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TESTING TWO FAILURE RATES NITH
RANDOMLY RIGHT CENSORED DATA AND
UNCENSORED DATA

By

Michael George Biake

A DISSERTATION
Submitted to
Michigan State University
in partiaI fquiTTment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Statistics and Probability

1981

 

 

1'
ARA-

in. «a m\\0

ABSTRACT
TESTING TNO FAILURE RATES WITH

RANDOMLY RIGHT CENSORED DATA AND
UNCENSORED DATA

By

Michael Goerge Blake

We study the problem of testing H0: h](t) = h2(t) almost
everywhere Lebegue measure (u) versus the alternatives
H}: h2(t) 3 h1(t) almost everywhere (u) and h2(t) # h1(t)
almost everywhere (u) with randomly right censored data and un-
censored data where hi is the failure rate function for the right

sided distribution function F1, i = 1,2. We show that the functional

m t
a(o; F1. F2) = 6 g {F](t)F2(s) - F](s)F2(t)}dm(s)d¢(t)

where ¢ is a strictly increasing continuous function has the property

that for (F1, F2) 6 H0 U H1 then A(¢; F1, F2) = 0 if and only

if (F1, F2) 6 H0 and A(¢; F1, F2) > 0 if and only if (F1, F2) 6 H1.
In the uncensored case, we give sufficient conditions so

that the estimator A(¢) = A(¢; Fm, F") is asymptotically normal

where Fm, Fn are the right sided empirical distributions of

F1, F2. In the randomly right censored case, we give sufficient

A A

conditions so that the estimator 2(m) = A(m; Fm, Fn) is asymptotically

A

I normal where Em, Fn are the Susarla-Van Ryzin estimators of
F1, F2. We also derive a consistent estimator for the asymptotic

variance of 3(m) under HO.

Kochar (1979) studied the above problem only for the un-

censored case where he used the functional

5(F F

0‘s 8
O‘fir‘l’

{F](t)F2(s) - F](s)F2(t)}d(x‘F‘1(s) + (M)? (s)d(T=‘ (t)

l’ 2) 2 l

+ (l-A)E2(t))

where O < A < l and E} = l-Fi, i = 1,2. In this paper, we show

A A

that the estimator 5(Fm, F") is asymptotically normal under

appropriate conditions. We also derive a consistent estimator for

its asymptotic variance under H0.

ACKNOWLEDGEMENTS

I wish to express my sincere thanks to Professor Hira Koul
for his guidance in the preparation of this dissertation. The
advice and encouragement he gave are greatly appreciated.

I wish to thank Dr. V. Susarla for several very helpful
discussions about my dissertation. I would also like to thank
Professors Dennis Gilliland, Joseph Gardiner, and Sheldon Axler for
their review of my work. The excellent typing of the manuscript
was done by Mrs. Clara Hanna.

Finally, I would like to thank my parents, Mr. and Mrs.
George A. Blake, for their constant encouragement during my years

as a graduate student.

11

INTRODUCTION AND SUMMARY .......................................
Chapter
I . £N(¢) IN THE UNCENSORED CASE .......................
l. Asymptotic Normality of /N'(8N(¢) - A(¢))....
2. Pitman A.R.E. ................................
II THE ESTIMATOR ZN(¢) ..............................
1. Notation and Preliminaries...z ................
2. Asymptotic Normality of VN'(AN(¢) - AN(m))....
3 Consistent Estimators of a? and oz
Under H0 ....................................
4. Computational Aspects ........................
III THE ESTIMATOR SN ................................
l Notation and Preliminaries...z ................
2. Asymptotic Normality of /N (6N - 6N) .........
3. A Consistent Estimator of as .................
. «2 «2
4 Computational Form of OO,m and 00’” ........
APPENDIX A .....................................................
APPENDIX B .....................................................
BIBLIOGRAPHY ...................................................

TABLE OF CONTENTS

Page
1

7
7

12
I3

24
28
32
32

42
47
57

INTRODUCTION AND SUMMARY

1,...,Xm be iid G]; Yl""’Yn

be iid 62’ where F1, F2, G1, G2 are

Let X1,...,Xm be 11d F]; X
be 11d F2; and Y],...,Yn
right sided absolutely continuous distribution functions, i.e.,

F](t) = P(X1 > t), such that F](O) = F2(O) = G](O) = 62(0) I. All

the random variables are independent. Also the life distributions

F F2 are on [0,m) with densities -f1. -f2 TGSPECthEIY-

1!

We get to observe Ui - min (Xi, Xi)’ 61'= [Xi < Xi] for

l f i f m and Vj = min (Yj’ Y3), yj = [Yj < Yj] for l 5 j 5 n,

where [A] is the indicator of the set A.

This model is called the two sample randomly censored model.
In it the random variables X} and Y3 are the censoring random variables.
The Gi's may have all their mass at m which corresponds to the
uncensoring model.

On the basis of the above observations, we wish to test

h = h2 a.e. Lebesgue measure (H) vs. H]: h2(t) 3 h](t)

O‘T
a.e. (u) and h2(t) # h1(t) ’a.e. (u) where h1, h2 are the failure

rates of F1, F2 respectively, i.e., for i = l, 2,

fi(t)/Fi(t) if F1.(t) > O

(l.l) h.(t)

+ m if Fi(t) = 0

2

Our plan of constructing a test statistic is the following.
Let F = {(F1, F2): F1, F2 are right sided absolutely continuous
distribution functions and F](O) = F2(O) = l}. For (F1, F2) e F,

1et

(1.2) D(s,t) = F](t)F2(s) - F](S)F2(t) for t 3 S 3 O
and

(1.3) T = sup {tz F2(t) > D}.

