1,2.
=
-~
-2
~~,
k=1,2, where
being replaced by
If lim
N~
~
and
~
~ , and Pk(~'~ ) = lim P{Q~ ~
~
P{~ ~ x~k)IHO}
-~
JI(2)
.!
Jxa(2),
~
= a, k = 1,2, and
= {~
3
are defined as
~(1) ~
+aO
€ E
;
(2)
~j
~
j=l, ... a }.
O
Proof. The proof is similar to the Theorem 4.5.1 and hence is omitted.
5.6
A numerical illustration with the anesthesia data
In this section, we consider a numerical illustration of the
UI-LMPR tests against an ordered alternative, i.e.
H in (5.1.4), in an
2
analysis of covariance (ANOOOVA) model with a real data set (Boyd and
Sen (1986».
Here,
•
95
TIME 95
= the
(XO)
dependent variable. time from completion of the injection
of removal mixture until the first twitch in the train of four
responses reached 95% of the control height.
(For the sake of simplicity. this variable may be called TIME or REVTlME
time elapsed from administration of treatment to completion of
reversal. )
=
Depth 1 = depth of neuromascular block at time of reversal (time
(Xl)
treatment administered).
Age (X )= the age (in years) of the patient.
2
[Depth 1 is expressed as a percentage. Le .. 100 (height of the first
twitch/control height)]
Observation Treatment
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
2S
29
30
X
1
1
1
1
1
1
1
1
1
1
4.8
13.2
5.8
4.6
6.0
2.9
5.2
5.6
3.9
5.6
Xl
17.0
6.0
30.0
40.0
30.5
48.0
50.0
20.0
40.0
24.4
2
2
2
2
2
2
2
2
2
2
6.0
9.6
15.5
8.7
7.9
5.2
6.6
2.7
5.4
8.2
30.0
25.0
25.0
56.0
51.0
61.0
26.0
45.0
45.0
21.0
3
3
3
3
3
3
3
3
3
3
16.4
6.7
8.6
7.9
6.0
19.4
19.0
2.8
6.6
10.4
15.0
6.0
20.0
15.0
32.0
0.0
15.7
59.0
26.0
21.0
o
X
2
27
41
27
33
34
21
43
26
36
23
25
58
55
43
36
45
34
41
29
25
41
32
49
2S
46
45
24
23
29
35
96
In this anesthesia data set, thirty observations are recorded for
the three different treatments (labeled as 1,2,3) so that we have three
samples, each of size ten.
We are interested in comparing the efficacy
of 3 treatments used for the reversal of anesthesia.
o) ,
primary variate, TIME 95 (X
In addition to the
we would like to include the covariates
Depth 1 (Xl)' and Age (X ) in the study.
2
Since the plot of the
residuals reveals the non-linearity of the regression of the primary
variate on the covariates.
Hence the classical parametric
ANOCOVA methods are not appropriate here.
In the rank analysis, Boyd
and Sen (1986) have made conclusions that the treatment effects are
different at the conventional 5% significance level and their proposed
UI-rank statistic based on the Wilcoxon scores is superior to the other
tests such as the global rank test, the Kruskal-Wallis test, the
modified Page test (based on the covariate adjusted rank statistics) and
the multivariate version of rank-sum test.
However, as we mentioned
earlier, the proposed UI-rank statistic of Boyd and Sen (1986) was
considered on an ad hoc basis, so we proceed to consider the UI-LMPR
test as in Section 5.4.
We first use the Normal scores to explain this.
It may be noted that in actual practice, even if the parent cdf's are
continuous, the data may have ties among the observations (as is the
case with this anesthesia data).
strictly applicable.
Therefore the UI-LMPR tests are not
To generalize the UI-LMPR tests for this kind of
data, it can be done precisely as in Section 8.5 (Puri and Sen (1985»
with the modification in Section 5.4 of this Chapter.
As such we will
take for the tied observations average score as the corresponding
scores.
Thus we have
o
~1
= (-0.543566,
0.006000, 0.537566)'
97
TO
,:.N2
= rO.024344.
and the covariance matrix
=
~
0.543702. -0.568046)'
L-0.422246. 0.380879. 0.041368
~
defined in (5.4.8)
0.917551
-0.581345
[0.338786
-0.5813447
0.916157
-0.029963
0.338786]
-0.029963
0.915663
Using (5.4.14). the covariate-adjusted rank statistic vector is
~
= (-0.380779.
