NONPARAMETRIC TESTS FOR INTERC~GEABILITY
UND~R COMPETING RISKS
By
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 905
JANUARY 1974
NONPARAMETRIC TESTS FOR INTERCHANGEABILITY UNDER COMPETING RISKS*
By PRANAB KUMAR SEN
University of North Carolina, Chapel Hill
Abstract
Some nonparametric tests for the hypothesis of interchangeability of the
elements of a (stochastic) 2-vector under competing risks model are proposed
and studied here.
Both fixed sample and sequential procedures are studied.
The case of progressively censored nonparametric procedures is also presented.
Along with some martingale theorems on allied rank statistics, their weak convergence results are considered and incorporated in the study of the asymptotic
properties of the tests.
The choice of locally optimal score function is also
considered.
AMS 1970 classification Nos:
Key words and phrases:
62G10, 62G20 & 60B10.
Asymptotically optimal score function, Bahadur
efficiency, hazard rate, interchangeability, invariance principles, (joint)
survival function, nonparametric tests, progressive censoring and sequential
tests.
*Work supported by the Aerospace Research Laboratories, U.S. Air Force Systems
Command, Contract No F 33615-C-71-1927. Reproduction in whole or in part
permitted for any purpose of the U.S. Government.
~.
For a two-component system, let F(x,y) be the joint distri-
We
bution function (df) of the survival times X and Y of the two components.
desire to test the null hypothesis that X and Yare interchangeable, 1. e. ,
F(x,y) = F(y,x) for all (x,y)~E2,
(1.1)
k
where E , k>l, stands for the k-dimensional Euclidean space.
Nonparametric
tests for (1.1) are due to Sen (1967), Bell and Smith (1969), and others.
In
competing risks problems, instead of (X,Y), the observable random vector is
(Z,Q), where
(1.2)
Z = min(X,Y) and Q=l, 0 or -1 according as Z=X, Z=X=Y and Z=Y.
For an exposition of joint survival functions under competing risks, we may
refer to Thompson et a1. (1972, 1973) where other references are cited.
based on a set of observable random vectors (Zi,Qi)'
test for (1.1) against suitable alternatives.
l<i~n,
Thus,
our problem is to
Nonparametric tests for this
problem are proposed and studied here.
Three different types of tests are considered:
(i) the conventional fixed
sample size procedure based on all the n observations through a single statistic,
(ii) the first sequential procedure based on the observations when the Z.care
~
observable sequentially, and (iii) the second sequential procedure suitable
under progressive censoring.
The first sequential procedure is suitable when
the observations are not available at the same time, so that if the null
hypothesis (1.1) may be rejected based on fewer than n observations, there is
a reduction of the total time to perform the test.
In the context of life-
testing problems, when n independent systems are subject simultaneously to a
continuous time-observation process and the (Z.,Q.) are observable only at the
~
~
expiry of the lives of these systems, one may naturally be interested in
2
monitoring the experiment with the objective of rejecting the null hypothesis
with the minimum sacrifice of the lives of the units, that is, stopping the
experiment at a time point where, for the first time, the accumulated evidence
leads to the rejection of H.
o
Unlike the other case, here the ordered random
variables corresponding to Zl"",Zn are observed sequentially, and the scheme
is known as a progressively censored scheme.
applies to this situation.
Our second sequential procedure
Thus, for both the sequential procedures, the stopping
times are random variables, and the procedures may lead to reduction of time and
cost of experimentation.
We shall discuss these in greater detail in section 2.
The test procedures along with the preliminary notions are introduced in
section 2.
Some martingale theorems, invariance principles and certain basic
invariance structures for the allied rank statistics are studied in section 3.
Section 4 is devoted to the study of the properties of the fixed-sample and first
I
sequential tests based on appropriate rank statistics.
Their asymptotic relative
efficiency (ARE) results are also considered.
res~lts
Parallel
sequential procedure are presented in section 5.
for the second
The last section is concerned
with the choice of optimal scores.
notions and the
roosed tests.
Let {(Z.,Q.), i>l} be a sequence
~
~
-
of independent and identically distributed random vectors (iidrv), where the
(Z.,Q.) correspond to (Xi,Y.) as in (1.2).
~
~
density function f(x,y),
~
We assume that F(x,y) possesses a
V (x,y)EE 2 , so that (i) the density function (say, g(z»
of Z. exists, and (ii) P{Xi=Y } = p{Q.=O} = 0, Vi>l.
~
i
~
so that G(z) is absolutely continuous in ZEE.
Let G(z) be the df of Z.,
Hence, ties among Zl'.'.'Zn can
be neglected with probablity 1.
Let c(u)=l or 0 according as u is > or <0, and ,let Rni = Lj:l c(Zi-Zj) be
the rank of Z.~ among Zl""'Z n , for l<i<n.
Thus, R
-n=(Rn1, ••• ,Rnn ) is some
~
4It
3
permutation of (l, .•. ,n).
For every n(>l), consider a set of real-valued rank-
scores a (l), ••• ,a (n), defined by
n
(2.1)
n
a (i) = E¢(U .) or ¢(i/(n+l»,
n
n.
~
where U 1 < ••. < U are the ordered random variables of a sample of size n
n
nn
from the rectangular (0,1) df [so that EU .=i/(n+l), l<i<n], and the score--
n~
function ¢(u), O<u<l, is assumed to be square-integrable and non-degenerate,
so that
o < A2
(2.2)
=
f
1
¢2(u)du <
00.
o
Consider first the fixed-sample size test.
Define the rank statistics
n>l.
(2.3)
As we shall see in section 3 [cf. Lemma 3.4] that under H in (1.1), gn=(Ql, ••• ,Qn)
o
and R are stochastically independent and Q assumes the all possible 2n realiza-n
-n
n
tions, each with the equal probability 2- • Thus, under H , E(T )=0 and
o
n
On the other hand, when H does not hold, n
-1
o
T
n
estimates a quantity which may be positive or negative depending on the df F(x,y).
Thus, for a one-sided test, we may consider the critical region specified by
(2.4)
while for the two-sided test, our critical region is given by
(2.5)
n~IT
n
IfA ->
c(2), where P{n-~IT If A > C(2)} = a.
n,a
n
n - n,a
In section 4, we shall see that the tests sketched above are genuinely distribution-free, so that C(i) i=1,2, depend only on the level of significance
n,a'
a (not on F), and there exist suitable c~i), i=1,2, such that
(2.6)
limn400
C~~~
=
c~i),
i=1,2, for every O<a<l.
4
Let us next consider the first sequential test.
