-
.
SEQUENTIAL DECISION PROCEDURES FOR TESTING
HYPOTHESES CONCERNING GENERAL ESTIMABLE PARAMETERS
By
Mahmoud Riad Mahmoud
Department of Biostatistics
The University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 881
AUGUST 1973
...
..
SEQUENTIAL DECISION PROCEDURES FOR TESTING HYPOTHESES
CONCERNING GENERAL ESTIMABLE PARAMETERS
by
Mahmoud Riad Mahmoud
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 Biostatistics
Chapel Hill
1973
Approved by:
~/
D4f14 Q.,tatb ..
Reader
.....
ABSTRACT
MAHMOUD RIAD MAHMOUD. Sequential Decision Procedures for Testing
Hypotheses Concerning General Estimable Parameters. (Under
the direction of DR. P. K. SEN.)
An invarianceprinciple, due to Strassen, is extended here to
continuous functions of several estimable parameters.
A sequential
procedure for discriminating between two close hypotheses concerning
these functions is introduced.
The invariance principle is used to
justify the proposed procedure and to study its properties.
The
procedure is extended to discriminating among three hypotheses concerning the same functions· is proposed and studied.
A simulation
study of an example is used to verify the proposed procedure.
w
•
...
ACKNOWLEDGMENTS
It is a pleasure to acknowledge my sincerest gratitude to
Professor P. K. Sen, my adviser, for proposing the problems in this
dissertation and giving me help in numerous ways.
It was his patience
and encouragement that supported me all the time.
I wish also to thank Professor
N.
L. Johnson for going through
the manuscript and making many helpful suggestions.
Thanks are also
due Dr. D. Quade and Dr. M. J. Symons for proofreading the manuscript.
I also thank Dr. R. P. Walker, the other member of my doctoral committee, and Dr. D. G. Kelley for sitting on my oral exam.
I wish also to thank Professor A. E. Sarhan of the Institute of
Statistical Studies and Research at Cairo University for his help in
many ways.
It was through him that I was first introduced to Statistics
and he has been supporting me since.
I wish to acknowledge the Department of Statistics for financial
assistance during the first two years of my graduate program and the
Department of Biostatistics for financial assistance in the rest of my
graduate studies.
Finally, thankful appreciation is due Mrs. Gay Hinnant for her
excellent work in typing the manuscript.
•
011
..
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ••
LIST OF TABLES
it
..
iv
Chapter
I.
INTRODUCTION AND REVIEW ••
1.1.
1.2.
1.3.
1.4.
II.
-e
ASYMPTOTIC PROPERTIES OF U-STATISTICS AND VON MISES'
DIFFERENTIABLE STATISTICAL FUNCTIONS •
2.1.
2.2.
2.3.
2.4.
2.5.
III.
..
Introduction • • • • • • • • • • •
Notations • • • • • • • • • • .
Weak convergence of U-statistics. •
Functions of several U-statistics
Estimation of y2 • . . . . . . . •
TESTING HYPOTHESES CONCERNING ESTIMABLE PARAMETERS •
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
IV.
Introduction. • •
Review of literature ••
Some definitions and previous results •
The problem and results • • •
Introduction • • • • • • • • • • • •
Two hypotheses problem • • • • •
OC function • • • • • •
ASN function • • • • • • •
Three hypotheses problem • . • • • .
Probability of correct decision
Bounds for the ASN function . . .
EXAMPLES AND NUMERICAL INVESTIGATIONS.
4.1.
4.2.
4.3.
4.4.
4.5.
REFERENCES • •
Introduction • • • • • • •
Correlation coefficients.
Rank correlation• • • • • • •
Comparison of two variances
Numerical results
.....
1
1
2
5
9
12
12
12
14
16
21
29
29
30
32
35
44
45
47
50
50
50
54
55
56
65
LIST OF TABLES
Table
Page
1. ASN (In Pairs) for the Correlation Coefficients
go = -.1 and g = .1.
1
..................
2. ASN (In Pairs) for the Correlation Coefficients
go = -.2 and g1 = .2.
..................
58
59
3. ASN (In Pairs) for the Correlation Coefficients
go
= -.
1. . . . . . . . . . . . . . . . . . . . . . . . .
60
4. ASN (In Pairs) for the Correlation Coefficients
go
= -.
2. . . . . . . . . . . . . . . . . . . . . . . . .
61
5. ASN (In Pairs) for the Correlation Coefficients
go =
.1 . . . . . . . . . . . . . . . . . . . . . . . . .
62
6. ASN (In Pairs) for the Correlation Coefficients
go
=
.2 . . . . . . . . . . . . . . . . . . . . . . . . .
63
e-
CHAPTER I
INTRODUCTION AND REVIEW
1.1.
Introduction.
Let X be a random variable (r.v.) with distribu-
tion function (DF) F(x).
On the basis of a finite number of observa-
tions on X we want to draw some inference concerning the (unknown) DF
of X.
Frequently we are interested in the values taken by some
parameter 8(F) of F(x).
we
We shall be concerned with testing statistical
hypotheses about the parameter 8
= 8(F).
We are only interested in
sequential procedures where the number of observations depends, in some
specific way, on the data obtained in the course of study.
Suppose that 8 takes values in some space ~, called the parameter space, and we want to test the hypothesis He:
alternative hypothesis HI:
e£
e
£
We against the
wI' where we and wI are subsets of ~.
In sequential methods of testing statistical hypotheses, He vs HI' a
rule is given for making one of the following decisions at any stage
of the experiment:
(1) to accept He and terminate the procedure,
(2) to accept HI and terminate the procedure, or (3) to continue the
experiment by making an additional observation.
With any sequential
procedure for testing He vs HI there will be associated two numbers
a and
B between
zero and one such that the probability is a that we
reject H when it is true and is
O
B that
we accept He when it is false.
2
We call the pair (a,S) the strength of the testing procedure.
Since the number of observations N at which we decide to stop
sampling depends on the outcome of observations, it is a random variable.
One particular property we desire in a testing procedure is that
it will eventually terminate with probability one.
If a testing pro-
cedure has this property we shall say that it "terminates."
functions of the parameter
e are
of special interest.
Also, two
These are,
(1) the operating characteristic (DC) function defined as the probability of accepting H when
O
e
is the true value of the parameter, and
(2) the average sample number (ASN) function defined as the expected
value of the number of observations N required by the procedure before
stopping.
A good discussion of the relevance of these functions is
found in Wald (1947).
Let Sl and S2 be two tests for testing H vs HI
O
which have the same strength.
8 E
QD,
If E
(N;8) < E
(Nj8) for some
S
S
1
2
we say that Sl is more efficient than 8 at 8. We say that
2
S* is uniformly most efficient (UME) for testing H vs HI if
O
inf
E * (Nj8) = 8 E (Nj8) for all 8 E
, where the infimum is taken
S
8
®
over all 8 of the same strength.
Uniformly most efficient tests do not
generally exist, and in such cases we confine ourselves to optimality
within a class of tests.
1.2.
Review of literature.
Let the distribution of X be known except
for the value taken by the unknown parameter 8.
Wald (1947) gives a
sequential test for testing H vs HI when both W and WI contain only
O
o
one point.
The test is based on the log likelihood and is known as the
sequential probability ratio test (SPRT).
The test is rather simple,
possesses some optimal properties and has a wide range of applicability.
.
3
However, in the majority of cases the DF involves unknown parameters
other than the parameter under investigation.
For example, if we are
testing hypotheses about the mean of a normal population, the variance
is usually unknown.
On the other hand, the SPRT is designed for testing
a simple hypothesis versus a simple alternative.
So if the hypotheses
do not specify completely all these parameters, the SPRT is not, at
least directly,applicable.
not known the
S~RT
Furthermore, if the form of the DF is
is not at all applicable.
Wald (1947) suggested the use of some weight functions to integrate 8 over the ranges W and WI or to integrate out the nuisance
o
parameters.
In principle, the choice of a weight function could be
made to satisfy some kind of optimal properties, but there is no
general method available for doing that.
An alternative is to consider
transforming the original sequence of observations; the transformation
to be so chosen that the new sequence does not depend on the nuisance
parameters.
Then we apply the SPRT to the new sequence.
Cox (1952)
gave sufficient conditions for the existence of such transformation and
gave an example on using 1t; see also Armitage (1947).
Since the
transformed observations are in general dependent, it is difficult to
know what properties the OC and ASN functions have.
Cox (1963) developed a method due to Bartlett (1946) which follows
from the asymptotic sufficiency of the maximum likelihood estimator
8 vs HI: 0 = 0 Cox (1963) suggested
1
0
A
1
A
using the statistic T = n{O - 2(8 + 0l)}' where 0 is the mle of O.
n
0
(mle).
For testing H :
O
8
It was assumed that °0,6
l 2
amounts of order n- / •
1
~
and the value
e of
the parameter differ by
Although the validity of Cox's test depends
on the convergence of T
n
to a random walk, this result was rigorously
4
justified later by Breslow (1969).
The statistic T
n
is asymptotically
1
a normal process with mean increment per unit time 8 - 2(8
0
+ 81 ) and
variance per step
and
where L
~)
is the likelihood function after
n observations, and $ is the nuisance parameter.
Cox (1963) then
suggested using Wald's SPRT for normally distributed observations as
an
approximate
test for testing H vs HI'
O
In the methods mentioned above it was assumed that the DF
F(x) of the r.v.
~
under consideration is known except for the value
of some parameters (8,$).
We shall be interested in cases where the
parameter of interest is a measure of some characteristic of the DF
without requiring the DF itself to be known.
For example 8 may be
the mean or the variance of the population.
So far we have been talking about testi.ng H :
O
HI:
e=
8 ,
1
8
=
8
0
vs
But we may be interested in choosing one of three
hypotheses about 8.
Let a
l
2
a
2
be two given numbers and suppose that
we want to choose one of the three hypotheses
(1.1)
Sobel and Wald (1949) modified the problem a little by subdividing the
parameter space into preference zones and indifference zones.
they suggested using two SPRT's simultaneously.
Then
Armitage (1950)
5
suggested the use of three SPRT's simultaneously.
Sobel and Wald (1949)
studied their procedure in detail when 8 is the mean of a normal population with known variances they gave an approximation to the DC function and some bounds for the ASN function.
Also Simon (1967) studied
(1.1) when 8 is the mean of a normal population with known variance; he
n
used a Wiener process approximation for the discrete process Sn = r x.
1
1
and treated it using a procedure similar to that of Armitage (1950).