If D(s,t) 3 O for all t 3 s 3 0 then F2(t) f F](t) for all t
since F](0) = F2(O) = l. Thus if D(s, t) 3 O for all t 3‘s 3 0
then h2(t) 3 h](t) for all t 3 T. Now for t < T and D(s, t) 3 0
for all t 3 s 3 0, we have that

F1(t) F](S)
(1.4) ‘F—z'mz—Z—(ET f0T311T>t2520.

F (t)
But (l.4) implies that -1 is a nondecreasin function for t < T.
F210 9

F
Since F1, F2 are absolutely continuous and 4:1 is nondecreasing for
2

t < T, we have that
Emma) - f1(t)F2(t)
F3 (t)

 

> 0

(1.5)

for almost all t < T where almost all is with respect to Lebesgue
measure. But (1.5) is equivalent to h2(t) 3 h](t) for almost all
t < T. Thus, if (F1, F2) 6 F and D(s, t) 3 O for all: t 3 s 3 O
. then (F1, F2) 6 H0 U H].

Now let (F1, F2) 6 H0 U H], then for t 3 T we have that

D(S’ t) = F](t)F2(S) 3 0 where S 5 t. Also notice that if
(Fl’ F2) 6 Ho U ”1’ then F](t) = 0 implies that F2(t) = 0. Thus

if t < T then F](t) > 0 and F2(t) > 0. Since (F F2) 6 H U H],

l’ O
h2(t) 3 h1(t) for almost all t < T then implies that (l.5) holds

for almost all t < T. Since F1, F2 are absolutely continuous and
F

(1.5) holds for almost all t < T, we have that 'F1 is a nondecreasing
2

function for almost all t < T. But F1, F2 are continuous, therefore,

F

'F1 is nondecreasing for all t < T. Hence (l.4) holds for all
2

T > t 3 s 3 0. Therefore, if (F1, F2) 6 H U H then D(s, t) 3 0

O 1
for all t 3 s 3 0. Thus for (F1, F2) e F, we have that

(1.6) (F1, F2) e Ho U H1 iff D(s, t) 3 O for all t 3 s 3 0.

Now let w: [0, w) + [0,w) be a nondecreasing right continuous

function such that w(0) = 0 and let

t
(1.7) Ms») = M93 F1. F2) = (I) 0(5. t)d‘P(S)d‘P(t)-

0‘58

If (F1, F2) 6 H0 then A(¢) = 0 since D(s, t)5 0. But if A(m) = O
and (F1, F2) e H0 U H1 then D(s, t) = 0 a.e. with respect to
“W x pcP where ucP is the measure induced by m. Suppose D(s, t) = 0

a.e.with respect to u and there exist a point (x0, yo) such

‘qu‘P
that xo < y0 and D(xo, yo) > 0. Then D continuous implies there
exists (a, b) x (c,d) such that (x0, yo) 6 (a, b) x (C, d), and for
all (x, y) E (a, b) x (c, d), D(x, y) > 0. Thus if w is strictly
increasing and continuous then “W x u? [(a, b) x (c, d)] = (m(b)-¢(a)) x
(¢(d)-w(c)) > 0. Therefore, D(x, y) = O for all y 3 x 3 0 if

A(w) = O and m is a strictly increasing continuous function. Hence

for (F1, F2) e H U H1 and for those m which are strictly increasing

0
and continuous we have that A(m) = 0 iff (F1, F2) 6 H0 and A(m) > 0
iff (F 1, F 2) e H. Therefore, A(¢) is a reasonable measure of departure
of H1 from H0 for a given pair (F1, F2).

Observe that H1 implies that Fé 3 F1 where F'= l-F. From
Doksum (l969) we recall that Fé is tail ordered with respect to
F3 (F2< t 1'] Fé(x) - x is nondecreasing in
x 3 0. Observe that F2 <t F1 and F1'1 Fé(0) = 0 imply F1"1 F (x) 3 x

for all x 3 O which is equivalent to saying F 2(x) 3 F1(x) for all

F1) if and only if, ‘F

x 3 0. On the other hand F2< t F1 and h1 nondecreasing and f1 > O
a.e. imply h2 3 h1. To see this observe that FE <t F1 implies

f2(x)
h2(X)

IV

f1(F'1-1 Fé(x)) for all x 3 O which in turn yields
f1(T1"T‘2(x))/n-T1 (E1'1 (F2(x))}= n1(E1"T 2(x))>h1(x)

for all x 3 0. Thus the alternatives H1 are smaller than the slippage

IV

alternatives Ha: Fé 3 F1 with strict inequality somewhere, and larger
than the alternatives F2 <t F1, F1 1. F. R and f1 > O a.e.

In Chapter 1, we study the uncensored model where we use
ONkp)= A(Q Fm, F") as an estimator of A(m) where Fm, Fn are the
right sided empirical distribution function for F1, F2 and N = m + n.
We obtain the asymptotic normality of /N'(£N(?) - A(¢)) and obtain a
consistent estimator of its asymptotic variance. Also a member of
this class of tests, one based on 9(x) = x, is found to have asymptotic
Pitman efficency relative to the locally most powerful rank test equal
to .84375 at Fe(x) = e-px exp {- 9-p2x2}, e > 0, equal to .7875 at

2
F6(X) = (1 - (3)6.x + ee'x(l - (l-e'x)3), e > O, and equal to l at

F6(X) 1 -exp (-{x + e(x + e-x-l)}), e > O, the Makeham distribution.

Whereas, the asymptotic Pitman efficeny of the Wilcoxon test relative
to the locally most powerful rank test (L.M.P.R.) are .0938, .35 and
.25 at the above alternatives. Also the asymptotic Pitman efficency
of the Savage test relative to the L.M.P.R. test are .5, .7292

and .75 at the above alternatives.