0.211633. 0.1691463)
Also. by (5.4.15). we have
= 0.135580
where D
st
=1
determinant of
2
-1
-1
[
-1
2
-1
-1]
-1
2
or 0 according as s is equal to t or not.
~
and VNOO is the cofactor of v NOO
in~.
I~I is the
So we also
have
::N
~ ~
= 4.832407
If we let
(5.6.1)
then we have
~ = (13.1084426.
-0.9401129)'.
~ = 7.375743 [~3 -~)
and
,
-1
~ ~
2
Hence Q
reduces to
N1
2
= ~~CA {~(J:J')
- - 0
-1
ONI
~
= 4.832407.
-1
~(JJ:J') ~(J:J')} 1{~(J:J') > ~.
2
~(J'J') ~(J') ~ ~} = (12.638386) /33.190844 = 4.812436
98
2
In order to evaluate the significance level of QN1' we use the
expression
(5.6.2)
where
p(J)
= ~~ P{~(J:J') > ~IHO} P{~~J'J') ~(J') ~ ~IHO}
For each J
($ S J S
AO
= {1.2}).
1
= p({2}) = 3.
,
by symmetric arguments and after some
calculations. the multinormal orthant probability p($)
p({1})
(5.6.3)
= p(AO) = ~.
2
Hence the asymptotic significance level of Q
N1
based on the Normal scores is 1 -
[~ (1
+ 0.907483) +
0.035067. which supports the alternative hypothesis.
~ x 0.970529]
=
However. the use
of the Normal scores achieves the high efficiency only for nearly normal
distributions.
For this lack of normality. the use of Normal scores is
not very favorable. so the use of Wilcoxon scores which are much simple
in computations as well as robust against heavy tailed distributions is
advocated.
Q~1
Similar arguments as above yield an asymptotic p-value of
(based on the Wilcoxon scores) of 1 -
2
3(0.989144)]
= 0.013912.
[~(1
+ 0.9559951) +
Thus we observe that the UI-LMPR tests based
on the Wilcoxon scores perform relatively better than on the Normal
scores for this anesthesia data.
Since the form of density function is
2
not known. the asymptotic significance level of Q based on each of
N1
these choices (Wilcoxon scores, Normal scores, Median scores) is more or
less below the optimal.
To overcome this practical difficulty. we need
to determine a consistent estimate of the true score functions from the
sample itself and then use these adaptive scores.
for future study.
We propose this area
e-
aIAPTER VI
TOPICS FOR FlTfURE RESEARCH
In clinical trials and life testing problems. it is not possible
to continue the experiment until all the individuals respond. so the
experiment is carried out either for a fixed period of time {truncated}
or for a fixed proportion of observation {censored}.
There is some loss
of efficiency due to such truncation or censoring schemes. often. a
statistical monitoring is advocated.
We will consider the hypothesis
testing problems against restricted alternatives in a progressive
censoring scheme {Pes} which allows us to monitor the experimentation
from the beginning until a statistically valid decision is reached.
For
nonparametric testing problem against global alternatives under a Pes.
Chatterjee and Sen {1973} have formulated a general class of tests based
on linear rank statistics. censored at successive failures.
Sen {1976}
has studied the asymptotic optimality of such rank order tests.
For
nonparametric testing against restricted alternatives under a Pes. Sen
{1984a} has proposed a class of rank order process {based on suitable
linear rank statistics and the DI-principle} and studied their
asymptotic distributions under null hypothesis {referred as a Bessel-bar
process}.
However. the asymptotic optimality properties and asymptotic
distribution theory under non-null hypothesis of the proposed Pes tests
are not totally known.
Those leave us some spaces for further study.
In a multivariate setup. as we have seen in Chapter 2 the
conditional expectation of LMP test statistics given the matrix of rank
100
may not be a simple linear functions of rank scores anymore.
The
condition (2.4.7) is a sufficient condition to deal with that, however
it is a stringent condition.
The quest to find the necessary and
sufficient condition or a less stringent condition instead of (2.4.7)
should be continued.
Since the condition (2.4.7) is crucial, so the
optimum nonparametric procedure is not appropriate for most discrete
data (such as binary data).
To work for a wider class of models, we
need to go through the parametric procedures available at the present
stage.