This way he was able to get some better approximations to the DC and
ASN functions. However, as before, these procedures are applicable only
to cases where the DF is known except for the value of parameter 8.
1.3.
Some definitions and previous results.
Let Xl' X2 ' ••• be a
sequence of independent and identically distributed random variables
P
(iidrv) with Each Xi having the same DF F(x), x £ R , the
dimensional Euclidean space.
P
in R •
p(~
1) -
Let F be a subset of the set of all DF's
If to any F £ F a quantity 8(F) is assi~n 8(F) is called
a functional of F defined on
F. Suppose that for some sample size n
8 admits an unbiased estimate for any DF F
E
F. That is, if
(Xl' ••• , X ) is a random sample of size n there exists a function
n
••• , x ) such that
n
8(F) =
(1.2)
J ... J f(x I ,
~p
for every F
E
••• , xn ) dF(x l ) ••• dF(x n ),
F. A functional 8(F) of the form (1.2) is referred to
as regular over
F. A regular functional is also referred to as esti-
mabIe parameter.
Let
ill
> 1 be the smallest sample size for which there exists an
unbiased estimate f(x I , "', x ) of the regular functional 8(F) over
m
F.
6
Then
e(F) =
(1.3)
for every F E
J
F. We call m the degree over F of e(F). Any function
f (xl' ••• , xm) satisfying (1. 3) is called a ,kernel of e (F) •
For any
regular functional e(F) we can always find a kernel that is a symmetric
function in its m arguments.
Therefore, whenever we speak of a regular
functional we shall assume that its kernel is symmetric.
The simplest
nontrivial example of a regular functional is the mean of a r.v. X,
where m = 1 and f(x)
= x.
P
Let Xl' ••• , X be iid r.v.'s with DF F(x), x E R.
n
••• ,. xm) be a symmetric function in its m arguments.
Let
A
U-statistic is defined by
(1. 4)
Un
= Un (Xl'
••• , X)
n
L:
C
... ,
n,m
where the summation is over all permutations (aI' ••• , am) such that
1
~
a
l
< ••• < am
~
n.
The function f(x , ••• , x ) is referred to as
I
m
the kernel of the U-statistic.
If f(x l , ••• , xm) satisfies (1.3) then
U is the minimum-variance unbiased estimate of e(F).
n
For a random sample (Xl' ••• , Xn ) of n iid r.v.'s with DF F(x),
x E RP the empirical DF is defined as
(1. 5)
where c(u) is equal to one if all of its arguments are nonnegative and
elsewhere it is equal to zero.
The differentiable statistical function
(DSF) of the regular functional 8(F) is defined as
e·
(1.6)
8(Fn ) =
f··· f
mp
R
7
f(x l , ••• , x n) dFn(x l ) ••• dFn(x n)·
Therefore, from (1.2), we have
n
(1.7)
... ,
r
8(F )
n
a =1
m
The U-statistics were introduced by Halmos (1946) and thoroughly
studied by Hoeffding (1948).
The DSF's were introduced by Von Mises
(1947) and they are closely related to U-statistics.
m
n 8(F )
(1.8)
=
n
where the summation
r*
2
-e
n
f(x
al
, ••• , x~ ),
~m
extends over all m-tuples (aI' .•• , a m) in
which at least one equality a
ance a (8(F
r*
(n) U +
m n
We may write
i
=
a
j
i
~ j
is satisfied.
If the vari-
» exists then 8(Fn ) - Un = ;1 Dn , where the expected value
E(D 2 ) is bounded for n tends to infinity. [See Hoeffding (1948)].
n
Then
In
[8(F ) - U ] tends to zero in probability, as n tends to
n
n
infinity which implies that the limiting distribution of
8(F)] is the same as that of In [U
n
Let
~
n
In
[8(F ) n
- 8(F)].
be the a-field generated by the unordered collection
(Xl' ••• , Xn ) together with Xn+l , Xn+2, •.. , so that Cn is
increasing in n > 1.
non-
The following lemma was proved by Berk (1966).
Lemma 1.1.
... ,
X ) I} <
m
00,
then {U., C , n > m}
n
n
is a reverse martingale.
The following theorem was proved by Sen (1960) and Hoeffding
(1961) by considering the representation --of
-a-~e-atis·tic
combination of uncorrelated random variables.
by a linear
In view of lemma 1.1
it was proved again by Berk (1966) using the martingale convergence
theorem (cf Doob 1953).
... , X--) I} <
Theorem 2.2.
m
almost surely (a.s.) as n
~
8
00
then
00.
1
Sproule (1969) has shown that for A > 2' (Un - e(F» =
0(n- 1 / 2 (log n)A) a.s. as n ~ 00 and this result was improved upon by
Ghosh and Sen (1970) who showed that (U
n
a.s. as n
~
~
log n)1/2)
They also showed that if
00.
*=
= 0(n- 1 / 2 (10g
- e(F»
max
E[f(xa '
1<a <••• <a <.m
1
1-
-
... , xa
)]2 <
00
m
-In-'
then
[e(F) - 8(F)] = 0(n- 1 / 2 (10g log log n)1/2) a.s.
(1. 9)
as n
n
~
00.
Miller and Sen (1972) proved that under the same condition
for every £ > 0
lim p{ max k
n~
m<k<n
(1.10)
as n
~ 00.
I
8(F ) - U
m
n
I>
E
1 2
n / } = 0,
Grams and Serf1ing (1973) proved the following results:
(i) If f2r <
(1.11)
00,
p{sup
k>n
(if) If E{
I
f(x
r ~ 1, then for every E > 0
I Uk
- e(F)
, ••• , X )
a1
am
Ir } -< A <
00
for all
< am< m and r a positive integer, then
-
(1.12)
E{
I Un
-
e(Fn ) Ir
} = 0 (n-r) .
In fact (ii) is also true for any r > 1.
These results show how closely related the U-statistics and
9
DSF's are.
Therefore, the convergence of one can be obtained from the
convergence of the other with the aid of the above results.
With this
in mind we shall develop our results in the following chapters for
U-statistics where similar results·for DSF's may be easily obtained
if we replace conditions on the moments of f(X , ••• , X ) by similar
m
l
ones on f (X
al
, ••• , X ),
am
I -< a l
< ••• < a
-
-
< n.
m-
As we mentioned, Berk (1966) showed that U-statistics form a
reverse martingale.
Loynes (1970) and Scott (1971) proved invariance
principles for reverse martingales.
Miller and Sen (1972) proved the
convergence of U-statistics and of DSF's to Wiener processes.
Sen
(1972) generalized an invariance principle due to Strassen (1967) to a
class of U-statistics.
-e
1.4.
The problem and results.
Suppose that we want to test (1.1).
If the analytic form of the DF of the random variable under consideration is not known, we cannot apply any of the methods mentioned above.
We introduce a statistic similar to that of Cox (1963) and notice that
'"
1
Cox's statistic T = n{6 - 2(6 + 6 )} is a measure of the discrepancy
n
l
0
'"
between the mle 6 of 6 and the midpoint ~(60 + 6 ), of the hypotheses
1
H :
O
6 = 6 and HI:
0
6 = 6
r
We define the statistics
1
M = n[6- - -(6
+ 61 ) ] ,
n
2 0
and
1
M* = n[6(F ) - 2(6 + 6 ) ],
n
n
1
0
where
e is
the V-statistic estimate of 6 and 6(F ) is the nSF cor-
responding to O.
n
As in Tn the exact distributions of Mn and M* are not
n
known and we use their asymptotic dlstributlons to dl'velop a sequential
10
procedure for testing hypotheses about e.
totic distributions of
Sen (1972)].
e and
As we. mentioned, the asymp-
e(F )have been developed [cf Miller and
n
But here we are more interested in testing hypotheses
about continuous functions of one or several estimable parameters.
Let g(&) = g(e(l), ••• , e(q»
be a continuous function in the
regular functiona1s e(l), ••• , e(q), q
the variance ratio
g(Q) is
°12/° 22 of
~ 1. For example g~) may be
two random variables.
estimable then we have the same situation as in the previous
paragraph.
We are interested in cases where
That is there may not exist a function f(x ,
1
g(])
If the function
=E
f(X , ••• , X ).
n
1
g(~
may not be estimable.
... ,
x ) such that
n
Now, we want to choose between the three
hypotheses
where a
and a
1
2
are given.
In Chapter II we study the asymptotic distribution of the random
variables g(U(l), ••• , u(q»
n
n
. estimate of e(i), i
conditions g(U )
-n
= 1,
as n ~
••• , q.
= g(u(l),
n
is the U-statistic
, where U(i)
n
We prove that under some regularity
••• , U(q»
n
. h
'
var~ance
per un i t step y 2 •
w~t
00
converges to a Wiener process
Then we find a consistent estimate of
2
) is similar to that of
Y . The invariance principle we prove for g(U
-n
Strassen (1967).
g (e
(1)
e(i),
We also show that same principle holds for g(O(F »
(F), ... , e(q)(F
n
i
= 1,
-
n
»,
n
=
where e(i) (F ) is the nSF corresponding to
n
•.. , q.
It is to be pointed out here that we are not interested in transforming a problem from testing hypotheses about some parameter to
testing corresponding hypotheses about a function of that parameter.
11
For one thing the problem is not invariant, in general, under continuous transformations.
However, we are interested in testing hypotheses
about a function that is of interest in itself, but the function is not
estimable and it is a function of several estimable parameters.
In Chapter III we develop a sequential procedure similar to that
of Sobel and Wald for the problem (1.13).
We study the operating
characteristic of the procedure and obtain a simple formula for some
limiting process of the OC function.
We also give some rough bounds
for the ASN function.
In Chapter IV we give some examples of the proposed test and
study, numerically, the ASN function of the test.
-e
CHAPTER II
ASYMPTOTIC PROPERTIES OF U-STATISTICS AND VON MISES'
DIFFERENTIABLE STATISTICAL FUNCTIONS
2.1.
Introduction.
As mentioned in Chapter lour main purpose is to
develop sequential procedures for testing more than two hypotheses.
We
proposed to use the U-statistic estimates of some regular functionals
as a basis for our test.
For the development of the test we need
certain mathematical tools.
These include:
(1) the asymptotic distri-
bution theory of continuous functions of several U-statistics,
(2) almost sure convergence of estimates of the variances of these
functions, and (3) almost sure convergence of sequences of these functions to Wiener processes.
These results will be studied in this
chapter.
,
2.2.
Notations.