In Chapter 2, we study in the randomly CEM;OEEd model the asymptotic
normalitKNo: {N (36(m) - Aq(¢))3 where AN(9) = 6 630(5, t) d 9(s)dm(t),
Zfi(¢) = A .6 {Fm(t)Fn(s) - Fm(s)Fn(t)} d ¢(s)d m(t); Fm and Fn are
the modified Susarla and Van Ryzen estimators of F1 and F2; and
Mn + T = sup {tz F2(t) > O} at an appropriate rate, under the assumption
that T 5 sup{t: 62(t) > 0} and sup{t: F1(t) > 0} 5 sup{t:

G1(t) > 0}. We also obtain a consistent estimator for the asymptotic
null variance of JN'XN(@).

Kochar (1979) studied the above problem for the uncensored case

using

t
6(F1. F2) = I D(s. t)d(AF1(s) + (T-I>T,(s)>d<>.f1(t) + (l-x)‘F’2(t))

0%8

O
and 3(Fm, Fn) instead of A(9) and £(m) where F(t) = l - F(t).
He found that the asymptotic Pitman efficency of the 3 test relative ‘

6+1)x’ e > 0’

to the L.M.P.R. test is equal to .8203 at Fe(x) = e'(
equal to .9843 at Fe(x) = (l-e)e'x + e e'x(l-e'x)2), e > O and equal
to .7813 at Fe(x) = (l-O)e'x + e e'x(l-(l-e’x)3), e > 0. Whereas,

the asymptotic Pitman efficency of the Wilcoxon test relative to the
L.M.P.R. test is equal to .75, .625, and .35 at the above alternatives.
, In Chapter 3, we will study the asymptotic normality of “fi-(EN - 5N)

for the randomly censored case where

M
N t __ ._ __
EN = 6N (F1, F2) - g A O(s, t)d(ANF1(S) + (l-IN)F2(s)d(ANF1(t)
+ (I'AN)F2(t)),
EN = 5N(Fm, Fn), AN = %-, and MN + T at an appropriate rate.

In what follows ”3“ means ”converges in probability to"

and "5” means "converges in distribution to".

CHAPTER I
8N(¢) IN THE UNCENSORED CASE

1. Asymptotic Normality of /N(3N(O) - O(O)).

 

In this section, we will restrict the model of Chapter 1 to
the uncensored case, i.e., G1 and G2 put all their mass at w. Hence

U1 2 X1, 61 E 1, Vi a Yj and y. s l for l f 1 < m and n1 < j < n.

j m
- - l. =._
Let N - m + n, Fm(t) - m .2] [X1 > t], Fn(t)nj)1 [Yj > t].
y

< x] + O(x)[y 3 x1). Also assume

that 0 < A < l, where X- -1im %-.

g X h (X.; Y.). Thus 8N(¢)
i=1 j=l

is a two sample U-statistic of degree (1,1) with symmetric kernel

Observe that AN(¢)=

h(P and E(£N(O)) = A(O). So by the theoEy of2 U- statistics (c. f.

pm and Sen), x/fi(3N(O) - Am) 3 MO, 3—1 ) where
a? = Var(O(X) ] O dF' + O 2(X) F2( (X) - jY O2 sz- O(X) Z O dfé) and

a; Var( (O(Y) ] O dF1 + O 2(Y)F1(Y) - OIOZ dF1 - O(Y) 7 O dF1), provided
0 Y

2

E h ¢(X; Y) < m. The following lemma gives a necessary and sufficent

condition so that E hi (X; Y) < w.

< w and f O2 dFé < m if and only if
0

8

Proof: E h (X;Y) = f 92 dF1 I W2 dFé + I R4 F) dFé + I T4 F2 di}
0 O O O

{N

m X °° X
- 2 ) O(x) f O3 sz dF1 - 2 ) O(x) ) O3 df1 dFé. Thus if
0 O O O
2 , m 2 —- m 2 ._
E h (X, Y) < m, then f O dF < w and f O dF < w.
W 0 l O 2
Conversely, if f O2 dF1 < m and f O2 dFé < m then
0 O
O?(x)F1(x) + O and O?(x)F2(x) + 0 as x + w. So letting
A - sup {O?(x)F1(x)} and B = sup {O2(x)F2(x)}, then 0 5 A < w,
0§x<m Ofx<m
OfB<oo 611d

E h2 (x; Y) < f O? dF1 f O2 dFé + A f O2 dFé + B f O2 dF1 < e.
I ' O O O O

2 2

Thus if f O df' < e and f O < m then E h2(X; Y) < e. D
O 1 O I

2

So by the theory of U-statistics and lemma 1.1, we arrive at

the following.

 

Thoerem 1.1: If f O2 dF1 < m and f O2 dFé < m, then
0 O
2 2
m“ - ())+~(031+°2) h
(AN(W) A W D O 1 T:X' w ere

x —- 2

x m
a? = Var(qXX) (1O «N, + O (x)F2(X) - A O2 d‘F‘2 - W) {O 61-3)

and

2 Y -- 2 Y 2 - ” -—
02 = Var(O(Y) I O dF1 + O (Y)F1(Y) - f O dF1 - O(Y) f O dF1).
0 O Y

n
Now let VN 1(X1, 9) = l- 2 h (X.; Y.),

3

Sl—i
ZN Lo-
—_I

.(2

V

n

3|

[.4
—l
TIME

m
1;] hW(X1’ Y.), S

VN’2(YJ.; O) = J

and
52 (W) = 1 E {V (G ) - A ( )12 8 Pro osition 4 of Sen
N.2 n-1J.=1 N, 2 "P N ‘P ° V p

2 2 2 1 2 .
(1960), we have that S (O) + o and S (O) + o prOVTded

E h$(x; Y) < w. Thus we have the following theorem.