The LR technique has been proposed for order restrictions in
exponential families.
For any fixed alternative, the power of any
reasonable test tends to one as
N~.
Neyman (1959) proposed a locally
asymptotically most powerful test for composite statistical hypothesis
for global alternatives.
This test is particularly applicable when
either standard testing procedures are not available or when they are
too difficult to compute.
On account of the complexity of the
restricted LR technique, we generalize the result of Neyman to the case
of testing q-dimensional parameter for restricted alternatives.
The
proposed statistics are shown to be asymptotically power-equivalent to
the restricted LR statistics under a sequence of contiguous restricted
alternative.
Further, under some additional conditions, the exact
distributions of the proposed statistics are shown to be distributed as
a mixture central chi-square distribution under such a sequence of
contiguous alternative.
A further study of the asymptotic distribution
theory, by limit distribution based on approximation or Edgeworth
expansions, of the proposed statistics will be also made later.
For a multivariate mixed model which involves the nuisance
parameters, in Chapter 5, in exchange for the loss of information we
101
achieve the simplicity in analysis resulting from the conditional
arguments and the elimination of nuisance parameters.
of nuisance parameters is high, the application of
(5.3.22)) may lead to misleading results.
Q;o
If the dimension
(defined in
To overcome this, the concept
of partial likelihood of Cox (1975) plays a vital key for it.
The
advantage of using partial likelihood is to reduce the loss of
information due to ignorance factor as small as possible by a suitable
factorization of full likelihood functions.
The Cox proportional hazard
model which is a special case of a mixed model is a semi-nonparametric
model with the regression of survival time being characterized in a
parametric set up but the hazard function being of arbitrary form.
We
will use the Cox proportional hazard model as an explanation for the
testing of subhypothesis against restricted alternatives based on the
partial likelihood scores (PLS) which form an adapted martingale
difference sequence.
The asymptotic distribution theory for the PLS was
given by Tsiatis (1981) and Sen (1981) using martingale properties.
Sen
(1984b) also considered tests against restricted alternatives based on
the Cox PLS and established that the asymptotic distributions of those
tests come out in the form of a chi-bar distribution under null
hypothesis.
Since in Cox model the partial likelihood is proportional
to the projection of the likelihood onto the space of rank statistics.
Therefore we will study the asymptotic optimality and distribution
theory of the corresponding tests based on the rank partial likelihood
statistics for restricted alternatives.
be considered under a PCS.
Further, the same problem will
Moreover, in order to allow for complicated
censoring patterns and time dependent covariates, an extension of
partial likelihood methods for proportional hazard models to a model
102
where covariate process have a proportional effect on the intensity
process of a multivariate counting process will be studied too.
The contiguity of probability measure under local alternatives to
those under the null hypothesis may be established.
Under such
•
contiguous alternatives. the asymptotic power functions of the
corresponding optimal test statistics based on rank PLS should also be
studied.
The pictures are quite comparable to the cases mentioned in
Chapters 4 and 5.
To deal with these. an extensive amount of simulation
work would be helpful. we are intended to do so in a follow-up study.
e-
BIBLIOGRAPHY
Abelson. P.R. and Tukey. J.W. (1963). Efficient utilization of
non-numerical information in quantitative analysis: general
theory and the case of simple order. Ann. Math. Statist. 34.
1347-1369.
Bahadur. R.R. (1960). Stochastic comparison of tests.
Statist. 31. 276-295.
Ann. Math.
Bahadur. R.R. (1965). An optimal property of the likelihood ratio
statistics. Proceedings of the Fifth Berkelev Symposium on
Mathematical Statistics and Probability 1. 13-26.
Bahadur. R.R. and Raghavachari. M. (1971). Some asymptotic properties
of likelihood ratios on general sample spaces. Proceedings of
Sixth Berkeley Symposium on Mathematical Statistics and
Probability 1. 129-152.
Barlow. R.E .. Bartholomew. D.J .. Bremner. J.M. and Brunk. H.D. (1972).
Statistical Inference Under Order Restrictions. John Wiley and
Sons. New York.
Bartholomew. D.J. (1959). A test of homogeneity for ordered
alternatives. Biometrika 46. 36-48.
Bergstrom. H. and Puri. M.L. (1977). Convergence and remainder terms
in linear rank statistics. Ann. Statist. 5. 671-680.