Let Xl' X2 , ••• , X be n iid r.v.'s having DF F(x),
n
Let f(x , ••• , x ) be symmetric in its arguments.
l
rn
Let 8(F) satisfy (1.3) and define U as in (1.4).
n
For c
= 1,2, ••• ,m
define
(2.1)
2
Suppose that E f (Xl'
(2.2)
••• , X ).
m
... , Xrn )
0,
~
c
= t
<
c
00
(F)
and define
13
If for some F
F = {F:
£
/S(F)
(2.3)
0 <
I
< oo}
~l(F)
<
00
If U(l) and U(2)
8(F) is said to be stationary of order zero at F.
... , xm )
are two U-statistics with kernels f(l) (xl'
~
(2.4)
= 1,
and
m
2
1
c
n
1
) d e fi ne
f (2) (x , ••• , x
n
(1,2) _.
(1)
c
c
- E{f
(2)
(X , ••• ,X) f
1
c
c
(X , ••• ,X )},
1
c
2, ••• , min(m ,m ).
1 2
~
For every h(l
h
~ m)
define
(2.5)
so that
n
(2.6)
n V 1·
n,
= . L1
[f (X ) - 8(F)].
1
J=
j
Write
(2.7)
= Vn,l ,
Un,l
and for 2 < h < m define
(2.8) Un h = n-[h] Lp
,
where Pn,h = {1
~
i,
n,h
~
f f
•••
RPh
•.•
~ i
h d[c(x.-X .) fh(x1,···,~).7T
i
J=l
h
~
-[h]
}
nand
n
Then we can write
(2.9)
8(F ) - S(F)
n
and
(2.10)
U n
8(F)
=mU
n,l
+R,
n
J
J
F(x.)]
J
= [n ••. (n - h + 1]
-1
•
14
m
m
where R = E (h) U R. The decomposition (2.10) was given by
n
h=2
nJ
Hoeffding (1961) and he showed that {(n) U h n > m} satisfies the
nJ
h
martingale property.
Miller and Sen (1972) showed that {Un h J Sn n > m}
J
is a reverse martingale.
(2.11)
<S
=
h
... , m define
= lJ
For h
Eh- l (_l)c (h)· sh •
c=O
c-c
Sproule (1969) proved that if E[f(Xl'
(2.12)
Var(Un h)
=
2.3.
Cov{U
nl'
... , X )]2 <
-1
(n)
J
(2.13)
-
<Sh
h
h J Un , hI} =
2
Weak convergence of U-statistics.
m
= O(n-h )J
Var(Un , h)
l
00
then
h = lJ ••• , m,
h = hI = 1, ••• ,m
h t: hI
0
1, .•. ,m
e·
Let S be a metric space and
L be the a-field generated by all open sets of S. Let Pn and P be
probability measures on L.
I
(2.14)
If
ydP n
Iy
+
S
dP
S
for every bounded, continuous real function y on S, we say that P
n
converges weakly to P and we write P -> P.
n
This concept of conver-
gence is studied in detail by Billingsley (1968).
metric p(x,y)= Ix-yl =
I~
On RP consider the
(xi - y.)2 Il/2, and let RP be the
i=l
p-dimensional Borel sets in RP •
J.
On RP weak convergence of probability
measures and convergence of distribution functions are equivalent.
on general metric spaces this is not necessarily true.
Consider the space C[O,l] of all continuous functions on
But
15
I
=
[0,1], and associate with it the uniform topology
p(X(o), yeo»~
= suplx(t)
- Y(t)l,
tEl
where both X and Y belong to C[O,l].
Loynes (1970) obtained an
invariance principle for reverse martingales and cited without proof
U-statistics as an example to show that a process obtained by linear
interpolation from {n
1/2
~
[Uk - 8(F)]; k
Wiener process in C[O,l].
n} converges weakly to a
Miller and Sen (1972) have shown that under
"the same set of conditions, processes obtained by linear interpolation
-1/2
from {n
k[U
k
- 8(F)], m ~ k
~
n} converge weakly in C[O,l] to a
Wiener process.
Let S
·e
o<
= {Set)
(2.15)
S(k)
and Set)
= Sk
for k
~
t < oo} be a random process, where
= Sk =
{
o -< k -< m -
0,
k[U - 8(F)],
k
t < k + 1, k > O.
k
~
1
m,
Consider a positive and real
valued function yet), t E [0,00), such that
(2.16)
yet) is t but t
-1
yet) is
+ in
t
0 < t < 00;
and
r [y(cn)]-l E{[f~(X )]2 I([f~(X )]2 > Y(cn»}
(2.17)
n>l
n
*
for every c > 0, where flex)
cator function of a set A.
process.
n
= flex)
Let
<
00
- 8(F) and I(A) denotes the indi-
{~(t)
: 0
~
t < oo} be a standard Wiener
Sen (1972) proved the following extension of Strassen's
(1967) first a.s. invariance principle.
Theorem 2.1.
If 8(F) is stationary of order zero and
then under (2.16) and (2.17), as t
-+
00
~
m
(F) <
00,
16
(2.18)
where y
2.4.
g(x)
= Y ~(t) +
S(t)
2
= m2
~l(F)
o(t y(t)
1/4
log t) a.s.,
(> 0 by stationarity of 8(F) of order 0).
Functions of several U-statistics.
= .&(x (1) ,
••• , x
neighborhood of ~
=
(2)
), q
~
Consider a function
1, that does not depend on n and in some
(8(1), ""
8(q»
q is continuous and has continuous
first and second order partial derivatives with respect to iidrv's with
x
(1)
, ••• , x
(q)
•
= 1,
For i
functional of degree m. over
1
That is, for each i
= 1,
••• , q let 8(i)
F
= {F:
=
8(i) (F) be a regular
18(i)(F)1 < 00, i
=
l, ... ,q}.
••• , q, i f Xl' ••• , Xm are iidrv's with
i
DF F
E
F, then there exists a symmetric function f (i) (xl' ••• ,x )
m
such that
e·
(2.19)
For a sample (Xl' ••• , Xn) n
(2.20)
U(i)
n
~
... , mq ),let
m*= max (m l ,
= U(i)(Xl,
••. ,x )
n
n
,
am.
<
~
n},
... ,
1 ~ i ~ q.
1
x)
We define
c'
and (2.4) respectively.
(2.21)
o<
(2.22)
Write U = (u(l)
-n
n '
1
1
We assume that
~(i)(F)<
1
~(i) (F) <
m.1
... ,
~(i) and ~(i,j) similar to (2.1), (2.2)
u(q»
n
00,
00
for all F
E
F , and i = l, .•• ,q.
i = 1,
...,
q•
and 8 = (8 (l)(F) ,
... ,
e(q) (F».
17
By almost sure convergence of a U-statistic to its expected value
and (1.11) we can find nO large enough such that for n
~
~
nO'
lies in
the neighborhood of 6(F) where g is continuous and has first and second
ag(.§.)
order partial derivatives.· We shall write
ax(i)
Then for n
~
g(~)
nO we can expand
~
about
ag(lf)
ax(i)
to mean
x=8
to get
q
(2.23)
g(U) - g@
-n
= r
+
i=l
i i (U(i)
1
2 i=l j=l
where
i'
n
is some point in the neighborhood of 8 referred to above.
Then, by (2.10) we can write
·e
(2.24)
g (U ) - g @)
-n
ag (8)
q
= r
U
i=l mi ax(i)
-1
q
q
r
r (u
2 i=l j=l
(i)
n,l
(.)
1
-
mi
q
+·r
r
i=l R=2
a
(i)
n
)(u
(j)
(2.25)
<
E
h
ax(i)
(.)
+
n,h
2
- 8 J )
n
Assume that there exists E > 0 such that for every
lli - il
m dg(.§.) U(~)
( i)
...
a g@')
~~
ax(i)ax(j)
~
such that
we have
I<
ax
Iag(1Y
(i)
-
M
1
<
00
i
= 1,
... , q,
and
(2.26)
i
= 1, ... ,q,
j = 1, ... ,q,
I~
By (2.26) we have for all i,j = 1, ••• , q,
-
~I <
E.
18
I i ·i
(2.27)
i=l j=l
Ghosh and Sen (1970) have shown that under (2.21) and (2.22) we .have
for i = 1, ••• , q.
(2.28)
a.s. as n
Thus, as n
-+
00.
00
q
(2.29)
-+
q
e·
=
ILL
i=l j=l
O(n-1 log log n) a.s.
Therefore, we ·have just proved the following lemma.
Lemma 2.2.
zero and
(2.30)
~(i)
m
<
If e(i) (F), i = 1, ••• , q are stationary of order
i = 1, .'"
00,
q, then under (2.26) we have as
n -+
00
agee)
(i)
q
mi m agee) U(i)
mi ---(r.i~)- U 1 + L
E (hi) (0)
h
i=l
ax
n,
i=l h=l
ax 1
n,
q
g(U) - gee) =
-n
-
L
+ O(n-1 log log n)
a.s.
Now, recall the definition of the a-fields C cited in Section
n
1.3.
The following lemma is due to Miller and Sen (1972).
Lemma 2 ••
3
For every h (1 < h < m ) {U (i) C , n _> h} f onns a
1
n,h
n
reversed martingale sequence.
0
This implies that
,
19
(i)
mi
ag(~ u(i)
{U * = r (mi )
C , n > mil forms a reversed martingale
n
h=2 h
ax(i) ,n,h' n
-
sequence.
Let c be a sequence of positive numbers such that c k is
k
2
3/2
t in k and c = O(c
). We prove the following.
k
k
Lemma 2.4.
If E[f
(i)
max c U~;)
k>n k
(2.31)
Proof:
+
(Xl' ••• , Xm )]
c
2
<
00,
0 in probability as n
then for i = l, ••• ,q,
+
00.
By (2.21) and (2.22) we have
(1) 2
E[Un* J
mi
= r
h-2
m 2
(hi)
and
~i
E [u(1) _ U(i) ] 2 =
(mi ) 2
n*
n+l*
h=2 h
[ag (i)
@)]
2
ax
[a g ®]
2
(i)
ax
[Var U(i) _ Var U(i) ]
o,h
n+1,h
ag ®t
[(h)
[ag®f
(n)
h
rg(~f
h
(0)-1 °
o + 1 h
h
2
m
= r i (mi ) fax(i)
h=2 h
=
2
m
r i (mhi )
ax (i)
h=2
=
m
2
r i (mh i )
ax (i)
h=2
=
n -1
-2
(h)
0h = O(n ),
n -1
-1
0+1 -1
°h - ( h)
°h]
[1 _ n+1-h]
n+1
O(n-3 ).