(Zi,Qi)'
i~l,
Here the i.i.d.r.v.
are observable sequentially, so that it may be advisable to stop
at an intermediate stage i.e., when (Z.,Q.),
l<i<k, are observed for some k<n,
~
~
,
provided the statistical evidence up to that stage provo cates the rejection of
H •
o
l~k<n,
For every k:
define T by (2.3), and conventionally let To=O.
k
Let
then
(2.7)
so that corresponding to (2.4) and (2.5), we consider the critical regions:
> M+
where P{M+ > M+ IH} = a,
n - n,a'
n - n,a 0
(2.8)
M+
(2.9)
M
> M
where P{M > M
IH} = a.
n - n,a'
n - n,a 0
Operationally, the test procedure consists in observing sequentially the T , k~l,
k
~
-1
..b.<
I
,'+
until for the first time for some k=N~n), n A TN(or n 2 A TN) exceeds M
n
n
n,a
-11
(or M ), and rejecting H at that stage with the termination of the experiment.
n,a
I
0
If no such
N(~n)
exists, then H is accepted when (Zl,Ql), ••• ,(Zn,Qn) are observed.
o
We shall see in section 4 that the' test procedure is distribution-free, so that
M+
n,CI.
or M
does not depend on the underlying F, and further, there exists
n,CI.
suitable constants
M: and M
, such that
a
(2.10)
+
limn~
~M
n,CI.
= M+CI. and limn-+ooMn,CI. = Ma •
Finally, let us consider the progressively censored rank test.
In this
case, the experimentation starts with the continuous observation on n units and
their values are recorded as they are observed sequentially.
Thus, here the
order statistics Zn, 1 -< .•• -< Zn,n (corresponding to Zl' ••• 'Z n ) are observed in
a sequence; by virtue of the assumed continuity of G, ties among the Z i can be
n,
neglected with probability one.
(2.11)
Z
We may note that
. = Zs .' l~i~n,
n,1.,
n~
~
•
5
where 8 =(8 1, ••• ,8 ) is some permutation of (l, ••• ,n).
-n
n
nn
In view of the fact
that Z. = Z R ,l<i<n, we term 8 as the vector of anti-ranks. Also, we denote
J.
n , ni
- -n
the Q corresponding to Z8 . by Q . = Q(n,8 ), for i=l, .•• ,n. Then, we observe
ni
j
nJ.
. 8nJ.
that at the kth stage
with Q(n,8
nl
), ..• ,Q(n,8
(2.12)
when Z l""'Z k have been observed, we are provided
n,
n,
nk
), for k=l, ••• ,n.
We denote by
T = L : Q(n,8 )an (i), l<k<n.
i l
ni
nk
Note that, by definition,
(2.13)
n
T
= L.nl R(n,8 .)a (i) = L.
Qia (R .) = T , n>l.
nn
J.=
nJ. n
J.= l
n nJ.
n
Conventionally, we let T =0 and T =0, Vn>O, and define
o
no
(2.14)
k
D+ = { max T k}/(n 2 A ) and D = { max IT k/}I
n
n
n
O<k<n n
O<k<n n
(n~An ).
+
For an one-sided test, we use D and reject H when
0
n
D+ > D+
where P{D+ > D+ IH} =
n - n,OI.
n - n,OI. 0
(2.15)
01.,
and for a two-sided test, we use D and reject H when
n
(2.16)
0
D > D
where P{D > D
IH}
n - n,OI.
n - n,OI. 0
= 01..
Operationally, the test procedure consists in continuing the experiment so
4 -1 k (or n4A
-11
long as nAT
T k I), l<k<n, continue to lie below D+
(or D )
n n
n
n
-n,OI.
n,OI. '
l
and if N«n) is the smallest pQsitive integer for which n 4 A- T N is > D+ (or
n n
- n,OI.
n~A-lIT
NI
n
n
is > D ), the experimentation is terminated along with the
- n,OI.
rejection of H '
o
If no such
N(~n)
exists, H is accepted.
o
In section 5, we
shall see that the tests based on D+ and D are genuinely distribution-free, and
n
n
+ and DOl.' such that
there exist suitable constants DOl.
(2.17)
lim
D+
= D+ and lim
D
n-?OO n,OI.
01.
n-?OO n,OI.
=
D , V 0<01.<1.
01.
For the study of the various properties of these tests, we require to study
first some basic properties of {Tn' n~l} and {T , l<k~n}.
nk
This has been
6
accomplished in section 3.
The density function g(z) of Zi is
given by
00
00
(3.1)
J f(z,y)dy
g(z) =
+ J f(x,z)dx, -oo<z<oo.
Z
Z
Let n(z) = p{z=xlz=z} = p{Q=lIZ=z},
-00< z<oo ,
so that
00
n(z) =
(3.2)
[J
f(z,y)dy]/g(z), O<n(z)~l, -oo<z<oo.
Z
For every
(3.3)
l<i~n,
let
n-1
n(i,n) = n(i-1)
J n(z)[Gz)] i-1.
[l-G(z)] n-i dG(z),
00
n*(i,n) = 2n(i,n)-1;
-00
(3.4)
~
- -l '1
n a (')
*(' )
d * n - n L~=
n ~ n 1,n an ~ n -
Note that under H in (1.1),
o
n(i,n) =
(3.5)
~,
n(z)~
Lk =n l~k·
for all -oo<z<oo, so that
n*(i,n) = 0, l<i<n and
--
~ =~*=O,
n
n
V n>l.
For every n>l, let B be the a-field generated by (Q ,R ) where Q and Rare
n
~n -n
~-n
defined in section 2.
Theorem 3.1.
Note that B is t in
n
For an(i)=E¢(Uni ),
l~i~n
n~l).
Then, we have the following:
and ¢ integrable inside [0,1],
{Tn-~~,Bn;
is a martingale.
Proof.
(3.6)
By (2.3), (3.3) and (3.4),
E(T n -~*IB
n n-1) =
E(T1-~1)=0,
while for n>2,
[L~-llE{Q,a
1=
1 n (R ,)IBn- 1}-~*n- 1]
n~
+ [E{Qn an (Rnn )IBn- 1}-~].
n
Now for 1<i<n-1, given B l' R , can be either R l' or (R 1,+1) with respective
- nn1
n- ~
n- ~
conditional probabilities 1-n
(3.7)
-1
Rn-1i and n
{
E{Q,a
1 n (Rn~,) IBn- 1} = Q,~ (l-n
-1
-1
Rn-1"~
and Q,1 is fixed, so that
-1
Rn- l,.)a
1 n (Rn- 1i)+n Rn- l,a
1 n (Rn- 1,+1)}
1
= Q,a
l(Rn-1')'
1<i<n-1,
1 n~
--
n>l}
7
where the last step follows from the well-known and easily verifiable identity:
(3.8)
+ iE¢(Un .+l )] = E¢(U n_li), l<i<n-l.
1.
n-l[(n-i)E¢(U .)
n
1.
Thus, from (3.6) and (3.7), we have
E{T n-]1*IB
n n- I}
(3.9)
=
Tn-1-]1*n-1
+ E{O-n an (Rnn ) IBn-·1}-]1 n •
Now, given Bn- l' the possible values of 0-n an (Rnn ) are ±an (j), j=l, ••• ,n, and
and -an (j) with probability
Qn an (Rnn ) = an (j) with probability ~(j,n),
n
!(l-TI(j,n», for j=l, •.• ,n, so that
n
(3.10)
E{Q a (R )IB I} = L.nla (j)!{2TI(j,n)-1} = ]1 , by (3.4).
n n nn
nJ= n
n
n
Hence, the theorem follows from (3.9) and (3.10).