Theorem 1.2: If f O2 dF1 2 <
O

< m and f O2 dF
0

then
81102) - MO)

1 S2 52 D
J11, summer; sNzw)

Corollary,1.l: Under H0: F1 = F2 = F,

 

811W)

 

 

1_ S2 52 —+ MO. 1) provided [O2 dF < e,
O

5111(O)+-:1 511,2(991 D

2. Pitman A.R.E.

In this section, we will study the Pitman A.R.E. of the
AN(x) test relative to the W1 test of Kochar, the Savage test, the
Wilcoxon test and the L.M.P. rank test for the following alternatives

belonging to H1.

1+9.

(D

., F (x) = [F(x)]

h—(X)=(e+l)hf(X).i. e

O O

-(e+l)x

For F0(x) = e'x, this leads to Fe(x) = e which is the scale

1 alternative in the exponential case.

h— (x) = hf0(x)(l + 9 pm F(x)), i.e., Fe(x) = F0(x)exp{- gm?F(x)}.

10

When Fo(x) = e'px, then "F (x) = p + p2 O x, the linearly increasing

6
hazard rate.

1

, _ -x -x -x k 1_
H4. Fe(x) - (l - e) e + e e [1 - (l - e ) l, k > 2 .
H5: hfé(x) = l + O(l — e‘x), i.e.,

'Fé(x) = l - exp(-{x + O (x + e'x-l)}), the Makeham distribution.

The alternatives H2, H3 and H4 have been considered by
Chikkagoudar and Shuster (1974) where they have obtained the L.M.P.
rank test for these alternatives. The L.M.P. rank test for H5 can

be similarly obtained.

 

 

. d A(x; F0, F6)
Let A = d’e | e = O
on on ch
2 _
){x I F d - 2 f f F (s)ds dt} d F (x)
2 = O O 0 Y o t 0 °
°N N A(1 - A) :

 

C(AN) = lim A , C(L.M.P.) = the efficacy of the L.M.P. rank test
/N ON .
. . . . C(AN) 2
for the alternative being conSTdered, and let e(AN, L.M.P.) - (C L.M.P. ) ,

the Pitman A.R.E. of A to the L.M.P. rank test.

N
' - l. = - \ ’5 ‘ =.l -
2. A - 2 , o (3A(1 A,N) , C(AN) 2 A A ,
C(L.M.P.) = 72XTTZTT" and e(AN, L.M.P.) = .375.

= P‘2(3A(T-A)N)‘*, C(A

For H

. _ 1; g
FOP H3, A ' ON N): 4 (3A(]‘ A))

SJ
11w

C(L.M.P.) = “2 (1' 5’ and ke(AN, L. M. P.) = .84375.

 

 

 

 

For H4=1K11111+21 , ON =(3A(1-A)N)‘*,
C(AN) = 1R31)(Rig: , C(L.M.P.) = {A(1-A)(4kg-])}%, and
d%,LM.P .)- 3H32-U 2.
(k+1) (k+2)

11

For H5, A = 3,011 = (3A(T.A)N)‘£5 C(AN) =

Using Theorem 2.1 of 0-5 (1974),we have G(x; 0) = 1-e'x, G‘](x) = 1OXl-x),

d

MM 0) = e"‘. H(x; e) = EXPI-6(x + e‘x-m. O(u) 1%H(e"(u>; an,
_ (l-u) 32
g<G“(u); 0) 3" 39

where U is uniformly distributed over (0, 1). Therefore,

C(L.M.P.) = and e(AN, L.M.P.) =1.

The following table gives the Pitman A.R.E.'s of the Wilcoxon

 

H(O"(u); O)Ie=0 = (l-u) + 211, and Var(O(U)) = 337,

test, the Savage test, the W1-test of Kochar, and the AN(x) test with

respect to the corresponding L.M.P. rank test for the above alternatives.

Table l. Pitman Asymptotic Relative Efficiencies with Respect to the
L.M.P. Rank Tests.

 

 

 

 

 

Alternative Wilcoxon Savage N1 ANTX)
Hypothesis
H2 .75 1.0 .8203 .375
H3 .0938 .5 .2307 .84375
k = 1 1.0 .75 .70 .25
H4 k = 2 .625 .8333 .9843 .625
k = 3 .35 .7292 .7813 .7875
H .25 .75 .5353 1.0

 

CHAPTER II

~

THE ESTIMATOR AN(¢)

1. Notation and Preliminaries.

(In the previours Chapter, we defined an estimator AN(O)
of A(O) = T I {F1(t)F2(s) - F1(S)F2(t)}dO(s)dO(t), for the uncensored
case. In tgig Chapter, we will define an estimator XN(O) of A(O)
for the censored model. To define AN(O), we first need to introduce

some notation, Let

(1.1) N = m + n
+ m
(1.2) Nm(t) = 2 [U1 > t]
i=1
+ n
(1.3) Nn(t) = Z LV. > t]
i=1 3

+
A- m 2 + N (U ) [6 = 0’ U f t]
(1’4) G 1(t) = R] { m 2 } 1 2

 

 

 

 

l + N;(U£)
+ .,1 " 1
(1.5) §m(t) g Nm(t;Gm (t)
and
(1.7) en“, = NI<t)E;‘(t)

12

13

The estimator N(O) is defined by

t A . . A
N g {F (t1Fn(51 - F (S)F (t>1 dO<s1dO<t>

(1 8) ZN(O) = n

111

08392

where M 5 MN + T sup{t: F2(t) > 0}. Also let

M t
(1.9) AN<O) = g g {F1(t)F,(s) - F1151F21t11 dO<S) dO<t>

The following notation is used throughout this Chapter and

 