Then
If we now apply Theorem 1 of Chow (1960) [i.e. the Hajek-Renyi inequality
for submartingales] we obtain for every E > 0
20
· {max
I (i)
P
C U
k>n k k*
I
>
£
}<
£
-2 {<Xl
2
(i)
(i) 2}
r c E[Uk* - Uk+ l *]
k=n k
<Xl
0(n- 3 / 2)
= £-2 r
k=n
as n
-+
<Xl,
and the lemma follows.
From (2.30) and by lemma 2.4 we can write that
q
ag@) (i)
1
log log n)
U ) - g@)=rm
g(-n
(i)
U
+
O(C)
+
O(
i=l i ax
n,l
n
n
(2.32)
=
Wri te
G(X ) =
j
i
i=l
ag@) (i)
-1
r mi
(i) U 1 + O(C ), as n
i=l
ax
n,
n
q
m ag@)(f(i)(x)_e(i)(F»
i ax(i) 1
j
,
iidrv's
(2.33) y
2
= E[G(X )]
j
Thus, as n
-+
2
-+
<Xl •
j = 1, •..
,n.
with E G(X.) = 0 and
J
q
q
ag@) ag@)
= r
r
~(i,j) j
i=l j=l mim j ax(i) ax(i) 1
'
1, •.. , n.
<Xl
n
(2.34)
n[g(U) - g(8)]
-n
-
= r G(Xj ) + O(n C- l ).
j=l
n
We now prove a theorem similar to Theorem 2.1.
positive and real valued function yet), t
(2.35)
yet) is t but t- l yet) is
£
+ in
Consider a
[O,<Xl) such that
t : 0 < t < <Xl,
and
-1
(2.36)
r [y(cn)]
n>l
2
E{[G(X )]
l
> y (cn)} <
w
21
for every £ > O.
S(k)
: 0
~
t < oo} by
, 0< k < m-1 = max(ml, ••• ,m ) - 1
0
(2.37)
= {set)
Define a process Set)
-
= [. k[ ~
j=l
-
q
G(X.)]
J
k
~
m,
and Set) = S(k) for k < t < k + 1, k > O.
If a(i)(F), i = l, ••• ,q are stationary of order
Theorem 2.5.
~i)(F) <
zero and
as n
i = 1, ••• , q, then under (2.35) and (2.36)
00,
~ 00
Set) = y t(t) + o«t
(2.38)
where
~(t)
1/4
log t) a.s.,
is a standard Wiener process and y is defined in (2.33).
Proof:
-e
yet»~
Take c
= n[(ny(n»1/4
n
log n]-l, then from (2.34)
n
n[g<.!ln) - g@)]=
E
j=l
+ o«(nY(D»114 log n).
G (X )
1 j
Since G(X ) and iidr.v.'s the theorem follows directly from
j
Theorem 4.4 of Strassen (1967).
Now, if we choose yet) = t(log log t)2/(10g t)4 and (2.36)
holds, then we have
(2.39)
If
Set) = y
yet)
=
~(t) + 0(t 1 / 2110g log t)
(log log t)4
t - - log t
a.s.
and (2.38) holds, then
(2.40)
2.5.
a.s. as t
2
Estimation of y .
In Cox's (1963)
formu1ati~n·he
the m1e of a in the quantities I aa , I a ¢ and I¢¢.
2
need to find an estimate for y
q
= E
E m.m.
i=l j=l
1
J ax
substitutes
To use (2.38) we
dg(.!V
q
~ 00.
(')
1
ag<.~)
ax (i)
~(i,j)
1
•
22
Since y
2
depends on
i
~~i,j)
We can estimate
i
we substitute for
its U-statistic estimate U •
~
by its U-statistic but Sproule (1969) has shown
that the U-statistic estimate for ~(i,j) is hard to compute.
1
Sen
(1960) decomposed a U-statistic as follows:
1
(2.41)
U (X ' ••• ,X )
n 1
n
n
E
=-
n a=l
V (X ' ••• ,X ),
a 1
n
where V is defined by
a
* f (Xa , Xa
(2.42)
E
and the summation extends over 1
j
= 2, ••• ,m.
estimate of
~1.
2
< a
<
= 1, •.. ,n.
a
),
< n but a.
moj
# a for
We use a similar approach to obtain consistent esti-
= 1 , .•. ,q
For i
V (i)
a
=
n 1
(m .=1 )
-1
E*f(i)(X,X
a
~
2, ••• ,m i •
= 1 , •.. ,n
we d e fi ne f or a
where the summation extends over 1
f or j =
a
am
He utilized the decomposition to obtain a consistent
mates of s~l(i,j).
(2.43)
~
, ••• , X
2
Thus U(i)
=1
n
n
~
a
2
a
, ... ,X
a
2
).
m.
~
~
< ••• < am
n but a
i
~~,
V(i) i = 1, ••• ,q.
a
a=1
# a
j
Define
(2.44)
We propose to use
(2.45)
"2
Y
n
q
dgG!n)
i=l
dx(i)
L
=
.6 ••
~J
2
as an estimate for y •
dg (.!2.)
Now, by continuity of
(i)' i
dX
there exists 01 > 0 such that
= 1, ••• ,q,
for every
£
> 0
23
ag@)
ax(i)
where
II u
-n
-
&..11 =
<
-
II-n
U
whenever
E
IU(i) - ~ (1) I.
max
1<1<
n
--q
- 8 II < 0,
-
-
Therefore, for every
"2
2
there exist 01 > 0 and 02 > 0 such that IYn - Y
Illln - &..11 ~
max
l<i,j<q
01 and
II~ij
~~i,j) II ~
-
I~
E
E
> 0
whenever
~?,j) II =
02' where 114 ij -
~4i. - ~~i,j)l. Thus
J
-
(2.46)
The following theorem was proved by Grams and Serfling (1973).
-e
Theorem 2.6.
Let {U } be the sequence of U-statistics generated
n
by a kernel f applied to a sequence of observations {Xi}.
E f
2r
<
00
for a positive integer r.
(2.47)
p{sup
k>n
IUk
-
81
Then for every
> E}
£ >
Assume that
0,
= 0(n l - 2r ).
It can be easily shown [by Theorem 2.1 of Grams and Serfling (1973)]
that if E f2r <
00
then
(2.48)
We now utilize (2.47) and (2.48) to prove the following.
Theorem 2.7.
is finite for some r > 1.
(2.49)
(i)
2r
l, ... ,q E[f . (X1, ... ,X .)]
m
1
Then for every E > 0
Assume that for
.
~ =
24
Proof:
Then ~(i,j)
1
(2.50)
Let ~i;,j)
= E fi i ) (Xa ) fij){Xa ) for any
1
~ a ~ n.
= ~(i,j) _ e{i)e{j) and
1*
•
_ c-{i,j)
""ij
"'1
Therefore
(2.51)
If e{i)
= 0 but e{j) ~
0, then
e(by (2.48»
= O{n-r ).
If e(i)
= 0 and e(j)
c
0 then
25
= O(n-r ).
If e(i) ~ 0 and e(j) ~ 0, then
We have
(by 2.56)
= O(n-r ).
Also by (2.48) we have
Therefore
(2.52)
By
definition of v(i) and v(j) we have
Ct
Ct
26
where the summation
r extends over 1
c
a~ ~
a,
~
= 2,
~
a
< ••• <
2
n but
~
~
i
••• , m and the summation
i
am.
r
extends over
c.
J
... < am
~
n
b~t S~;
a,
~
= 2,
j
••• , m••
J
For c
= 0,1, ••• ,
ewhere
/
common.
The number of elements in A , for c
c
i
s equa 1 to (mn-1)
-1 ( n-m i ) (m~_l).
~
c
i
m -c-1
= 0 = 1, ••• ,min(m.,m.)
~
J
Thus, assume m < m.,
i -
j
(2.53)
o(c) U(i,j)
c
'
J
- 1
,
27
h
0()
were
c = (n-l)
m.-l
J
-1
n-m i
m -1
(m -c-l
) ( ic )
=
j
U-statistics with kernels
ij _ (i)
(j)
fc - f
(Xl, ••• ,X)
f
(Xl,···,Xc+l'Xm +l' .•• 'Xm.
m
1 ).
i
i
.
J-ci. r
Also E(f J) <
i
00
( . ) 2r
if E(f 1)
<
00
and E(f
(j) 2r
)
<
Therefore
00.
m.-l
t(i,j) = ~
1*
i=O
o(i) [u(i,j) _ EU(i,j)]
c
c
m -1
i
+ L
c=O
-e
o*(c) E U(i,j)
i
'
where 0*(0) = 0(0) - 1 = O(n- l ) and o*(c) = o(c),c=l, ... ,m.-l.
1
Therefore
Then, by (2.48), (2.52) and (2.54) we get (2.49).
Now, by (2.46), (2.47) and (2.49) we have the following.
Theorem 2.8.
then
If for i
= l, ••• ,q, E(f(i»2r
<
00
for some r > 1
28
A2
(2.55)
p{ly
Theorem 2.9.
n
2
- y I > E
If for i
}
•
I
= O(n-r/2 ),
= l, ••• ,q
E(f
(i) 2
)
<
00,
then
(2.56)
Proof:
By continuity of ag(x)
ax(i)
and convergence almost surely of
U-statistics we have
ag@)
+ -..,.-,..
(2.57)
ax(i)
a.s.
as n
+
00.
Also, from (2.50), (2.53) and convergence almost surely of U-statistics
then
(2.58)
Hence the result.
•
'"'ij
+ /:(i,j)
"'1
a.s.
as n
+
00.
e-
CHAPTER III
TESTING HYPOTHESES CONCERNING ESTIMABLE PARAMETERS
3.1.
Introduction.
Let X 'X ' ••• be a sequence of iidr.v. 's, with
l 2
P
DF F(x), x £ R , P ~ 1. For i = l, ••• ,q let e(i) = e(i) (F) be a
regular functional over some subset F
= {F:
le(i)
I
<
00,
i
= l, ••• ,q}
of the set of all DF's in the p-dimensional Euclidean space.
a function
Consider
g~) = g(x(l) , ••• ,x(q»that is continuous and has continuous
first and second ordered partial derivatives with respect to x
(1)
, •.• ,
(3.1)
The problem was studied by Sobel and Wald (1949), Armitage (1950)
and Simon (1967) when g@)
= e,
tion with known variance.
In this chapter we develop a similar pro-
and e is the mean of a normal distribu-
cedure to that of Sobel and Wald (1949) for (3.1).