Q.E.D.
Since Q~=l with probability 1, for every i>l, by (2.3) we have
1.
(3.11)
2
= I..\inl
= a n (i) + Ll<.4.<
_1.;J_nE{Q.Q.a
1. J n (Rn1..)an (RnJ.)},
where by (2.1) and (2.2), as n+oo,
(3.12)
=
I
1
o
We let TI*(z) = {2TI(z)-1}, -oo<z<oo, and for
(3.13)
TI*(k,q;n)
¢2(u)du.
l<k<~n(~l),
define
n!
II TI*(u)TI*(v) [G(u)] k-l [G(v)-G(u)] q-k-l
(k-l)! (q-k-l)!(n-q)! -oo<u<v<oo
=~---:,...--,---'=":".,....,-,..-,--~
[l-G(v)]n-qdG(u)dG(v).
Note that under Ho ' TI*(k,q;n) = 0,
V l~k<~n.
Since the Zl"",Zn are iidrv,
by some standard arguments, it follows that for l<i:fj<n,
8
J
I
(3.14)
a (s)a (t)
u=I. v=I'O(-l)U+V
E{Q.Q.a (R .)a (R .)} = .
~ J n n~ n nJ
s;'f=l n
n
)
o
.=s, R .=t, Q.=(-l) u , Q.=(-l) v ] }
nJ
~
J
P[R
n~
2
l<k<:~n
an (k):an (q)rr* (k,q;n).
Thus, from (3.11) and (3.14) and by Theorem 3.1
(~
ETn =~*),
we have
n
2
VeT ) = nA 2 + 2
a (k)a (q)rr*(k,q;n)-(~*)2.
n
n
1<k<~n n
n
n
(3.15)
It readily follows from (3.5) and the fact that under H , rr*(k,q;n)=O, that
o
VeT IH ) = nA2 and n-lV(TIH )
non
n 0
(3.16)
-+ A2
as n~.
To simplify (3.15) for large n when H is not necessarily true, we assume
o
that the fol,lowing conditions are satisfied:
(I)
~(u)
=
~1(u)-~2(u)
where
~j(u)
is non-decreasing and absolutely con-
tinuous inside [0,1], and
(3.17)
and (II) rr(z) is absolutely continuous in z for all O<G(z)<l.
Let us then define
(j2 = (j2 (F) =
(3.18)
0
2[
JJ
-oo<u<v<oo
00
1
J ~2(u)du
-
( J rr*(z)~(G(z»dG(z»2
+
-00
rr*(u)rr*(v)[G(u){l-G(v)}~I(G(u»~'(G(v»+~(G(u»{l-G(v)}~'(G(v»
-
G(u)~'(G(u»~(G(v»]dG(u)dG(v)].
Note that !rr*(z)I .~ 1, ~ -oo<z<oo, so that some standard computations yield that
a 2 (F)<00 for every F.
Theorem 3.2.
Then, we have the following.
Under (2.1), (3.17) and conditions I and II,
9
. n -1 V(T ) -+- q 2 as
(3.19)
Proof.
n~.
n
By virtue of (3.17), we obtain, on proceeding as in Hoeffding (1973),
that
(3.20)
Consequently, if we prove the theorem for an (i) =
E~(U
'r
.), l<i<n,
the result
--
n~
applies as well to the other case of an(i) = <p(i/(n+1», l<i<n.
We let T =0,
o
and for n>l,
Ln = Tn -Tn- 1-~'
n
(3.21)
2
0
11
= E(L2n IBn-1)' n>l.
-
Then, by Theorem 3.1, we have n-1V(T ) = n-1~.n1E[q2], so to prove (3.19), it
L~=
n
suffices to show that as
n
n~,
(3.22)
By (2.3), (3.21) and a few steps we obtain that
2
2
I
2
qn = EaR
[n( nn) Bn-1] - ~n +
(3.23)
+
I
- 1E a R
.
Ini=l
B
{[ n( ni) -an-1( Rn-1i)] 2 In-I}
Q.Q.E{[a (R .)-a l(R l·)][a (R .)-a l(R l,)]IB I} +
J
n n~
nn- ~
n nJ
nn- J
n-
1~i~j~n-1 ~
2
I. n- I}·
I , - 1Q,E{Qn an (Rnn )[an (Rn i)-an-l(Rn-1i)]B
n 1
~=
~
Now, as in the proof of Theorem 3.1, we have
(3.24)
=
1
o
<P
2 (u)du as n~,
Bn-l} =
E{[an (Rn~.}-an- l(Rn- l,)]2I
~
(3.25)
[n
(3.26)
f
-2
2
Rn- 1.(n-R
~
n- 1.)][a
~
n (Rn-1.+1)-a
~
n (Rn-1')]'
~
1<i~n-1,
E{[an (Rn~,)-an- l(Rn- l·)][a
~
n (RnJ,)-an- l(Rn- l,)]IB
J
n- l' i~j}
-2
= n WI (n-w ) [an (w 1+1)-an (wl )] [a (w +1)-a (w )] ,
n 2
2
n 2
10
E{Qn an (Rnn )fan (Rn i)-an- l(Rn-I')]
18 }
1.
n
(3.27)
= L~,n1a
(j )1".*
n (j ,n){[an (Rn-I'1.+c (Rn-I'1.-j ) )-an-1 (Rn- I'1. ) ]}
J= n
.
. -2
IRn-1i
= [an (Rn- l,+l)-a
(R
1i)]{n
(n-R
I')
'1
1.
n nn- 1. J=
TI*(j,n)an (j)
2
R
+1 TI*(j,n)an (j)}.
- n- Rn- 1'~.~
l. LJ = ,
n-11.
Also, note that for
(3.28)
1<i~j~n-1,
E(Q,IR
1)
1. ~-
(3.29)
= 2TI(Rn- l"n-1)-1
= TI*(Rn- l"n-1),
1.
1.
E(Q,Q,IR
1) = TI*(Rn- l"R
if Rn- I'1. < Rn- I'J
1. J -n1.
n- l,;n-I),
J
= TI*(Rn- 1j,Rn- l,;n-1),
if Rn- Ii > Rn- I'·
1.
J
Thus, writing E(q2) = E{E(q2IR I)}' and using (3.23) through (3.29) that
n
n -n(3.30)
2
2
2
TI*(i,j,n-I)n- i(n-j)[a (i+1)-a (i)][a (j+I)-a (j)] +
1<i<j~n-1
n
n
n
n
I
L~:iTI*(i,n-1)[an(i+1)-an(i)]{n-2(n-i)Ij:1TI*(j,n)an(j) . ~n
*('J ,nan
) (')
}•
n -2 l.Lj=i+1TI
J
Note that TI*(z) = 2TI(z)-I is a bounded and absolutely continuous function of z,
so that by the well-known bounds for expected order statistics, we have
(3.31)
(3.32)
TI*(i,n) = TI*(G
TI*(i,j;n) = TI*(G
-1
-1
(i/(n+1») + o(n
(i/(n+1»)TI*(G
-1
~
),
l~i<n,
~/-
(j/(n+1») + o(n 2),
l~i<j<n.