 

Chapter 3:
(1.10) H1 = F161 for 1 = 1, 2
‘ S
(1.11) H1(S) = P(61 = 0, U1 gs) = 6 F1 dG1, s 3 0
N S —
(1.12) H2(S) = P(y1 - 0, V1 5 s) = 6 F2 dGz, s 3 0
S
(1.13) H1(S) = P(61 ‘ 1, U1 5 s) = g 61 dF1, s 3 O
as S
(1.14) H2(S) = P(y1 - 1, v1 5 s) = g 62 dFé,s 3 o
Ngm
(1.15) Hm(t) - m
- hm
(1.16) Hn(t) - n
N 111
(1.17) m Hm(t) = 1;] [61 = 0, U1 5 t]
and
~ n
(1.18) n Hn(t) - jZ1EYj - 0, vJ 5 t]
Also F1], 6;] , and Hg] stand for 1/F1, 1/61, and 1/H1 respectively.

2. Asymptotic Normality of /N’(ZN(O) é AN(O)).

 

We start this section by quoting some inequalities from Koul,

Susarla and Van Ryzin (1981) and state them as Lemmas for convenience.

14

Lemma 2.1:

(2.1) E{(1 + N:(Vi))'r|(vi, 11)} 5 c h'r H'" (v.)

(2.1)' {E (1 + N;(u1))'”|(u.

Q2)EM1+fi(wMTMO

J . Y.). (v.. v.)1 g c n'r Hg“ v.)

1 1 J J

mzr HO+N§OMTMA.OL(%.%M_

A
O
S
1

I
—-a 1
1
A
c:
v

1 5 i, j 5 n and where C depends only on r.

 

Lemma 2.2:
. 2d
(2.3) E{( en(v1) - 62(v1)) |(v1, )1 < C(n “T 1154de2
-2d -4d
may H(G(U)-G(UHNHU a 11<uh “UTWH dG
' m i 1 i i’ 1
+m'2d(1 - 61) H14d (U111
(2 4) E{( O (t) - G 2(t))2d} <c h‘d 1 H'4d F dG
' n - 0 2 2 2
whenever H2(t) > 0
. t ._
(2.4)' E{( Gm(t) - G1(t))2d} < c m'd ) H14d F1 dG1
0

whenever H1(t) > 0

for d = 1, 2,..., and where C depends only on d.

 

Lemma 2.3: V.
~-1 -1 2d d -2d 1 4d
(2.5) E{(Gn (v1) - 62 (v1)) |(v1, 1)1 < c n H2 (v1){£ H2 F2 do2
v.
1
- d -— I - -4d
+ (A 112‘5 F2 deg)”E + h d(1 - Y1) H2 (v1)1

15

U.
. ~-1 -1 2d -d -2d 1 -4d -
(2.5) E{(Gm (U1) - (31 (U.)) |(ui, 61)} 5 c m H1 (Ui){é H1 FldG]
Ui
+ (A Higd Fldfii)€ + m d(1 - a.) Hi4d(ui)}

t
-8d - k
+ (6 H2 F2 d62) }

t
. ~-1 -1 2d -d -2d -4d -—
(2.6) E(Gm (t) - 62 (t)) 5 c m H1 (t){g H1 F1 do1
t
+ (g Hi8d F] dE.)*}

whenever H](t) > 0, H2(t) > 0 and for d = 1, 2,..., where C

depends on1y on d.

Lemma 2.4:

(2.7) E{(%n(v,) - F2(v,))2d((v.. .1), 5 c n-d

-6d

”2 (V1)

w)Fhmvsp}<cddm“w)

(2.7)' E{(?m(ui) - F 1 _ 1

1

2d -d H-6d

(2.8) E{%n(t) - F2(t)} 2

f C n (t) whenever H2(t) > 0

(2.8)' E(Em(t) - F](t))2d 5 c m‘d H;5d(t) whenever H1(t) > o,

for d = 1, 2,..., and C depends on1y on d.

We wi11 henceforth assume that T = sup{t: F2(t) > 0}
f sup{t: 62(t) > 0} and sup{t: F](t) > 0} f sup{t: G](t) > 01.
A150 without 1055 of genera1ity, we wi11 assume that T = + m. Now

1et F G , G and M = M satisfy the fo11owing conditions:

1’ F2’ 1 2’ AN N

(2.9) f F1 d v < e for i = 1, 2
o

16

H13 d m)2

(2.10) N‘*( 1+ o as N + e

2

(3?1 HT4 d d)2 + o as N + e for 1 = 1, 2,

-1
(2.12) N ( 1 1

M
I
o
-5 M -3 2
(2.11) N (f H d m) -+ O as N + w
o
M
I
o
(2.13) N'1 q?(M) H12(M) + o as N + e for 1 = 1, 2

(2.14) AN =

2mg

+ A as N + m with O < A < 1, and M = MN + m.

Observe that

M t . .
ZN(¢0 - AN(¢0 = £ £ (Fm(t) - F1(t))(Fn(s) - F2(s)) dw(s)d v(t)
M t . .
- 6 g (Fm(s) - F1(s))(Fn(t) - F2(t)) d ((5) d ¢(t)
M .
(2 15) - é 91,m(t)(Fn(t) - F2(t)) d ¢(t)
M .
+ g 92,n(t)(Fm(t) - F](t)) d ¢(t)
t M
where g]’n(t) = (g F](s) d 9(5) - { F1(s) d ¢(s))Et f M]
t M
and 921M(t) = (6 F215) d ¢(s) - { F2(s) d ¢(s))[t 5 M].