First we develop a
testing procedure for
(3.2)
H' •
1"
gee) = g o -
6.
vs
H'2·•
where 6. > 0 and go are specified, and study its properties.
utilize this testing procedure to study (3.1).
Then we
In fact we are going
to study (3.1) only when go' gI and the true value of the parameter
g(8) differ by small amounts.
30
The sequential procedure for testing (3.1) may be simply described
as follows.
We establish a testing procedure R for testing (3.2) and
l
a similar procedure R for testing
2
(3.3)
vs
H3·
' •
H4·
' •
g(~ =
gl + 6..
Both procedure R and R are computed at each stage of inspection until
2
l
Either:
one of R and R leads to a decision to stop before the other;
l
2
then the former is no longer computed and
it leads to a decision to stop,
Or:
th~
latter is continued until
both R and R lead to a decision
l
2
to stop at the same time; in this case both procedures are discontinued.
We are going to show later that under some proper choice of constants
a and b it is impossible for R to accept
l
Hi
while R accepts
2
H;'.
Then there are three possibilities for the final decision R.
R accepts
l
R accepts
R accepts
2
.If
g
- 6.
and
gl - 6.
then
HI
If
go + 6.
and
g1 - 6.
then
H2
If
go + 6.
and
gl + 6.
then
H3 ·
3.2.
0
Two hypotheses problem.
functional of degree m , i
i
functions f(i)
= l, ••• ,q
Suppose that e(i)(F) is a regular
= l, ••• ,q.
Then there exist symmetric
such that (2.19) holds for i
= l, ••• ,q.
Let U(i) be defined by (2.20) and suppose that (2.21) and (2.22) hold.
n
We want to test
(3.4)
where 80 and 6.(>0) are specified.
31
For a single parameter
e Sen
(1973) gives a method for construct-
ing sequential procedures for testing
e = eo = b,
His test is based on the U-statistic estimator of
b, known.
e,
and it enjoys the
optimality of Cox's (1963) sequential likelihood ratio test as b, tends
to zero.
The same technique is used here to formulate our proposed
sequential procedure and to study its various properties.
,We start with initial sample size nO
for small b,.
We shall take nO(b,)
= b, -1 ;
= nO(b,)
moderately large
then for n
~
nO we use the
following procedure.
Procedure
b < 0 < a.
~
f.
We choose two real numbers a and b such that
Define a stopping variable N(b,) as the smallest integer
n (6) for which one of the following inequalities is violated.
O
(3.5)
Accept H_ l i f for N(6)
for N(6)
= n,
n
=
ll[g(~)
"2
and accept HI if
n, n ll[g<J!J - gO] -< by,
n
"2
- gO] ~ a Yn •
Theorem 3.1.
Under conditions (2.21) and (2.22) we have
Proof:
~
(3.6)
•
For n
(3.7)
If g@)
= go
then
nO(b,) we have
32
g(8»<
-
a Y
b.
n n- 1 / 2}
By Hoeffding (1948), the convergence of Y to Y (in Theorem 2.9) and
n
~
with the Slutsky theorem (cf Cramer (1946»
Yn
[g(U
) -n
g(~)]
is
asymptotically normally distributed with mean zero and unit variance.
By continuity of the normal distribution and Theorem 2.9 we obtain
that
b Yn -1/2
IIi
P8 { b.
n
< "" [g <.Yn)
Yn
+
0 as n
+
00.
Then P8{N(b.) > n}
+
- g® ]
0 as n
~
< a Yn n-l/2}
IJ.
00.
On the other hand, if
g@) :f: go' then by (3.7) we have
(3.8)
By Theorem 4 of Sen (1960) we have g(U ) a.s. g(8) as n ~
-n
~
-
00.
Hence
for g@ :f: go' [g(.!!n) - gO] tends, almost surely, to a value different
from zero.
Then by Theorem 2.9 and (3.8) we have P {N(IJ.) > n} con8
verges to zero as n
~ 00.
3.3.
We consider the asymptotic situation where we let
b.
~
DC Function.
O.
Hence the theorem.
The results should provide a good approximation for practical
situations where IJ. is fixed but small.
A reasonably large initial
"2
sample size nO(IJ.) is needed to insure the accuracy of Y as an estimate
n
2
of Y for all n
(3.9)
~
nO(IJ.).
We assume that
=
00
but
33
Let F_I,O(X) be the unknown DF of the Xi's under the null
tive HI:
= go
g(,ID
= go
g~)
hypothesis H_ I :
+~.
~
-
Let
and FI,O(x) be the DF under the alterna~@) denot~
the OC function of the
proposed test; Le. LF®is the probability of accepting H_
g<.~.>
is the true value.
LF<'~'>
A rough approximation for
I
when
would be
where
This approximation is rough because it involves three approximations;
one is approximating g <.!!n) - g~) by a Wiener process, another is
2
"2
approximating y by Y and a third is neglecting shooting over the
n
boundaries.
Moreover, it can be shown that for every g@) ;. go'
the OC function converges, as
~ +
0, to
°or 1 according as g is
>
or < gO.
To avoid such limiting degeneracy we let
(3.10)
g(..@.)
LF(4),~)
We write
g(..@.)
go + 4>~,
=
= go +
4>
~
4> e: I
= {4>
and we write
Theorem 3.2.
£
F4>,~
to denote the DF in this case.
~(4),~)
as
~ +
o.
Under conditions in Theorem 3.1 we have, for
I,
e
(3.11)
00.
to denote the OC function of the proposed test when
now study the limiting behavior of
every 4>
: 14>1 < k} for some 1 < k <
e
-2a4>
-2a4>
a
a - b
- 1
- e
-2b4>
4>
= 0.
We
34
Proof:
We have
(3.12)
/
g~)
Since we do not know the DF of
to a Wiener process.
exists a to
(3.13)
we now make use of its convergence
By (2.40), for every E > 0 and n > 0, there
= to(E,n)
such that
2
t- 1 / IS(t) - y ~(t)1 > E/2} <~.
p{sup
t>t
o
Also, by Theorem 2.9 there exists an nO(E,n) such that
(3.14)
P {max
n>nO(E,n)
~O
Thus we can take a
Iy"2
- y
2
n
= ~O(E,n)
I > '21
d <~ •
such that
(3.15)
Thus t
~
2
~
E for all t E[O,
o > 0, to > 0 and
~
nO(~l)]'
where
~ E[O,~O].
Now, for every
E I, define
~(t) 2 o-~ + to~ for smaller t ~ to
(3.16)
p(~,o,a,b,tO)
= P than any other t
~
to for which
~(t) ~ o-l a + to~.
Then, by Anderson (1960) we have
e2~a
(3.17)
P(~,
0, a,b,O)
=
e
2~a
a
a - b
_ 1
- e
2~b
~ =
O.
35
Define
Thus, from (3.12) - (3.16) and (3.18) it follows that for every
g(~)
!1 £[O,Ll O]' and
= go +
epLl, ep
£
1.
By the Levy inequality, the probability that
{~(t);
0
~
t
~
nO(Ll)}
crosses either of the two lines (Y b (j) _ t ep Ll) or (Y a (i) _ t ep Ll)
Ll £
Y·
Ll £
Y
for i,j
= 1,2
can be made smaller than n'(>O), where n'
+
0 as £
+
O.
Thus by (3.19) we get
·e
+ n + n'.
Now, P(ep,c,a, b,O) is continuous in a and b, so by choosing £ and
n
sufficiently small, both sides of (3.20) can be made arbitrarily close
to (3.17).
3.4.
ASN Function.
Now, for
(3.21)
Hence the theorem.
£
By (3.9) we have, as Ll
O.
+
> 0,
P{N(Ll) > n} < p{~2 b < nil[g(U ) _ g ] < ~2 a}
-
n
-n
0
n
36
. A2
A2
A2
2
2
+ P{yn < n.6[g(U
) - gO] < Yn a, Y - Y _< Y d
-n
~
. {A2
2
2}
.{ 2
P Y - Y > Y € + P by (l+€) <
n6[g(~)
- g6]
2
< ay (l+€)}.
Let g@
= go
+
eM, ep €
Then
1.
A2
2
2
Pa {N(6) > n} ~ Pa{y - y > y €}
(3.22)
2
+ Pa{by (l+€) - n ep 6
2
< n6[gGl.n) - g(.§.)]
2
2
< ay (l+€) - n ep 6 }.
Consider the case of ep # 0 and let for every 6 > 0,
n
(3.23)
ep,6
=
2
2
o<k
k X
6 2 ep2
<
e-
00.
2
By proper choice of k, y (a-b)(l+€) can be made smaller than ~lepI6.
Thus the second term on the right hand side of (3.22) is bounded above
by
By continuity of g@), for every € > 0 there exists 0 > 0 such that
Ig (.!!n)
- g (@)
I<
E
whenever
Ilu-n - -e II
<
-
o.
Therefore,
p{/g(U)
- g(i>! > d
-n
<
-
p{llu- n
- -ell> a}.
By Theorem 2.6 i f we assume that E(g (1) ) 2r <
then p{sup I~
k>n
- ~I
l 2r
> o} .0(n -
).
00,
i
= 1, •.. , q, r
> 1
38
(3.28)
R
n
= gfU
~
) - g(]) -
q
E m
i=l i
dg(!>
dx(i)
U(i)
n,l
* 1) ,
= g (U
) - . g (U
~
--n, 1) + (g (U
--n, 1) - g <..~) - Un,
(r)
where
*
(9)
dg(8)
m
i=l i dX ()
i
q
U 1 = (Un, l'···'Un, 1) and Un, 1
-n,
=
U(i) .
n,l
E
Write
(3.29)
and
(3.30)
e-
Then
R
n1
(3.31)
=
q
dg(A)
E (U(i) _ U(i» ---~
i=l
n
n,l dx(i) ,
where A is some point between U and U 1.
41
--n,
Also,
(3.32)
where!' is some point in the neighborhood of continuity of g(.)
around 8.
Let
A~,£ = [II~ - Jill ~
Bn,£
= A'n,£ n
2
An,£ •
P{Bn ,£}
~
£],
A~,£
=
[11~,1
-
Jill ~
£], and
Then
p{II.Yn,l - ~II
> d + p{II~,l
- ~II
> d.