Thus, by (3.4), (3.31) and Hoeffding (1953, 1973), we obtain that
00
(3.33)
~ +~(F) =
n
f
{2TI(z)-1}¢(G(z»dG(z) as n~.
-00
Also, by the recent results of Hoeffding (1973), the third term on the rhs
11
(right hand side) of (3.30) converges to 0 as n+oo.
By (3.32) and some standard
steps, the fourth term on the rhs of (3.30) converges to (as n+oo)
(3.34)
2
JJ n*(x)n*(y)G(x)[l-G(y)]¢'(G(x»¢'(G(y»dG(x)dG(y),
_oo<x<y<oo
and similarly, the last term converges (as n+oo) to
(3.35)
2
JJ n*(x)n*(y){¢(G(x»[l-G(y)]¢'(G(y»-G(x)¢'(G(x»¢(G(y»}dG(x)dG(y).
-oo<x<y<oo
The proof of (3.22) follows from (3.18), (3.12), (3.28), (3.33), (3.34) and
(3.35).
Q.E.D.
Now, by virtue of Theorems 3.1 and 3.2, for every
O<s~t~l,
(3.36)
In the sequel, it will be assumed that rr2 is strictly positive, so that
(3.37)
Let 1=[0,1], Wn·
(0)=0, n>l,
and define
(3.38)
.{
k
k+l
Consider then a stochastic process, Wn= Wn (t),t£I}, where for -n< t -< --,
n
(3.39)
k
k+l
k
Wn (t) = Wn (n)+(nt-k)[Wn (7)-Wn (n)]' k=O, ••• ,n-l.
Thus, for every
n~l),
W belongs to the space C[O,l] with which we associate the
n
uniform topology specified by the metric
(3.40)
p(x,y) = sup Ix(t)-y(t)I , x,y£C[O,l].
t£I
Finally, let W={W(t),t£I} be a standard Brownian motion on I, so that EW(t)=O
and E[W(s)W(t)]=min(s,t) for every s,t£I.
Theorem 3.3.
Under (3.37) and the conditions of Theorem 3.2,
12
W
n
(3.41)
Proof.
V
~
W, in the uniform topology on e[O,l].
As in Hajek (1968) and Hoeffding (1973), for every n>O, there exists
[under (3.17)] a decomposition
(3.42)
¢ (u) = ¢ (1) (u) + ¢ (2) (u) - ¢ (3) (u), O<u<l,
where ¢(l) is a polynomial, ¢(2) and ¢(3) are non-decreasing, and
{rj~2 (I~(j) (u)! {u(l-u)}-<'du } <nr(IHu)! {u(l-u)}-<'du.
(3.43)
Now, in (3.18), on replacing ¢(u) by ¢(j)(u) everywhere and denoting the corresponding quantity by a~, j=1,2,3, it can be shown that (3.43) implies that
J
(3.44)
where n'(>O) depends on n, and
l~~nln~{Li~l Qi~(~i)
by (2.1).
-
n'~O
as
n~O.
Also, by virtue of (3.20),
Li~l Qi¢«k+l)-l~i)}1
= 0(1), for
~(i)
defined
Hence, here also, it suffices to work with an(i)=E¢(U ni ), l<~n(>l).
Suppose now in (2.3) and (3.4), we replace the score function a (i) by
n
a
n,
j(i) = E¢(.)(U .), l<i<n, j=1,2,3, and.denote the corresponding quantities by
T .,
n,J
n1
J
~
. and
n,J
--
~*
., respectively, for j=1,2,3.
n,J
Similarly, in (3.38)-(3.39), we
and a by Tk "~k* . and a., respectively, and define the resulting
,J,J
J
process by W . = {W .(t),t£l}, for j=1,2,3. Then, by (3.42), we have
nJ
nJ
replace
Tk'~*k
(3.45)
Note that Theorem 3.1 applies to each of {T
.-~* .,B
n,J
n,J
;n>l}, j=1,2,3, and by
n -
definition, SUPt llw .(t)l= max, n-~ITk .-~~ .lla., so that by the Kolmogorov£
nJ
O<k<n
,J
,J
J
inequality for martingales, we have
(3.46)
p{suplW .(t) I ~ K} = p{ max ITk .-~~ .1 ~ Kvna.}
t£l nJ
O<k<n
,J,J
J
< (nK 2 )-lE[T
n,
j-~
.]2/a~ ~ K- 2 , as n-+<x>, j=1,2,3.
n,J
J
13
by Theorem 3.2.
By virtue of (3.44), (3.45) and (3.46), for every £>0 and E'>O,
there exists an
n>o,
(3.47)
p{suplw (t)-(rrZ/cr)w 2(t)+(rr3 /rr)W 3(t)I>E'}<E.
tEl n
n
n
such that under (3.17) and (3.42),
Consequently, by (3.44) and (3.47), it suffices to prove that as
(3.48)
W
n1
V
+
n~,
W, in the uniform topology on C[O,l],
and for this purpose, we use a functional central limit theorem for martingales
[cf. Theorem 3 of Brown (1971)] according to which it suffices to show that as
n~,
for every E>O,
n E{L,2 II ( IL. 1 I > Err Vn)}
n -II ~=
'·l
~,
~,1
(3.49)
~ n
(3.50)
2
2
(£'-1
q,~, 1)/(nrr1 )
~-
P
+
+
0,
1,
where Ln ,l and qn,l are defined by (3.21) for ~=~(1) and I(A) stands for the
(r)
indicator function of a set A. Let ~(l)(u)
= (d r/ du r )~(l)(u), r=0,1,2. Since
~(1)
is a polynomial and is absolutely continuous, we have
sup 1~«lr»(t)1 = K «00), for r=0,1,2.
O<t<l
r
(3.51)
< IT 1 1/ + la1 1(1)/ =
,
,
(3.52)
ILn, 11 ~ I~=11Ia n, l(R
~-
2/10
1
~(l)(u)dul < 00, and for n>2,
,) - an
-1, l(Rn -I')
I + lan, l(Rnn ) 1 +
~
n~
IlJn, li-
Note that Rn~, is either Rn- Ii or Rn- 1,+1,
so that on using (3.8) and (3.51),
~
(3.53)
lan, l(Rn~.)-an-,
11(Rn-'1,)1
~
-< lan, l(Rn- l,+l)-a
~
n, l(Rn- 1,)1
~
~
max
la 1(k+1)-a 1 (k)/ =
1<k<n-1
n,
n,
o (n-1 ),
as under (3.51), n[an, 1(i+1)-an, l(i)] = an, l(i) is bounded, and
14
Ian,l(i)-<P ~g (i/ (n+l) I -+
(3.54)
0, as n-+oo, 'V
l~i<n.
max /a l(k) I = 0(1) and I~ 11 = 0(1). Consequently,
l<k<n n,
n,
by (3.52), (3.53) and the above; we have that for every £>0, there exist an
Similarly,
lan, l(Rnn ) I ~
integer n£, such that
(3.55)
On the other hand,
(3.56)
n
-I, n£
2
I I
Cl
_l,n£
2
Li=lE{Li,lI( Li,l > £crlynI ~ n Li=lE(Li,l)
= n-Lv[T
n ,
£
1] ~ cr 2 (n In)
1 £
-+
0 as n-+oo.