Bhattacharyya. G.G. and Johnson. R.A. (1970). A layer rank test for
ordered bivariate alternatives. Ann. Math. Statist. 41. 1296-1310.
Bhattacharya. R.N. and Rao. R.R. (1976). Normal approximation and
asymptotic expansion. John Wiley and Sons. New York.
Boyd. M.N. and Sen. P.K. (1984). Union-intersection rank tests for
ordered alternatives in a complete block design. Commu. Statist.
A 13. 285-303.
Boyd. M.N. and Sen. P.K. (1986). Union-intersection rank tests for
ordered al ternatives in ANCXJJVA. .T. Amer. Statist. Assoc. 81.
526-532.
Brown. L.D. (1971). Non-local asymptotic optimality of appropriate
likelihood ratio tests. Ann. Math. Statist. 42. 1206-1240.
Chacko. V.J. (1963). Testing homogeneity against ordered alternatives.
Ann. Math. Statist. 34. 945-956.S.K. and De. N.K. (1972).
Chandra. T.K. and Ghosh. J.K. (1978).
Bahadur efficiency.
Comparison of tests with same
Sankhy~ Ser. A 40. 253-277.
Chandra, T.K. and Ghosh, J.K. (1982). Deficiency for multiparameter
testing problems. In Statistics and Probability: Essays in honor of
C.R. Rao (eds.: G. Kallianpur, P.R. Krishnaiah and J.K. Ghosh),
North Holland, Amsterdam, 179-191.
Chatterjee, S.K. and De, N.K. (1972). Bivariate non-parametric
location tests against restricted alternatives. Calcutta
Statist. Assoc. Bull. 21, 1-20.
Chatterjee, S.K. and De, N.K. (1974). On the power superiority of
certain bivariate location tests against restricted alternatives.
Calcutta Statist. Assoc. Bull. 24, 73-84.
Chatterjee. S.K. and Sen, P.K. (1973). Nonparametric testing under
progressive censoring. Calcutta Statist. Assoc. Bull. 22, 13-50.
Chatterjee, S.K. and Sen, P.K. (1964). Nonparametric tests for the
bivariate two-sample location problem. Calcutta Statist. Assoc.
Bull. 13, 18-58.
Chinchilli, Y.M. and Sen, P.K. (1981a). Multivariate linear rank
statistics and the union-intersection principle for hypothesis
testing under restricted alternatives. Sankhya. Ser. B. 43,
135-151.
Chinchilli, Y.M. and Sen. P.K. (1981b). Multivariate linear rank
statistics and the union-intersection principle for the orthant
restriction problem.
Cox. D.R. (1975).
Sankhya. Ser. B 43, 152-1171.
Partial likelihood.
Biometrika 62, 269-276.
Csorgo. M. and Revesz, P. (1981) Strong Approximations in probability
and Statistics, Academic Press, New York.
De, N.K. (1976). Rank tests for randomized blocks against ordered
alternatives. Calcutta Statist. Assoc. Bull. 25. 1-27.
Hajek, H. (1962). Asymptotically most powerful rank order tests.
Math. Statist. 33, 1124-1147.
Hajek, J. and Sidak, Z. (1967).
New York.
Theory of rank tests.
Ann.
Academic Press,
Herr, D.G. (1967). Asymptotically optimal tests for multivariate
normal distributions. Ann. Math. Statist. 38. 1829-1844.
Hodges, J.L. Jr. and Lehmann, E.L. (1962). Rank methods for
combination of independent experiments in analysis of variance.
Ann. Math. Statist. 33, 482-497.
Hoeffding, W. (1951). Optimum nonparametric tests. Proc. Second Berkeley
Symp. Math. Statist. and Prob., 83-92.
Hoeffding, W. (1965a). Asymptotically optimal tests for multinomial
distributions. Ann. Math. Statist. 36, 369-401.
•
Hsieh, H.K. (1979a). On asymptotic optimality of likelihood ratio
tests for multivariate normal distributions. Ann. Statist. 7,
592-598.
Hsieh, H.K. (1979b). Exact Bahadur efficiency for tests of
multivariate linear hypothesis. Ann. Statist. 7, 1231-1245.
Huskova, M. (1977a). Rate of convergence of linear rank statistics
under hypothesis and alternatives. Ann. Statist. 5, 658-670.