39
If we assume that E(f(i»2r < 00, r
=2 +
6, 6 > 0 then by Grams and
Serfling (1973)
(3.33)
and
(3.34)
Thus
piAn,E } = O(nl - 2r ) '
(3.35)
and
·e
(3.36)
Now,
1 2
P{k / /11t1 1 >
n}
=
1 2
P{k / 111t1 I >
1 2
+ P{k / /11t1 >
and therefore
n,
Bk,E}
Bk,E}
2
P{k1/2/11t11 >
n,
Bk,E} + P{Bk,E}
=
P{k1/2IR11
>
-It
n,
Bk,£ } + O(k l - 2r ).
ag(~)
In Bk
(i)
,£ ae
n,
is bounded and we have
40
= O(k- r ),
r =
2
+ 0.
Thus
(3.37)
Similarly
= P{kl/21~21
n, ~,E} + O(nl - 2r ),
i (U(i) i=l k,l
< P{k l / 2 1
-
>
8(i»1 2 >
n'}.
Now we can write
o < 01
-1/2+2°1
For large k, k
can be made arbitrarily small and we have
< 1/4.
41
q
n
()
i=l
j=l
= K L p{1 L
(f i
(X) _
( )
e i )1
> n"'k
1/2+0
1}.
j
Since E(f(i»4+20 < 00, then we can apply Theorem 2 of Katz (1963) to
get that
(3.41)
Therefore, by (3.38)
(3.42)
We now combine (3.28) - (3.30), (3.37) and (3.42) to get
(3.43)
Proof of Theorem 3.3.
Define two stopping variables N~1){6) and
N~2){6) as the smallest positive integers ~nO(6» for which
q
agee)
2
2
n[ L m
- u(i) + w 6] is not contained in {X (l-£)b_ X (l-£)a) and
i=l i ae(i) n,l
6
6
Then, similar to (2.26) we have
(3.44)
Now
(3.45)
2
q
ag{.Q.)
P {N{6) > n} ~ Pe{O bi 1+£) < k r fi
U(i) + k w A
e
i=l i dx(i) k,l
<
2
X a(l+£)
6
' nO _< k _< n }
..
42
for at least one k:
(2)
= P{N~
nO(6)
"2
k
~ n~,6'
6 k
1/2
~
n},
2
v £
nO ~ k ~ n},
for at least one k:
~
k
(6) > n} + Pe{Y k - Y > ~
for at least one k:
Now, for n O(6)
2
~
nO ~ k ~ n}.
is bounded and therefore by (3.43)
we have
(3.46)
P{6kl~1 > £/2 for at least one nO ~ k ~ n}
= O(n-0'
), 0'
O
Also, by (2.55) we have
(3.47)
Then~
P{~~ -
y
2
>
X:£
for at least one k:
nO(6)
~ k 2 n}
by (3.45) we have
Similarly,
P~{N(6)
(3.49)
Since 6
6
-+
0
2
n~,6
> n}
~ P~{N~l)(6)
> n} -
-0
is bounded and nO (6)
-+
O(n~0(6».
0 as 6. -t--O, we have for
> O.
43
(3.50)
where n can be made arbitrarily small by choosing
~
small.
N~i)(~) is based on a summation over independent
Now, since
random variables, then if we neglect the excess over the boundaries for
small
~,
and use Theorem 3.1 then by Wald's (1947) fundamental identity
we have
liml~2E~N~j)(~) -
(3.51)
+ (-l)j£) ~ (~,y)1
(1
~-+O
~(~,y)
where
=
= 0, j = 1,2,
y2~-1 [bP(-~) + a(l - P(-~)],
~ ~ 0
_y2p '(0) (a-b),
~ = O.
Then, the theorem follows for e~o by choosing ~ small to make, by (3.51)
(3.52)
For
~
=0
we note that
neighborhood of
~
= 0,
sequence of values of
P(~)
is continuous and differentiable in some
so that p'(O) exists.
~,
say
~
=±
£~,
~ ~
Hence, by considering a
1 where
£~ ~
0 as
~ ~
and then using the above proof, we obtain by L'Hospital rule that
(3.53)
lim{~2EON(~)} = lim ~(~~,y)
~~O
~~
= -y 2p'(O)
Hence the theon!l1l.
(a-b).
00,
44
3.5.
Three hypotheses problem.
Suppose we want to choose one of the
hypotheses
where go and gl differ by a small amount.
Consider the following problems for 15. > o.
that go .. -gl·
(3.55)
There is no loss in assuming
Testing
H' •
1·
g@
= g0
- 15.
vs
H' •
2·
g@)
= go + 15.
Testing
H'3·•
g@)
= gl
- 15.
vs
H' •
g(Q.)
= gl + 15..
and
(3.56)
3·
Let R and R be two procedures similar to procedure P described on
1
2
page31with (a,b) and (a',b') respectively.
Let N and N be the
1
2
stopping variables corresponding to R and R respectively.
1
2
to test (3.?5) and R to test (3.56).
2
We use R
1
We define a procedure R as
follows.
Procedure R.
Both R and R are computed at each stage of the
1
2
inspection until, Either:
stop before the other.
one of the procedures leads to a decision to
Then the former is no longer computed and the
latter is continued until it leads to a decision to stop.
R and R lead to a decision to stop at the same stage.
1
2
both computations are discontinued.
Or:
both
In this case
Then we take a final decision as
in the following table.
Rl accepts
H'1
If
R2\ accepts
and
If
H'2
and
If
H'
and
2
H'
3
H'
3
H'
4
. (final) R accepts
then
H_ l
then
H
O
then
HI"
45
If we take a < at and b < b' it can be shown that it is impos-
-
-
4.
sible for R to accept Hi while R accepts H
l
2
From Theorem 3.1,
Rl and R2 both terminate with probability one.
Then it follows that R
will also terminate with probability one.
Then
There is no loss in assuming that go ='-gl =
c
~
for some c > 1.
We shall see later that if c is large we can simplify the probability of correct decision.
= ep
Let g@)
~,
ep e: 1.
Write
L(Hilep.~;R) to denote the probability that the procedure R will accept
Hi when
g(~)
= ep
~
is the true parameter value.
L(Hilep,~;R)
= Pe{accept
go +
= gl +
(ep+c)~
HiIR}, g(~)
(ep-c)~.
= ep~.
That is,
We can write g@)
= ep
e- 2a (ep+c) _ 1
-2a(ep+c)
-2b(ep+c)' ep ~ -c
e
-e
a
a - b
L(H1Iep,~;R)
= L(H4Iep,~;R2)
, ep
e
+
as
-e
~+o
Probability of correct decision.
, ¢ = c.
Let L(¢,~IR) be the probability
of correct decision when the true value of the parameter is
ep e: I.
= -c.
l-e-2b I (ep-c)
-2a ' (¢-c) -2b'(ep-c)' ¢ ~ c,
-b '
at-b'
3.6.
=
Then, by Theorem 3.2 we obtain the following
relations.
(3.59)
~
In virtue of (3.57) we define L(ep,~IR) as follows.
g(~
=¢
~,
46
(
<P
l-L(H 1 I<P,Li;R)
~-(c+l)
-(c+l) < <P <-(c-l)
l-L(H_ 1 I<P,Li;R)
(c-l) < <P < (c+l)
<P
~
(c+l)
....
It is to be noted that we take L(<P,LiIR) to be the smaller values at
points of discontinuities.
L(<P,LiIR).
Let L(<pIR)
be the limit, as Li ~ 0, of
Then
(i) L(<pIR) is monotonically decreasing with negative curvature
for <P
~
-(c-l).
(ii) L(<pIR) is monotonically increasing with negative curvature
for <P
~
(c-l).
(iii) L(<pIR) has negative curvature for -(c-l) < <P < (c-l).
We now want to choose (a,b) and (a',b') to insure that
L(<P/R) ~ 1 - Yl for <P ~ -(c+l), L(<pIR) ~ 1 - Y2 for -(c+l) < <P < (c+l)
and L(<pIR) ~ 1 - Y2 for <P ~ (c+l); where Yl , Y2 and Y are positive
3
constants.
From the monotonic properties of L(<pIR) it is enough to
insure that
(3.61)
L(-(c+l)IR) = l-y l , L(-(c-l)IR)= L«c-l)IR)
L«c+l)IR) = l-y •
3
That is, we require
1 _ e 2b
(3.62)
1 - L(-(c+l)IR) =
2a
e
- e
2b = Yl ,
l-Y
2
and
47
(3.64)
1 - L«c-1) /R)
= 1 - e
2b '
~2a~'~~2~bT,
e
+
- e
t
e2b {2C-1) (e Za {Zc-1) -1)
2a{Zc-1)
2b{2c-1)
e
- e
1
=
•
Y2
and
I
1 - L{c+11~)
(3.65)
If c is sufficiently large that we can neglect the bracketed
terms then we have
~e
(3.66)
2a
=
Za'
=
e
1 - Y2
YZ
Y2
- Y1
,
Y3
- YZ
.
2b
=1
Zb'
=1
e
and
e
(3. 67)
1 - Y3
YZ
If Y1
= YZ = Y3 = a
3.7.
Bounds for the ASN function.
of N when g@)
(say), then a
= ~ tJ., ~ £
~
£
= a' = -b = -b' =
1-0 1/2
log ( 6 )
Let E{N,~IR) be the expected value
I is the true value of the parameter and R is
the sequential procedure employed.
that for all
e
From the definition of R it follows
I,
(3.68)
For
~
< -(c-l) the probability of coming to a decision to stop first
with R is high and therefore
Z
48
(3.69)
E(N,<pIR) "" E(N,<pIR1 ) for
<P
< -(c-l).
Similarly
(3.70)
E(N,<pIR)
r-'
E(N,<pIR ) for
2
<p >
(c+l).
Formula (3.68) gives a lower bound for E(N,<pIR) for all
However, these bound by (3.69) and (3.70), will be close for
only.
<p E
I.
(c-l)
1<p1 >
It is not easy to find close bounds for 1<p1 ~ (c-l), but
noticing that this corresponds to Ig~)I~ (c-l)~, and that we study
the case where
~ +
0, this is a small interval.
study the ASN over the whole range of
<p
Furthermore, we shall
numerically in Chapter IV.
We now follow Sobel and Wald (1949) in deriving upper bounds for
the ASN function.
*
Let R be the following rule:
l
until Rl accepts go -
~."
"Continue sampling
Since this implies that R2 accepts gl -
~
at the same or a previous stage, it follows that
(3.71)
Similarly i f we define R* to be the following rule:
2
until R2 accepts gl
+~.
"Continue sampling
Then
(3.72)
Upon neglecting the excess over the boundary then we can prove,
similar to Theorem 3.3, that
(3.73)
and
<p < -
(c+l)
49
(3.74)
~
> (c+l).