Hence, (3.49) follows from (3.55) and (3.56).
To prove (3.50), we use (3.23) through (3.27) for <P=<P(1) i.e., an ,l(i),
l<i<n.
Writing then w"1.J = mineRn- l"R
1. n-I')
J and
w~,
1.J = max(Rn- li,Rn.,. lj)' we have
(3.57)
,
2
L
QiQ,n
1<i<j<n-1
J
-
-2
-
w,,(n-w~,)[a
1.J
1.J
n,
l(R l,+l)-a l(R li)][a l(R lj+1)-a l(R 1j)]
n- 1.
n,
nn,
nn,
n-
R
n- 1
-2
n-1i n* (J,n)a
'
,
(n-Rn- 1')"
+ 2I i =1Q1.,[an, l(Rn- l,+l)-a
1.
n, l(Rn-l,)]{n
1.
1. L = 1
n, l(J)
J
2
-n- R -I'
n 1.
I~=R
J
+1 n*(j,n)a l(j)}·
n,
n- I'1.
1
The first term on the rhs of (3.57) converges to
second term to ~i(F) = (
f
f <P(1) (u)du
0
n*(z)<p(1)(G(z»dG(z»2.
as
n+oo,
and the
00
By (3.53), the third term
-00
goes to 0 as n+oo, while the fourth term can be written as
(3.58)
where the Sn1., and Q(n,S n1.,) are defined by (2.11) and shortly after that.
that as in (3.31),
(3.59)
E[Q(n-1,S n-l,)Q(n-l,S
I')] = n*(i;j;n-1), for 1_<i<J'<n-1,
1.
n- J
Note
15
so that by (3.54), (3.32) and (3.59), the expected value of (3.58) converges
(as n-+oo) to
(3.60)
2
JJ ~*(x)~*(y)G(x)Il-G(Y)]d¢(l)(G(x))d¢(l)(G(y)).
-oo<x<y<oo
.
On the other hand, Q(n-l,Sn_li)' l<i<n-l, are interchangeable and bounded (by 1)
random variables, so that on evaluating the 4th moment of (3.58), using the
Markov-inequality and the Borel-Cantelli Lemma, i t follows that (3.58) converges
almost surely to (3.60).
In a similar manner, it follows that the last term on
the rhs of (3.57) converges almost surely (as n-+oo) to
(3.61)
JJ
2
~*(x)~*(Y)I¢(l)(G(x))Il-G(y)]¢~i~(G(Y))
-oo<x<y<oo
Thus, q~,l
Remark.
-+
of
G(x)¢~i~(G(x))¢(l)(G(Y))]dG(X)dG(Y).
almost surely as n-+oo, and this implies (3.50).
Q.E.D.
On using (3.42)-(3.43) and the recent results of Hoeffding (1973), it
can be shown that under (3.17), (3.37) can be improved to:
k
In2I~n-~(F)]1 -+ 0 as n-+oo,
(3.62)
so that { max Ik~(F)-~~l/l:no} -+ 0 as n-+oo. Consequently, in (3.38), it is
l<k<n
possible t~ replace ~~ by k~(F) for l<k<n.
•
jl
. jn
Let us now consider the situation when H holds. Let J =«-1) , ••• ,(-1) )
o
-n
where j1.' is either 0 or 1, l<i<n,
and let J-n={j_n : j.=O,l,
l<i<n}.
Also, let
- 1.
-S
n
be the set of all possible n! realizations of S , defined after (2.11).
-n
Finally, let
g(~n)
= (Q(n,Snl), .•. ,Q(n,Snn))·
Then, we have the following.
Lemma 3.4.
Under H in (1.1) Q =(Ql, ••• ,Q ) and R =(R 1, •.. ,R ) are stochastically
o -~
n - - -n
n
nn
independent, and for every S £S ,
-n n
P{Q(S ) = j }
(3.63)
for every j_n £J-n •
- -n
-n
Proof.
Now Zl, ••• ,Zn are iidrv, so that gn can have all possible n! permuta-
tions of (l, ••• ,n) with the common probability lin!.
On the other hand, if r
-n
16
is any permutation of (l, ..• ,n), then
p{Q =j , R =r IH0 }
(3.64)
~n
=
f ••• f{
( ~n )
~n
~n
~n
n
l-ji
ji
IT {g(Zi)[TI(Z.)]
[l-TI(z.)] }dz.,
i=l
~
~
~
~
where the n fold integration extends over the domain {~Z
~n
H ,
o
is the anti-rank
TI(z)~
ve~tor
<oo} and
snn
Since, by (3.1) and (3.2), under
corresponding to En'
<••• <Z
for all Z, (3.64) reduces to
(3.65)
2
n
Hence, p{Q~n=j~n IR~n=r~n ,H0 } = 2- ,
and Q •
snl
Vr
~n
-n
.
P{R=r}.
~n
~n
, and this implies the independence of R
~n
Hence
~n
p{Q
-n
(3.66)
=j
~n
= 2-n , V j
IH }
0
~n
EJ •
~n
By virtue of the fact thatS-n is the anti-rank corresponding to some R
, we
~n
have p{Q(S )=j
~
~n
~n
IH0 }
= pro =j IR ,H }, and hence, (3.63) follows from ("3.66)·
~ ~n -n
0
and the independence of Q
and R.
~n
-n
(k)
Let S~n
(k)
=(S n l""'S
. nk)' Q(S
'" n
(k)
the a-field generated by (S
-n
Lemma 3.5.
Proof.
(3.67)
Q.E.D.
,Q(S
~
) = (Q(n,S n l),···,Q(n,S n k»' and let B*k
n be
(k)
n
»,
when H holds, for k=l, ..• ,n.
0
For every n>l, {Tnk,B~k' l<k~n} is a martingale.
By (2.12), for every k>q,
E(T nklB*nq ) = E{L.kl
an (i)Q(n,S n~.)IB*
~=
nq }
k
= Tnq + L.~=q+1 a n (i)E[Q(n,S n i)IB*nq }.
By Lemma 3.4, for every i>q, E[Q(n,S
for every k>q.
E(T klB* ) = T
n
nq
nq
W*(k/n)
n
)/B*]
= 0, so that by (3.67),
nq
Q.E.D.
We let T =0, and for l<k<n,
no
(3.68)
ni
4
-1
= nAT
n n k'
17
and by linear interpolation between [k/n,(k+l)/n], for k=O,l, .•• ,n-l, we
complete the definition of W*={W*(t),tEI}.
n
n
.
Theorem 3.6.
Under (1.1), (2.2) and the condition that
(3.69)
W*
n
max {Ia (k)I/i'n)
l<k<n
n
g W,
+
0 as n+oo,
in the uniform topology on C[O,l],
[Note that (3.17) implies (3.69) but the converse is not true.]