Huskova, M. (1977b). Rate of convergence of quadratic rank statistics.
J. of Multivariate Anal. 7, 63-73.
Hu~kova,
M. (1980). The rate of convergence of simple linear rank
statistics under alternatives. Contributions to Statistics, J.
Hajek Memorial Volume. Academic, Prague, 99-108.
Jureckova, J. (1971). Nonparametric estimate of regression coefficients.
Ann. Math. Statist. 42, 1328-1338.
Jureckova, J. and Puri, M.L. (1975). Order of normal approximation for
rank test statistic distribution. Ann. Probab. 3, 526-533.
Kallenberg, W.C.M. (1980). ASYmototic optimality of likelihood ratio
tests in exponential families. Mathematical Centrum, Amsterdam.
Koziol, J.A. (1978). Exact slopes of certain multivariate tests of
hypothesis. Ann. Statist. 6, 546-558.
Kudo, A. (1963). A multivariate analogue of the one-sided test.
Biometrika 50, 403-418.
NeYman, J. (1959). Optimal asymptotic tests of composite statistical
hypotheses. In Probability and Statistics, ed. Grenander, U. New
York, John Wiley and Sons, 213-234.
NUesch, P.E. (1966).
On the problem of testing location in
multivariate populations for restricted alternatives. Ann. Math.
Statist. 37, 113-119.
Oosterhoff, J. and Van Zwet, W.R. (1970). The likelihood ratio test
for the multinomial distribution. Proceedings of the Sixth
Berkeley Symposium on Mathematical Statistics and Probabilities,
1. 31-50.
Perlman, M.D. (1969).
One-sided testing problem in multivariate
analysis. Ann. Math. Statist. 40, 549-567.
Puri, M.L. and Sen, P.K. (1969). A class of rank order tests for a
general linear hypothesis. Ann. Math. Statist. 40, 1325-1343.
Puri, M.L. and Sen, P.K. (1971). Nonoarametric methods in multivariate
analysis. John Wiley and Sons, New York.
Puri. M.L. and Sen. P.K. (1985). Nonoarametric Methods in General
Linear Models. john Wiley and Sons. New York.
Robertson. T. and Wegman. E.j. (1978). Likelihood ratio tests for
order in exponential families. Ann. Statist. 6. 485-505.
Roy. S.N. (1953). On a heuristic method of test construction and its
use in multivariate analysis. Ann. Math. Statist. 24. 220-238.
•
Schaafsma. W. and Smid. L.j. (1966). Most stringent somewhere most
powerful tests against alternatives restricted by a number of
linear inequalities. Ann. Math. Statist. 37. 1161-1172.
Sen. P.K. (1968). On a class of aligned rank order tests in two-way
layouts. Ann. Math. Statist. 38. 1115-1124.
Sen. P.K. (1976). Asymptotically optimal rank order tests for progressive
censoring. Calcutta Statist. Assoc. Bull. 25. 65-78.
Sen. P.K. (1981). The Cox regression model. invariance principles for
some induced quantile processes and some repeated significance tests.
Ann. Statist. 9. 109-121.
Sen. P.K. (1982). The UI-principle and L.M.P. rank tests.
Mathematica Societaties Janos Bolyai 32. 843-858.
Colloqua
Sen. P.K. (1983). A Fisherian detour of the step-down procedure.
Contribution to statistics. 367-377. North Holland.
Sen. P.K. (1984a). Nonparametric testing against restricted alternatives
under progressive censoring. Univ. of North Carolina. Mimeo Series
1473.
Sen. P.K. (1984b). Subhypotheses testing against restricted alternatives
for the Cox regression model. j. Statist. Plan. Infer. 10. 31-42.
Sen. P.K. and Puri. M.L. (1977). Asymptotically distribution-free
aligned rank order tests for composite hypothesis for general
multivariate linear models. Zeitschrift fur
Wahrscheinlichkeitstheorie and Verwandte Gebiete. 39. 175-186.
Tsiatis. A.A. (1981). A large sample study of Cox's regression model.
Ann. Statist. 9. 98-108.
Waldo A. (1943). Tests of statistical hypothesis concerning several
parameters when the number of observation is large. Trans.
Amer. Math. Soc. 54. 426-482.
Wong. W.H. (1986).
88-123.
Theory of partial likelihood.
Ann. Statist. 14.
e-