Again, it is difficult to fi~ upper bounds for E(N,~IR) for I~I < (c+l)
and we refer this to the numerical studies in the next chapter.
In the following chapter we give some examples and make some
numerical investigations of the proposed procedure.
CHAPTER IV
EXAMPLES AND NUMERICAL INVESTIGATIONS
4.1.
Introduction.
In the last chapter we developed a sequential
procedure fdr discriminating between two hypotheses concerning continuous functions of several
estimable parameters.
The procedure is
essentially valid for cases where the two hypothetical values and the
true value of the function under consideration are close to each other.
Based on this procedure we developed a test for discriminating among
three hypotheses concerning the same function.
We studied in some
detail the OC function of the proposed test; we developed some approximating formula for the OC function.
However, we were not able to study
the ASN function in as much detail.
We only gave some rough bounds for
the ASN function over some intervals of the parameter space.
This
calls for some numerical investigations to see how close to actual
values these approximating formula
are.
In this chapter we carryon
a simulation study of one of the cases where the proposed test is
applicable.
4.2.
Correlation coefficients.
with DF
Let Xl' •••
F~),
2
61 = cr (XiI)'
are regular functionals of F(x) such that:
'~
be a sequence of iidr.v.'s
51
r
\ 8
1 = E(X11 - EX21 ) 2 =
~ JJ(xII
- x 2l ) 2 dF(X1 ) dF(x 2 )
8 2 = E(Xll - 'EX21 )(X12 - X2Z ) =
(4.1).J
~
Jf(xII -
x Z1 )(x12 - x 21 )
dF <1£1) dF(x 2)
[ 8
3
= E(X12 - X22 )2 =
~ JJ(X12
-
X2Z )Z
dF(x1 ) dF(x 2)
l
XII aod X •
12
go -
~
Suppose that
>-1 aod go +
~
~
> 0 aod go
ar~
given real values such that
< 1 aod that we waot to test
r-
vs
(4.2)
Let Uoi be the U-statistic estimate of 8 , i
i
rI
1
U =
20(0-1)
01
0
0
L
L
i=l j=l
= 1,2,3.
Z
(xi! - x j1 )
I
(4.3)
,j
~
l
1
UOZ =
20(0-1)
1
U03 = 20(0-1)
0
0
2:
2:
i=l j=l
0
0
2:
2:
(xiI -
X
j1 ) (x iZ - x j2 )
Z
(xi2 - x j2 )
i=l j=l
We base our test 00 the statistic
U
02
g(U ) = g(U l'U 2'U 3) =
l/Z .
-n
0
0
0
[U
U ]
01 03
(4.4)
For a
= 1, •.• ,0
we defioe
Theo
52
r
1
Val = 2(n-l)
n
2
E (x
- xil)
al
i=l
1
V =
a2
2(n-l)
n
E (x
al - xil) (xa2 - x i2 )
i=l
1
V =
a3
2(n-l)
n
2
E (x
- x ) •
a2
i2
i=l
i
,
.I
(4.5)
For i,j = 1,2,3 we define
~iJ'
(4.6)
1
= -n
n
E V . V • - U . U .•
a=l (Xl. a]
n1 nJ
Then we define
"2
(4.7)
Yn
3
=4 E
-0.
ae.1
ag(U )
ae.J
-0.
i=l j=l
i
ag(U )
where
3
E
= (_1)i[3+(-1)]
=
e=u
--0.
4
~ i .,
J
g(U)
-0.
Uni
We choose two real numbers a and b such that b < 0 < a and for
n
~
nO (depends on 6) we use the following sequential rule.
Rule R :
l
Continue sampling until one of the following inequal-
ities is violated.
(4.8)
If the left inequality is violated first we accept H_
the right inequality is violated first we accept HI.
l
and if
If we desire that
the test have a strength(a,6) then by Theorem 3.2 we approximate a and b
by
(4.9)
1
1-6
1
S
a = -2 log ~ and b = - log - u.
2
I-a
53
Now, suppose we want to choose one of the following hypotheses
(4.10)
iiI:
g(~) >
o.
Since the three hypotheses are very close to each other it would be
difficult to discriminate among them with a high probability of making
a correct decision.
Instead we choose a small number g > 0 and let
go = -gl and find a testing procedure to discriminate among the three
hypotheses
(4.11)
Then if we accept. Hi we accept the corresponding hypothesis Hi'
i
= 1,2,3.
Now, problem (4.11) is the same as (3.54) and we choose a small
~
> 0 such that
~
< gl and use procedure R in Chapter III.
Since, in
our case here, g~) is a correlation coefficient then Ig(~) I
2
1 and
we choose gl and ~ such that /gl + ~I < 1.
Going back to Chapters II and III, see Theorems 2.7, 2.8 and 3.3,
we find that for the validity of our test we require E
for some 0 > O.
But since, in our case here,
g~)
possibly we may not need such a powerful condition.
xi+o to
exist
is bounded then
This condition is
satisfied for many bivariate DF but it still puts some limitations to
our procedure.
On the other hand alternative measures of association
such as Kendall's tau or Spearman's rho are valid for a much wider
class of distributions.
For normal distribution both product moments
correlation and rank correlation are valid and we may like to
their performances.
compar~
Tests for rank correlations will be discusspd in
the following section.
54
4.3.
Rank correlation.
DF F<.!.), 1£
2
e: R •
Let Xl' ••• '~ be a sequence of iidr.v.'s with
We define the rank R of x ' i
ai
ai
= 1,2
by
(4.12)
where
c(u)
=
o
ifu<O
1
2
if
1
if u > O.
u = 0
= l, ••• ,n,
If in the sample {(x ' x )}, a
al
a2
all xal's and all xa2 's
are different, the rank correlation coefficient is given by
K'
(4.13)
12
= ....,..:-=-3
n -n
~
(R
a= 1
al
- n+l) (R"'2 - n+l).
2
2
v-
We have
3
(4.14)
n
n
n
K' = -3--- E
E
E
n -n a=l S=l y=l
~(xal
- x S1 ) ~(xa2 - xy2 )
where
r
~
(u) = 2c(u) - 1
=
I
I
-1
if u < 0
=0
0
if u
1
if u >
1
o.
We can write
K'
(4.15)
where
t •
n(~-l) a~S ~(xal
=
(n-2) K + 3t
n+l
- xS1 )
...
~(xa2
- x S2 )'
55
andK
= --::--.;3~~~
n(n-l) (n-2)
diff~rent
Et' being over all
a and
subscripts a,
y.
The quantities
K and tare U-statistics Qf degrees 3 and 2 respectively, and
(4.16)
T = Et =
(4.17)
K
ff~(Xll
= EK = 3
- x 2l )
JJJ~ (xII
~(x12
- x 21 )
- x 22 )
~ (x12
dF~l) dF~2)
- x 32 ) dF (xl) dF ~2) dF ~3)
Then T is the product moment correlation of the difference signs, and
K
is the grade correlation.
In the continuous case K' is a biased
estimate of the grade correlation K for finite n but it is unbiased
in the limit.
If we write 8
1
= K and
8
2
= t, then g~) = [(n-l) 8
1 + 38 2 ]/(n+l)
is a continuous function in 8 and 8 but it depends on n. However,
1
2
n-l
as n become large n+l 81 becomes a dominating factor and in testing
g~
hypotheses about
concerning 8 •
1
we may 'only consider testing similar hypotheses
We use the procedure R of Chapter III to test two or
three hypotheses concerning 8 .
1
All
~oments
of the kernel of
~
exist
and the procedure would have a wider range of applicability than for
the product moment correlation coefficient.
4.4.
Comparison of two variances.
2
2
Let X and Y be two random variables
with variances 01 and 02 respectively.
between the hypotheses
(4.18)
It is desired to discriminate
56
We have
oi and o~
as two regular functiona1s where
X )2
2'
and
2
O2
1 (
= 2E
~1
).2
- y2
•
As in the correlation coefficient problem, instead of (4.18) we study
the problem of discriminating among the hypotheses
2
-1.< 1 - g
2
1
O
2
0
(4.19)
H_ :
1
0
H :
O
l-gl ~
2
1
:2 ~
2
-1.> 1 + gl
2
O
2
0
1 + gl
HI:
O
2
where 81 is some constant > O. We choose 6. > o such that 1 - g1 - tJ. > 0
8
2
2
1
We let 8 = 0 and 8 = O and apply the proand write g(&) =
2
1
2
1
e .
2
cedure R of Chapter III to study (4.19).
4.5.
Numerical results.
In Chapter III we gave some bounds for the
ASN function of the proposed procedure over some interval of the
parameter space.
Namely we showed that
E(N,cpIR1*) ~ E(N,cpIR),
E(N,cpIR*) ~ E(N,cpIR),
2
where
6.
2
E(N,cpIR~) ~
:;:
for cp + c < -1
~~c
for cp - c > 1.
6.~O
and
6.
2
E(N,cpIR;)
~
6.~O
't'
We were not able to obtain such upper bounds for Icpl < c + 1.
However,
since the expected value of the maximum of two random variables is
greater than or equal to the maximum of their expected values, so we
can obtain some lower bounds of E(N,cpIR) as
57
We tested these bounds numerically to see how close they are.
took the correlation coefficient example of section 4.2.
We
We took X to
be a bivariate normal with mean vector zero and covariance matrix
We used a computer program to generate random numbers with the
above distribution.
different values of
We applied the proposed sequential procedure for
~,
gl and p.
was taken to be equal to
l/~.
In each case the initial sample size
Although our main interest as we
I
developed the procedure was for cases where the two hypothetical values
go and gl and the true value of p are close to each other we also considered some cases where the true value was far from the hypothetical
values.
For each case considered we applied the procedure 40 times and
took the average of the sample sizes.
to in the tables as "actual."
and 2.
This average is what is referred
The results are sununarized in Tables I
Among the cases considered there are two cases where 40 samples
were too costly and our results there are based on 25 samples in one I
and 20 in the other.
There are 4 other cases where the expected
sample sizes are too large and extensive costs prohibited us from
getting results concerning these cases.
These cases are marked in the
tables by question marks and we hope that we can investigate these
cases in a future time.
The tables show also the number of cases where
a wrong decision was made.
For cases where
~
= .025 or
~ =
.05 the results obtained by simu-
lation are "fairly close" to the expected results.
However, we may
58
need more than just 40 samples in each case to decide how close they
are.