Let ~nk = Tnk-Tnk-l' l<k<n.
Proof.
Then, by (2,12) and Lemmas 3.4 and 3.5, we
have
a 2 (k) ,
n
(3.70)
v~ = Lk~l E(~~kIB~k_l) = Lk~l a~(k) = n A~,
(3.71)
where A2
+
A2 =
1
J ¢2(u)du,
0
o(V ), for all k: l<k<n.
n
n
as n+oo,
Also, by (3.69), I~nkl = lan(k)I ~
max la (i)l=
l<i<n n
Hence, for every E>O,
(3.72)
The convergence of the finite dimensional distributions of W* to those of W
n
follows directly from (3.71), (3.72) and Theorem 2.1 of Dvoretsky (1972).
By
virtue of Lemma 3.5 and the Doob upcrossing inequality for semi-martingales,
the proof of the tightness of w* follows along the lines of Section 6 of Brown
n
(1971), and hence, the details are omitted.
44rf.fI/X)rR~mAjfAK!?,,,R/~fI;tR~tflA:KI?:;r!?,,,J?JWrJtf'lJm
r\.,
+
Tn' Mn I'r:J'\i"i;
~mr1 Mn'
By virtue of Lemma 3.4,
under H in (1.1), Q and R are stochastically independent, and p{Q =j IH }
o
-n
-n
-n -n 0
-n
= 2
for every j EJ • Thus, i f we let
-n -n
T = \.n a (i)U., n>l,
(4.1)
n
L.J.= l
n
J.
where U.,
l_<i<n, are iidrv, and P{U.=+l}
J.
J.-
=~,
i>l, we conclude that Tn has the
18
-
-
same distribution (under H ) as of T.
o
n
On the other hand, T involves a linear
n
combination of iidrv, and hence, its distribution can be traced without much
problem.
In fact, if one keeps in mind the classical one-sample problem, then
the corresponding rank order test statistic [viz. Hajek and Sidak (1967, p. 108)]
-
has the same distribution (under the null hypothesis of symmetry) as of Tn •
Consequently, the available tables for this situation [viz., Owen (1962)] for
various common scores and small sample sizes provide the necessary tables for
our case too.
-
Since, the distribution of Tn depends only on an (l), ••• ,an (n),
we conclude that under H in (1.1), T is genuinely distribution-free.
o
n
On the
other hand, by Theorem 3.3, it follows that for every real x,
(4.2)
lim pin
n~
where
~(x)
-k
2
x
1
Tn /An <xIHo } = I!TI
is the standard normal df.
J
exp{~t2}dt = ~(x),
-J::JO
Thus, if
~(Ta)
= I-a, O<a<l, we obtain
from (2.4), (2.5) and (4.2) that
lim C(l) = T and lim
a
n,a
n~
n~
(4.3)
C~~~
= T a/2 , O<a<l.
With a view to studying the ARE of the proposed test for various score
functions, we first consider the Bahadur-efficiency of the tests.
For this,
we first consider the following.
Lemma 4.1.
Under (3.17), n
-1
Tn +
~(F)
a.s., as
n~,
where
~(F)
is defined by
(3.33) •
Using (3.42), we rewrite Tn = Tn, 1 + Tn,2 - Tn, 3' where by the Schwarz
Proof.
inequality and (3.43), for j=2,3,
2
.(i»
In -1Tn,]. I -< (n-II . n 1Q2·) ~ (n-II i n lan,]
(4.4)
for n
].=].
~
=
no(n), where n'>O and n'+O as n+O.
k
I n
~ = [a 2 .(i)]2 < ~n',
I
n i=l
n,]
Thus, by choosing n (and hence, n')
sufficiently small, it suffices to show that n
-1
T 1 +
n,
~(F)
a.s., as
n~.
By
~
19
the same decomposition (i.e., (3.42», we can show that
Then, by (3.20) and (2.3), we have
(4.7)
Now, Xi = Qi¢(l) (G(Zi»' i>l, are iidrv with mean
~l(F),
and hence, by the
Kintchine strong law of large numbers,
(4.8)
a.s., as n-+oo.
Also, by theG1ivenko-Cante1li theorem, sup IG (z)-G(z) 1+0 a.s., as n-+oo, so that
z£.E
n
on noticing that IQil ~ 1, V i.?::.l and ¢(1) is a polynomial, we immediately conclude that the second term on the rhs of (4.7), being bounded by
max 1¢(1)( ~1 G (Z.»
l<i<n
n
n 1
.. proof is complete.
- ¢(l)(G(Zi»I, converges a.s. to 0 as n-+oo.
So, the
By (4.2), Lemma 4.1 and the definition of Bahadur (1960) efficiency [cf.
Puri and Sen (1971, p. 122)], we conclude that the BARE (Bahadur ARE) of {T }
n
based on the score function ¢ with respect to {T*} based on the score function
n
¢* is given by
(4.9)
where ~(F,¢) =
e1 (¢,¢*)
f
=
[~(F,¢)A(¢*)/~(F,¢*)A(¢)]2,
00
-00
n*(z)¢(G(z»dG(z), A2 (¢) =
for ~(F,¢*) and A2 (¢*) hold for ¢=¢*.
f
1
¢2(u)du and similar expression
0
Notice that one may rewrite
20
00
In* (z)] 2dG (z)}{
(4.10)
1
IJ
_00
o
1
=
_00
00
J
~2(u)du]I
}
In*(z)]2dG(z)]
-00
1
1
1
(J ~2(u)du){[J ~(u)~(u)du]2/[J ~2(u)du][J ~2(u)du]}
o
0
= p2(~,~).(J
1
o
where ~(u)
J n*(z)~(G(z»dG(z)]2
I
00
0
0
~2(u)du),
= n*(G-l(u» = 2n(G- l (u»-1),
O<u<l.
Thus, (4.9) reduces to
(4.11)
Thus, from the BARE point of view, the optimal choice of
~(u)
is
~(u),
O<u<l,
and as a result,
e(~,~)
(4.12)
= p2(~,~)
is always bounded by 1.
We could have also considered the Pitman ARE, where we conceive of a
sequence {Hn } of alternative hypotheses, such that under H • F(x,y)
n
= F(n) (x,y),
is such that Zl, .•• ,Zn are iidrv with a df G(n)(z) (dependent on n) and
n(z) = n(n)(z) also may depend on z, in such a way that
(4.13)
and
J
~::
G(n)(z)
= G(z)
exists, and n(n)(z)
= ~+n~y(z),
z£E,
ly(z)1 1~(G(z»ldG(z)<oo. Then, if we let
1
(4.14)
(4.15)
~*(u) = y(G-l(u», O<u<l, A(y) = J I~*(u)]2du;
o
p(~*,~) =
(J
. 0
1
~*(u)~(u)du)/[A(~)A(y)],
it follows by some routine steps that the Pitman ARE of {T } with respect to
n
{T*} is
n
(4.16)
21
In this case, the asymptotically optimal score function is ¢=ljJ*.
and M •
Let us now consider the tests based on M+
n
n
Note that here also the
n
null hypothesis distribution of M+ or M is generated by the 2 n., equally likely
n
n
realizations of (Q ,R).