We wi11 show that under conditions (2.9) - (2.14), /N(ZN(¢) - AN(¢))
converges in distribution to a N(0, oz) random variab1e in three steps.
The first step is to show that the first two terms on the right hand
side of (2.15) converges to zero in L2. This step is done in Lemma
2.5. The second step is to show that the 1ast two terms on the right
hand side of (2.15) can be approximated in L2 by a genera1ized U-
' statistic. This is done in Theorem 2.1. Fina11y, we show that this
2)

genera1ized U-statistc converges in distribution to a N(0, 0 random

variab1e.

17

Lemma 2.5: If conditions (2.10), (2.11) and (2.14) ho1d then

 

~ M . M .
N E{(AN(v) - AN(v)) - (g 921M(Fm - F) dv - é 9],M(Fn - F2) dcp)}2

+ 0 as N + w.

Proof: By (2.15), we have

M A M .
N EHKNW) - AN 2)) - I 92.11”)“ - F1) d? +111 91.11% - F2) d<p12
' M t . 2
S 2 N{E(I ((F m(t) - F (t))(Fn(s) - F2(s)) d¢(s) d¢(t))
o o

M t . 2

“6(1) (F m (S) - F](S))(Fn(t) - F2(t)) dcp(S) dcp(t)) }.
However,

M t . .
N 5(5 3 (Fm(t1 - F1(t))(Fn(s) - F2(s)) dv(s) demo)2

M . t . t . x .
= 2NE{g (Fm(t) - F1(t)) é (Fn(s) - F2(s)) dq(s)(g (Fm(x) - F](x)) g (Fn(y)

- F 2(y)) d¢(y) d¢(X)) d¢(t)}

C N

f 5-“ (f (“13(t))H£3(S) d¢(S) d<P(t))2 by Lemma 2.4 +10 by conditions
0 0

(2.10), (2.11) and (2.14).

Simi1ar1y,

M t . .
N E16 6 (F (S) - F](S)(Fn(t) - F2(t)) d¢(5) d CP(t)}2

m
M t
s fi-N-(I [1114(5) 1153(1) d (Ms) d 12(1))2 -> o
0 0
- by conditions (2.10), (2.11) and (2.14). D

Thus vN'(ZN(¢) - AN(w)) can be approximated in L2 by

18

A

M A M
/N(g 92,M(Fm - F1) do - é g1,M(Fn - F2) do). We W111 now proceed as
in section 3 of Susar1a and Van Ryzin (1980) with some modifications
to show that

M .

N E{g g1 M(Fn - F2) do - n‘*’(sn - E(Sn))}2 + o

where

+a M -1 -2 ~ M -
(2.16) Sn = n {6(2 H2 - Hn H2 )M2,Md Hn + 5 g1,M G2
M

1
Hn dv}

First observe that

M M

~ _ 2-1
g g1,M(Fn ' F2) d? ' g g1,M{Hn(Gn

-1 -1
- 62 ) + G2 (Hn - H2)} dm .

M
. -1 -1 “-1 -1
Now | 6 g1,M{Hn(Gn - 62 ) - Mn 62 ( Gz - 62 )} dm 1

” M “-1 -1 1 ~-1 -1
” M -1 ~-1 -1 2
5 4 (5 F2 do) 6 62 ( on - 02 ) do

by equation (3.5) and Lemma 3.1 of Susar1a and Van Ryzin (1980).

Therefore, 1
M . M -1 ~-1

N E{é g1,M(Fn - F2) dw - g g1,M(Hn 02 ( 02 - G ) +

-1 2
62 (Hn ‘ H2)) d@}

A

15 N M 2 M -1 2 2
(6 F2 d?) E{ g 62 ( 62 - on) d?)

IA

IA

c N M 2 M -1 -4 2 . .
‘—§'(I F2 d?) (I G2 H2 d?) + 0 by condition (2.12). But
n 0 0

19

M

A - '1 A
I 91 M M G2 ( G2 ‘ 62) d? ‘ é 91,M H2 62 ( Gz ‘ Gn)ch

M
-1
+ £ 91.M Gz (Hn ‘ M2)( Gz ' G ) d?
Since N E{ M (H - H ) G']( G - G ) d }2
0 91.M n 2 2 2 2 Q

C N '1 ‘4 )2

f -—§-(g 91 M G2 H2 dm + 0 by condition (2.12) we have that
n 3

M 6-1

M
. ~ 2
M E{ g g1.M(Fn ‘ ”2 d? ‘ I 91, M( G2 (Hn ‘ ”2) + F2( G2 ' Gn)) MM}

+ 0 if condition (2.12) ho1ds. It is a1so easiTy seen that

M A M t ~ t _ ~

N E {I 91 MF2( 62 - G)dv - f g] M(t) F2(t)(f --'-'—;-dHn - f Hzldemcp}2
0 ’ 0 ’ 0 1+Nn 0

converges to zero provided (2.12) hons. Now writing n(1 + N"')'1 -

(2 Hal- Hg Hn) as (1 + N :) 1H 2(Hn -H 2) - H2 (1 + N w) + n(1 + N3)"1

H2 (Hn - H2)2 , it can be shown that

-1
2

-2

+ H2

~ 2
Hn)d Hn d¢(t)} 2+ 0

                     

M
N E££ g1 M(t
n
provided condition (2.11) hons.

Therefore, if conditions (2.11), (2.12) and (2.14) ho1d then

M . -1
N E{g g],M(Fn - F2)d o - n F(sn - E(Sn))}2 + o.