Also we hope that in the future we may be able to improve upon
the expected values to get better approximations.
As for 6
expected.
=
.1 the number of wrong decisions made is larger than
This is more apparent in Tables 3 and 4 where the problem
of choosing one of two hypotheses was considered.
At first we thought
that this may be considered a random fluctuation and that a larger
number of samples may reveal different results.
However, it was sug-
gested that the difference between actual and expected values is too
large.
We investigated these cases by taking more samples and we
obtained the same type of results.
obtained by letting 6 tend
As the expected values were
to zero it is our belief that those large
differences are due to the fact that 6 = .1 may not be small enough to
obtain good approximations.
Another problem now open is to study the ASN function through
its moments.
In our simulation we took only one case and we used
U-statistics estimators.
In the future we hope to tackle other cases
using U-statistics as well as Von Mises' differentiable statistical
functions.
Although U-statistics cover a wide range of problems of
interest, they leave others that are as much important uncovered.
This
suggests that other estimators may be studied to see if some justifications could be proved to base similar procedures upon them.
How-
ever, this may require a completely different approach than we used in
proving Wiener process approximations if they hold.
should be considered.
Nevertheless they
59
TABLE 1
ASN (IN PAIRS) FOR THE CORRELATION COEFFICIENTS
a
~e
.10
= -.1
and gl
=
.1
SE
No. of wrong
decisions
41.41
0
?
?
?
542.72
512.25
45.25
0
424.08
32.14
2
.1
*294.44
*849.69
?
?
?
.2
271.36
197.00
23.96
0
.3
121.36
85.35
8.73
0
.4
69.25
59.80
7.30
0
.5
41.40
33.98
3.38
0
0
144.63
13.51
9
.1
*147.22
*212.42
159.13
20.39
1
.2
135.68
84.08
13.95
6
.3
60.96
44.82
6.46
0
.4
33.33
30.38
4.28
0
.5
20.70
22.05
2.46
0
Actual
tt837.96
.1
*588.89
*3398.78
.2
0
0
.05
.05, go
Expected
p
.025
=S=
*upper
bound
tt based on 25 samples
60
TABLE 2
ASN (IN PAIRS) FOR THE CORRELATION COEFFICIENTS
.025
and gl = .2
p
Expected
Actual
SE
0
*294.44
*530.15
*3198.97
360.70
12.18
0
t533.05
45.11
0
?
?
.1
.2
?
No. of wrong
decisions
197.63
9.40
0
280.10
30.30
1
.2
?
?
?
.3
243.80
263.73
34.55
0
.4
103.88
96.88
9.89
0
.5
55.20
52.85
5.93
0
0
88.85
8.37
5
90.35
10.72
6
.2
*73.61
*144.29
*199.75
161. 77
28.31
0
.3
121.90
80.10
10.98
0
.4
51.94
42.53
6.25
0
.5
27.60
19.75
2.06
0
.1
.1
.10
= .05, go
*147.22
*288.58
*799.00
0
.05
= 13
= -.2
a
*upper
bound
t based on 20 samples
e~
61
TABLE 3
ASN (IN PAIRS) FOR THE CORRELATION COEFFICIENTS
a
.025
~e
.05
.10
tt
SE
= B=
.05,
go = -.1
SE
No. of wrong
decisions
tt622.44
58.27
0
288.58
302.92
16.35
0
.2
180.91
165.68
10.17
0
0
294.44
310.73
26.36
0
.1
144.29
145.60
12.40
0
.2
90.45
57.80
5.40
0
.3
60.96
40.85
3.94
0
.4
41.55
35.65
4.07
0
.5
33.33
20.10
2.09
0
0
147.22
118.52
13.69
3
.1
72.15
59.42
12.08
2
.2
45.23
28.54
4.51
2
.3
30.48
22.78
2.43
0
.4
20.76
18.85
1.95
0
.5
16.67
14.93
1.31
0
p
Expected
Actual
0
588.89
.1
based on 25 samples
=
Standard Error
62
..
TABLE 4
ASN (IN PAIRS) FOR THE CORRELATION COEFFICIENTS
a.
.025
.05
.10
= 13 =
.05, go
= -.2
No. of wrong
decisions
p
Expected
Actual
SE
0
294.44
292.85
17.68
0
.1
192.39
188.95
14.76
0
.2
135.68
127.00
7.51
0
0
147.22
152.47
12.09
0
.1
96.19
78.36
5.09
0
.2
60.95
56.83
3.93
0
.3
48.86
44.18
3.57
0
.4
34.63
28.43
1. 75
0
.5
23.66
27.75
1.66
0
0
73.61
55.35
7.15
2
.1
45.23
40.33
5.01
0
.2
30.48
30.48
2.79
0
.3
24.38
18.28
1.66
0
.4
17.32
16.00
1.38
0
.5
11.83
12.05
.62
0
e~
63
TABLE 5
ASN (IN PAIRS) FOR THE CORRELATION COEFFICIENTS
a
.025
--
.05
.10
=S=
.05, go
= .1
SE
No. of wrong
decisions
53.20
0
7-
7
288.17
45.25
0
294.44
266.38
30.30
0
.1
849.71
?
?
?
.2
271.36
197.00
23.96
0
.3
121. 90
85.55
8.73
0
.4
69.25
59.80
7.30
0
.5
41.98
33.98
4.39
0
0
147.22
81.83
10.49
6
.1
212.24
153.00
20.15
0
.2
135.68
79.38
14.22
7
.3
60.96
44.30
2.43
0
.4
34.63
30.38
4.28
0
.5
20.70
22.05
2.46
0
p
Expected
Actual
0
588.89
tt596.96
.1
3398.78
.2
542.72
0
ttbased on 25 samples
7
64
TABLE 6
ASN (IN PAIRS) FOR THE CORRELATION COEFFICIENTS
a
.025
.05
.10
= e = .05,
go
=
.2
No. of wrong
decisions
p
Expected
Actual
SE
0
294.44
277.73
14.97
0
.1
577 .16
530.15
45.95
0
.2
3196.01
?
?
?
.3
487.60
426.50
32.68
0
0
147.22
151.08
9.28
0
.1
288.58
275.73
30.94
1
.2
799.00
?
?
?
.3
243.80
263.73
34.55
0
.4
103.90
96.88
9.89
0
.5
55.21
52.85
5.93
0
0
73.61
63.18
8.35
3
.1
144.29
79.83
11.20
6
.2
199.75
158.65
28.65
0
.3
121. 90
80.10
10.98
0
.4
51.95
42.53
6.25
0
.5
27.60
19.75
2.06
0
e·
...
REFERENCES
Anderson, T. W. 1960. "A modification of the sequential probability
ratio test to reduce the sample size," Ann. Math. Statist. 31.
Armitage, P. 1947. "Some sequential tests of student's hypotheses,"
Supp. J. R. Statistical Soc. 1.
1950. "Sequential analysis with more than two alternative
hypotheses and its application to discriminant function analysis,"
J. R. Statist. Soc. B. 12.
Bartlett, M. S. 1946. "The large-sample theory of sequential tests,"
Proc. Camb. Phil. Soc. 42.
Berk, R. H. 1966. "Limiting behavior of posterior distribution when
the model is incorrect," Ann. Math. Statist. 37.
Billingsley, P. 1968.
Wiley, New York.
"Convergence of probability measures,"
John
Breslow, N. 1969. "On large-sample sequential analysis with application to survivorship data," J. , Appl. Probability, 6.
Chow, Y. S. 1960. "A martingale inequality and the law of lathr
numbers," Proc. AIDer. Math. Soc. 11.
Cox, D. R. 1952. "Sequential tests for composite hypotheses," Proc.
Camb. Phil. Soc. 48.
Cox, D. R. 1963. "Large-sample sequential tests for composite
hypotheses," Sankhya A E.
Cramer, H. 1946. Methods of mathematical statistics.
University Press.
Doob, J.
1953.
Stochastic processes.
Princeton
John Wiley, New York.
Dvoretzky, A., Kiefer, J. and Wolfowitz, J. 1953. "Sequential
decision problem for processes with continuous ·time parameter.
Testing hypotheses," Ann. Math. Statist. 24.
Ghosh, M. and Sen, P. K. 1970. "On almost sure convergence of Von
Mises' differentiable statistical functions," Calcutta Statist.
Ass. Bult. 20.
66
Grams, W. F. and Serf1ing, R. J. 1973. "Convergence rates for
U-statistics and related statistics," Annal$ofStatistics 1.
Halmos, P. R. 1946.
Statist. 17.
"The theory of unbiased estimation," Ann. Math.
...
Hoeffding, W. 1948. "A class of statistics with asymptotically
normal distribution," Ann. Math. Statist. 19.
1961. "The strong law of large numbers for U-statistics,"
Inst. Statist., Univ. of North Carolina Mimeo Series, No. 302.
Loynes, R. M. 1970. "An invariance principle for reversed martingales," Proc. Amer. Math. Soc. 25.
Miller, R. M. and Sen, P. K. 1972. "Weak convergence of U-statistics
and Von Mises' differentiable statistical functions," Ann. Math.
Statist. 43.
Mises, R. V. 1947. "On the asymptotic distribution of differentiable
statistical functions," Ann. Math. Statist. 18.
Scott, D. J. 1971. "An invariance principle for reversed martingales,"
Z. Wahrschein1ichkeitsrheorie und Verw. Gebiete 20.
Sen, P. K. 1960. "On some convergence of U-statistics," Calcutta
Statist. Ass. Bu1t. 10.
.",
Sen, P. K. 1974. "Almost sure behavior of U-statistics and Von
Mises' differentiable statistical functions." To appear in
Ann. of Statistics.
Sen, P. K. 1973. "Asymptotic sequential tests for regular functionals
of distribution functions," Theory of Probability and Applications, 18. (in press)
Simon, G. 1967. "A sequential three hypotheses test for determining
the mean of a normal population with known variance," Ann. Math.
Statist. 38.
Sobel, M. and Wald, A. 1949. "A sequential decision procedure for
choosing one of three hypotheses concerning the unknown mean of
a normal distribution," Ann. Math. Statist. 20.
Sproule, R. N. 1969. "A sequential fixed-width confidence interval
for a mean of a U-statistic," Ph.D. Dissertation, Univ. of North
Carolina.
Strassen, V. 1967. "Almost sure behavior of sample sums of independent
random variables and martingales," Proc. Fifth Berkeley Symp.
Math. Statist. Prob. 2.
Wa1d, A.
1947.
Sequential Analysis.
John Wiley, New York.
...
~