-n -n
[It may be remarked that given Q and R , the vector
-n
-n
(Tl, .•• ,T ) assumes a particular value dependent only on the score function and
n
(Q-n ,R-n ).]
Thus, here, one can enumerate the distribution of M+ or M by direct
n
n
n
evaluation of all the 2 n! equally likely realizations of (gn'~n); by this
constitutio~, the statistics Mn and M+
H.
n are distribution-free under
..
0
The
process of evaluating the exact null distribution of M+ or M becomes pron
n
hibitively laborious as n increases.
However, for large n, by virtue of Theorem
3.3 and well-known results on the boundary crossing probabilities for a standard
Brownian motion, we obtain that for every x>O,
lim P{M+ < xlH }
n 0n-x>o
(4.17)
(4.18)
lim P{M ~ xlH o }
n
n-x>o
= 2~(x)-1,
Ik:_oo(-1)k[~((2k+l)X)-~((2k-l)x)].
=
Note that if W be the upper 100a% point of the df in (4.18), then by (2.8),
a
(2.9), (4.17) and (4.18),
(4.19)
lim M+
n-x>o n,a
=
T
a
/2 and lim M
n-x>o n,a
= W : O<a<l.
a
If we denote the rhs of (4.17) and (4.18) by H+ (x) and H(x), respectively, we
note that by (4.17), for large x,
(4.20)
Also, noting that l-H+ (x)
(4.21)
~
l-H(x)
~
2[1-H+ (x)], we have for large x,
-log[l-H(x)]
= ~x2{1+o(1)}.
Further, by Lemma 4.1 and (3.12), it follows that as n-x>o
22
~+
n M + ~(F)/A a.s., and n
(4.22)
n
~
2
M + I~(F)I/A a.s.
n
Hence, the efficacy [in the sense of Bahadur (1960)] of either M+ or M is
n
~2(F)/A2
(4.23)
where
~(u)
=([
1
o
n
1
~(u)~(u)du)2/(f ~2(u)du),
0
is defined after (4.10).
As such the BARE of M+ (or M ) with respect
n
n
to Tn in (2.4 [or (2.5)] is equal to 1, when the same score function
employed in both the cases.
~(u)
is
On the other hand, in (2.4)-(2.5), our sample size
is prefixed and equal to n, while in (2.8)-(2.9), it is a random variable N ,
n
and Nn can be smaller than n with a positive probability.
In fact, by Lemma 4.1
and (4.19), it follows that for every £>0,
(4.24)
P{Nn >
£nl~(F)+O} ~ p{n~T[n£]/An
= p{n-lT[ n£ ]/An
>
and a similar result follows for M.
n
>
M:,al~(F)+O}
n~M+I~(F)+O}+O,
as n+oo,
n,a
Consequently, when H is not true, one may
0
expect a considerable amount of reduction of the ASN of the 1st sequential procedure, without any loss of the BARE.
RNvJ'WW.~;tJt~~k~)~M~&J~jW~>Jr,
D:
~
Dn •
Note that by Lemma 3.4, under
n
H , Q(S ) assumes all possible 2 realizations j £3 , each with the equal
o - -n
-n -n
n
probability 2- • By a look at (2.12) and (2.14), we observe that the set of
realizations of (T 1, ••• ,T ), and hence, of D+ or D , generated by the set of
n
nn
n
n
n
2 equally likely realizations of Q(S ), can be traced, and the exact null dis- -n
tribution can be computed. By virtue of this constitution, the tests based on
D+ and D are distribution-free.
n
n
It follows from Theorem 3.6 that as n+oo,
)
(5.1)
P{D+ < x} + p{sup Wet) ~ x},
n t£I
v O.s.x<oo;
23
P{D ~ x} -+- p{sup IW(t) I ~ x}, V O<x<oo.
n
tEl
(5.2)
and M+ (or D and M ) both have the same limiting null distriAs a result, D+
n
n
n
n
bution given by (4.17) (or (4.18».
(5.3)
Consequently, as in (4.19),
= Wa :
lim D+ =T /2 and lim D
n+oo n,a a
n+oo n,a
O<a<l,
and (4.20)-(4.21) also apply to these statistics.
Let us now denote by
(5.4)
T(X) =
f
X
TI*(z)¢(G(z»dG(z), -oo<x<oo.
-00
Lemma 5.1.
Under (3.17) and the conditions of Theorem 3.2,
(5.5)
n~D+ -+- sup T(x) / A a. s. , as n+oo,
n
x
(5.6)
n~D -+- sup IT(x) 1/ A a. s. , as n+oo.
n
x
Proof.
As in the proof of Lemma 4.1, we write, on using (3.42), T =
nk
Tnk ,l + Tnk ,2 - Tnk ,3'
Then, for j=2 or 3,
l<k<n I
(5.7)
max
n
-1
T k'
n ,J
I~
(n
-1, k
-1, k
2
[.'-la . (i»
1.- n,J
< [lI. n a 2 j(i)]~n',
- n 1.=1 n,
1.-
~(k/n){ll.n1a2
.(i»~
n 1.= n,J
-
2 ~
[.'-lQ.) (n
~
1.
where n'(>O) depends on n(>O) in (3.43), and n'-+-O as n-+-O.
Consequently, it
suffices to replace Tnk by Tnk,l and ¢ by ¢(l) in D:, Dn and T(X), respectively,
where by (3.42), ¢(l) is a polynomial, and hence, (3.51) holds.
For some arbitrary £>0, choose a set of m (=m) points
£
where £m£ > 1-£.
(5.9)
We also, denote by
knj = [nj£]+l, for j=l, ••. ,m, knm+l=n.
Note that TI*(z) is absolutely continuous and bounded, ¢(l) is a polynomial and
24
G is absolutely continuous with G(Zj)-G(Zj_l)
~
E,
l~~l.
Hence, for every
0>0, there exists an E>O, such that
(5.10)
IT(X)-T(y) I~o .for every x,ydZj_l,Zj], l~~m+l.
On the other hand, (3.17) insuring (3.69), and ITI*(z)l~l, imply that for every
knJ" l<k<n<k
"
-.....:: nJ
q
Ian, 1(1)"I
k 1) I~ n -1\L'=k+l
In-1 (Tnq, l-Tn,
1
(5.11)
1
k
~ {n- Li~k+l a~ 1 (i)}~ ~ {n- Li~~
11
,
~
(f
jE
(j-l)E
an2 1 (i)}~
nj-l
'
1
1
~(l)(u)dU)~ < ~o, for every l~<m+l.
Consequently, it suffices to show that
(5.12)
max I{ max [n-lT" lJ-T(Zj)
l<"<m+l l<i<k
n1,
~- - nj
1+0,
a.s., as n~.
The proof of (5.12) follows along the lines of Lemma 4.1, and hence, the
details are omitted.