SimiTar1y, we have by conditions (2.10), (2.12) and (2.14) that

A

M _% 2
N E£é 92,M(Fm ’ F])d¢ ' m (Sm ' E(Sm))} + 09

M M

u] M( =f 92 M F1 dwt M]. Hence we have the fo11owing theorem.

20
Theorem 2.1: If conditions (2.9) - (2.14) hon, then
m

N E{(ZN(v) - AN<¢)) - m‘*<s - E(sm)) + n‘lfisn - e<sn)>}2 + o

where Sn and Sm are defined by (2.16) and (2.17) respective1y.
Now 1et

¢n<(v1. 71). (v2. 2)) = u, M(V1) Hg‘mm1 -

I
O
o
<
—‘

A
:3
l-J

+

-1 _
“2,M(V2) hz (V2)[Y2 ’ 09 V2 5 M]
V1AM VZAM
1 -1 '
2 ((1) 91,M62 ”‘9 T (I) 91,M62 ‘1‘?)

+

(2.18)
n-1 -2 _
- (25-){EV1 > V21M2,M(v2) H2 (V2)[Y2 ‘ 0» V2 5 M]

-2 -
+ [V2 > V11U2,M(V1) H2 (V1)[Y] ' 0: V] 5 M1}-
Lemma 2.6: Var(on) = 0(¢z(M) H§2(M)).

Proof: As wi11 be shown be1ow, each term in the right hand side of
(2.18) has a second moment bounded by a constant mu1tip1e (C1’ 02,...
are constants) of ¢?(M) H52(M). In particuTar, the f011owing inequa1ities

hon,
E _ 2 -2 -2
([Y] ‘ 09 V] f MJUZ’M(V]) H2 (V])) f C] H2 (M),

- - 2 _ - - ‘
HUI 91," 62 dcp) 5 c2 Hg 62 dc?) - 2 c2 3 F2(v) $92 (u)<.:N> dwv)

5 c3 ¢?(M) H;‘(M).

~ and

21

M
2 -4 _ 3 -—
ENv1 > V21u2,M(V2) H2 (V2)[v2 - 0. v2 5 M1) 5 c4 5 H2 (u) F2(u)d 62(u>
-2
5 CS H2 (M)
A11 three inequa1ities comp1ete the proof of the 1emma. 0

Theorem 2.2: S - E(Sn) + N(0, 0%) where

 

" o
2_”-2 ” 2 e
(2.19) 02 - 6 H2 (t) ({ 91 F2 d m) d H2(t)
u w ‘ 2
and g (u) = f F d.w - f F d W . Provided o < w and
1 0 1 u 1 2

N" d?(M) H£2(M) + 0.

Proof: First notice that

V.AM
n-8 s = g-n (v ) H‘1(v )[ = o v < M] + l- [J g 6'1 do
n n § “2,M j 2 j Yj ’ j - n 1 0 1,M 2
_2 n n _2 _
n 321 kg] “2.M(Vd) H2 (Vj)[*j 0’ vi 5 MJEVk > VJ]

‘

(3)..| Z. ¢n ((vjs Yj): (vks Yk))

where 2' stands for summation over a11 (J, k) such that 1 f j < k f n.
Letting vn,]((V], v])) = E(¢n|(v1, v1)). 1n,2 = d“. and

9n = E(n'% S"), we then have
V1AM
-1 -1 -

- (flfil)u2’M(V1) Hé11V])LY] = 0, V1 5 M] +

vaM
-1
E(é 91,M 62 dv)

22

+

-‘l _
2 E(Uzm (V2 ) H2 (V2)[Y2 ‘ 0: V2 f M1)
Mr M
H-2
(T 1) of1A NZ M H2 d H2
V1AM
f 6'] do + (911) (v ) H'1(v )[ = o v < M]
0 g1,M 2 n u2,M 1 2 1 Y1 ’ 1 -
V1 AM VZAM
%) I u2,M H2 d H2 + 51% 91.M G2 d?)

+

-1 _
2 E(“2,M(M2) H2 (V2)[Y2 ‘ 0, V2 f M1)-

From Hoeffding (1948),
-% _ 4 n-2 2
Var(n Sn) - n n-1 Var(wn’1) + E(fi:T)'Var(Mn,2)’
and
“ 2

E{(sn - 5(sn)) - 2 n'M 1 (Mn,1((Vj’ Yj)) - en)}

%( $1 Var(wn 1) +— "2 MVar( 1n,2) + 4 Var(wn,1) -8 Var(wn’l)

2 _ 2

’ fi:T'

IA

Var(on) + 0 as N + w

by 1emma (2.6), since N'1 w 2(M) H? (M)-+ 0. Therefore, Sn — E(S")

can be approximated in L2 by 2 n 3 X {on ”(V ., yj )) - en }. Since
2 . 2

(2.21) 4 Var(dn1) + 02 (see appendix A) and 02 < m,

2n 5 1 {Mn 1((VJ , yj )) - en } can be shown to be asymptotica11y norma1.

Thus (Sn - E(Sn)) 0 N(0, 02). D

1 Theorem 2.3: S - E(S ) +~N(0, 02) where
m m D 1

(2.22) 02 W2 12(t) (292 F1 de)2 d fi1(t)

23
u w 2
and 92(u) = 5 F2 do - I F2 d m provided 01 < w and
u

N'1 o?(M) H;2(M) + 0.

Proof: By a simiTar arguement as in Theorem (2.2).

2 2
Theorem 2 4' /N (X ( ) - A ( ))1+ N(0 31-+ 02 ) if 2 < m 2 < m
' ' N ¢ N ¢ D ’ x TTT' ’ °1 ’ O2

and conditions (2.9) - (2.14) hold.

Egggfi. By Theorems (2.1), (2.2), (2.3) and independence.