Let us now denote by
(5.13)
T~
= sup
T(X) and TO
= sup
x
IT(X)I.
x
Then, by (5.1), (5.2), (4.17), (4.18), (4.20), (4.21), Lemma 5.1 and (5.13),
it follows that the efficacy of D+
(or Dn ) in the sense of Bahadur (1960) is
n
given by
(5.14)
Here also, we note that if we let for -oo<x<oo,
(5.15)
= 0,
and i f we let
otherwise,
25
=f
(5.16)
1
o
1jJ2(u)du,
x
-oo<x<oo
then we have
(T~)2/A2
(5.17)
= sup
[A~(1jJ)p2(1jJx'¢)]'
X
where p2(1jJ ,¢) < 1.
x
Note that A2 (1jJ) is non-decreasing, so that if we let
x
¢(u) = 1jJ00 (u) = ~*(G-l(u», O<u<l, the rhs of (5.17) is maximized; for any other
¢(u) (not
proportional to 1jJ00 (u», the rhs of (5.17) is bounded from above by
"
"
1
" A~(1jJ)
= I1jJ~(u)du,
o
so that
00
(5.18)
Hence, here also, maximizing the BARE leads us to the asymptotically optimal
sCore function 1jJ00 (u)
=
~*(G-l(u»,
O<u<1.
A similar result holds for D".
n
In
the next section, we shall study the optimal score function, in little more
details, for some important cases.
In some important special cases,
~*(z)
can be written in more explicit forms, and the optimal score functions
can be obtained in simpler forms too.
6.1.
Stochastically independent components.
Here X and Yare stochastically
independent, so that for all (X,y)EE 2
(6.1)
Let f
F(x,y)
l
and f
2
= F(x,oo)F(oo,y) = Fl (x)F 2 (y),
say.
be the density functions for F and F respectively.
l
2
Thenby
(3.1) and (3.2),
(6.2)
(6.3)
g(z) = f (z)[1-F 2 (z)]+f 2 (z)[1-F (z)], ~(z) = f (z)[1-F (z)}/g(z);
l
2
l
l
~*(z)
=
[f l (z) [1-F 2 (z)]-f 2 (z) (1-F l (z)]]/(f l (z)(1-f (z)]+f2 (z) [l-Fl(z)]]
2
26
where the hazard rates rl(z) and r 2 (z) are defined by
(6.4)
ri(z) = f.(z)/[l-F.(z)],
zEE, for i=1,2.
1
1
Now, under H in (1.1), F =F 2 , so that rl(z) = r 2 (z) for all z.
l
o
We consider
two special cases where F and F may differ in locations or scales.
l
2
First
consider the model
Then r 2 (z) = r (z-6), so that by (6.3),
l
For small 6, (6.6) yields (whenever rl(z) is differentiable)
(6.7)
r*(z)
~
d
(6/2)[dz log rl(z)], zEE.
Thus, for local translation alternatives, the asymptotically optimal score
function is
(6.8)
~oo(u)
= [(d/dz) log rl(z)]
-1
z=G
,O<u<l.
(u)
We may recall that the classical two-sample location problem [viz., Hajek
and Sidak (1967, p. 66)], the locally most powerful rank test corresponds to the
score function
(6.9)
In general, (6.8) and (6.9) are different from each other.
To show this, let
us consider the general exponential type of df's for which Fl , f l and fi exist
and the following hold:
(6.10)
d~ { 1;:~~;) }
+
0 as x+oo and
d~ { :~~:~ J
+
0 as
Note that (6.10) implies that -[l-F l (x)]fi(x)!ff(x)+l as x+oo and
as X+OO, so that as x+oo,
>+-00.
Fl(x)fi(x)/ff(x)+l~
27
d~ log r i (x) = fi (x) /f l (x) + f l (x) / [l-F1 (x)]
(6.11)
= [fi(x)/fl(x)]{l+f~(x)/[l-Fl(x)]fi(x)}
=
[fi (x)/fl(x)]{o(l)},
and as x+-oo,
d
(6.12)
{
dx log rl(x) = [fi(x)/fl(x)]
= [fi(x)/fl(x)]
f 1 (x)
f 1 (x) }
1 + l-Fl(x) • fi(x)
•
flex)
Fl(x)
}
.1+ l-F (x) • flex) [1+0(1)]
{
l
= [fi (x)/f (x)] {l+F (x) [l-F (x) ]-l[l+o(l)]}
l
l
l
= [fi(x)/fl(x)]{l+o(l)}.
Thus, l/Joo(u) behaves alike 1jJ(u) as u+ 0, but differently when u+L
In particular
for normal df, fi(x)/fl(x) = -x, so that it appears that l/Joo(u) attaches more
weight when u is small and less as u+l.
important too.
From one point of view this is quite
If the null hypothesis is not true, with greater weight for
small u, the T will be crossing the barrierD+
(or + D ) faster than the
n,a
- n,a
nk
other case.where l/Joo(u) would have attached more weight to the upper tail.
we would
~xpect
an early termination in such a case, and hence, the ASN for the
progressively censored test will be smaller when H does not hold.
o
Consider now the scale model where
(6.13)
In this case, r (z)
2
=
-1
8
r (z/8), ZEE, so that
l
(6.14) .
and hence for 8=1+0, 0 small, (6.14) tends to
(6.15)
Thus,
(-0/2){1+z(d/dz) log rl(z)}, ZEE.
28
Consequently, for local scalar alternatives, the asymptotically optimal score
function is
~oo(u)
(6.16)
= l+[z(d/dz)
.
log r 1 (z)]
-1
z=G
,0<u<1.
(u)
By arguments similar to (6.10)-(6.12), it follows that (6.16) is generally
different from the optimal score function for the classical two-sample scale
problem.
6.2.
Interchangeable components model.
Here we assume that (1.1) holds under·
Ho and under alternative, X and Y-6 are interchangeable for some real 6.
Thus,
under alternative,
F(x,y)
(6.17)
= Fo (x,y-6),
where F (x,y) - F (y,x) for all (x,y).
o
0
(x,y)£E 2 ,
Let us denote the joint survival function
by
(6.18)
F(x,y)
=1
- F(x,oo) - F(oo,y) + F(x,y), (x,y)£E 2 •
Then, note that under (6.17) and small 6,
00
(6.19)
TI*(z)
=
00
6[f (z,z) - J [(a/du)f (x,u)]
o
0
u=z dx]/[2J f 0 (x,z)dx] + 0(6),
z
where f
o
z
is the density function corresponding to F.
when f o (z,z)
0
= f2(z),
V z£E.
0
(6.19) reduces to (6.7)
For specific f 0 , such as the bivariate normal
density, (6.19) may be evaluated and the corresponding
In general, these are quite complicated.
~(u)
can be determined.
.29
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[1]
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Stochastic comparison of tests.
[2]
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[3]
BROWN, B.M. (1971).
~, 59-66.
[4]
DVORETZKY, A. (1972). Asymptotic normality for sums of dependent random
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HAJEK, J. (1968). Asymptotic normality of simple linear rank statistics
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~
V
~
v
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Theory of Rank Tests.
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PURl, M.L. and SEN, P.K. (1971). Nonparametric Methods in Multivariate
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