 

2 2
U U
. . = "‘ J. __2
CoroTTary 2.1. Under ”0’ F1 F2, «N AN(¢)-E N(0, A + ]_x )
provided a? < a, cg < m and conditions (2.9) - (2.14) ho1d.

Remark: Notice that if G1 and G2 put all their mass at + m, i.e.,

there is no censoring, then

w y w y
a; 2{6 g 91(5) F2(S)9](y)d¢(y)d¢NS) - g g 91(S)F2(S)91(y)F2(y)d¢(y)d¢(S)}

y __ y ._
(a g,(t)dq><t112 szm - <1 5 91(t)d<1>(t) d @1112
Y1 1 ._ 2

Var(g 91(t) d 2(t)) = Var(¢(Y1) 6 w dF1 + w (Y])F](Y1)
11 m
I <92 d'F—1 - cp(Y.l) ] (p dfi). Also
0 11 x
2 x F + 2 1 2 F °° d?

II II
CT*~8
O 8
.<

So the asymptotic variance of VN'(EN(¢) - AN(¢)), in this case wi11 be
. Y Y
-1 1 - 2 1 2 -— ” «—
(1-1) Var(?(Yl) f m dF1 + w (Y1)F](Y1) - I m dF1 - ¢(Y1) I v d 1)
0 0 Y

24

1 x1 x1 w

+ 1' Var(o(x]) é o dFé + m2(X])F2(X1) - é o2 d?é - P(X1) I v d—é)
x

which is the asymptotic variance in the uncensored case.

3. Consistent Estimators of a? and a: under H0.

 

Under H0: F = F2 = F, we have that

1

0% = Z {H£1(t) { F(s)do(s) E F(3)dcp(s)}2 d fi2(t). So 1et

E (s)dcp(s)}2 d fi (t)

(3.1) n

52 = M 1+n
°N,2 é

0%(‘1'

M .
{ Fn(s) d¢(s)

where H (t) = %- 2 EV. < t, y. = 11. We wi11 now show that

53 2 + US under H0 provided conditions (2.9), (2.11), (2.12)

p
and (2.13) hon.

Lemma 3.1: If conditions (2.9), (2.11) and (2.13) hon then
V.

1 1+n 3 ~ M 2
(3.2) HZ {—1—— I (Fn(t) - F(t))d<P(t) I F d<P} YjEV- g M] + 0
a 1+N (v.) o v. 3 P
n J J
T 1+n VJ M 2 2
(3.3) HZ {—7— f F dcpf (Fn(t) - F(t))d<p(t)} 7. EV. _<_ M] + 0
j 1+N (v.) 0 v. 3 J P
n J J
and
v
(3 4) 1‘2 { 1+" f3 i; - F)dcph}l (F - F)de} 7 [V < M] «+ 0
" J T+N+(V.) o " v. " 3 J P
n J J
Proof: V
E{ 1+" fj (E (t) F(t))d¢(t) ? F d¢)2[V M“}
- . < J
3 T+N+(V.) 0 " v. 3 '

25

M
< (f Fdo)2 E{y.[V. 5 nj £5 c——l$fl———
0 J J 3 l+Nn(v.)

Where Ej(°) = E(°l(vj9 Yj)),
M t
< c n“(é F do)2 6 H;2(t) (é H§3 d o)2 62(t)dF}(

3

t)

where we have used Lemma 2.1, Lemma 2.4, and the equality

t 2 t x
(6 b(dew(x)) = 2 f b(x) é b(yldm(y)d P(x).

_] M )2 M t H_3 2 ._
But n (g Fdo 1 H22 (t) (6H 2 do) 62(t) d F2(t)

M M
f (f F ch>2 ("'L2 H§1(M))nJ2 (f H23 dCP)2
0 0

M M
By conditions (2.9) and (2.l0), we have that n-%(f F dcp)2 (f H53 dcp)2 + 0.
0 0

Since w is strictly increasing, ¢(0) = 0 and MN + w, there is an

No, such that \fN 3 NO we have ¢(MN) 3 ¢(l) > 0. Thus

n‘la H;'(M) = o"(i) n'i oii) H5‘<M> g e"(i) n'

by (2.l3). Therefore,
V.

J A M
E(v- (~l:$-- f (Fn(t) - F(t))d¢(t) f chp)2 Ev. 5 M] + 0
J 1+N n(v. ) o v. 3
J J
and so we have that
vj M
91.2 4—1" f (an - F(t))d<p(t) f F dcplz i. tv. 5M1+ 0.
J 1+Nn (Vj ) 0 Vj J J P

By similar arguements, we have that (3.4) + 0 and (3.3) + 0
P P
provided conditions (2.9), (2.ll) and (2.13) hold. D

26

Lemma 3.2: If condition (2.l2) hold then
V.

 

M

 

_ J
l—N —l—-—+" 4121(V.))2 (f chpf chp)2 y. CV. 5 M] + o.
J 1+N ”(V) 3 0 VJ. J J P
Proof:
n+1 -1 2 VJ M 2
Ehrj (-—+—— -H2 (le) (f chPf FdCP) [vi 5 M71}
l+N (v.) 0 v.
M n J J
3 -2 + -2 +
. H . + . - . - .
5 (6 WP) E{YJ 2 (VJ)(l Nn(VJ)) ((n l)H2(VJ) Nn(VJ)
2 M
+ (2H2(Vj) - 1)) f Fdo (V. 5 Mi}
v. J
J
-l M —4 M —
g n c f H2 52 f Fdo dF
0 t
-l M -4
5n (2sz dq>+0 by (212)
0
Thus
1 n+1 V' M 2 '
Hz(—————--H (v.))2(f chpf chp) y.[V.§Ml->0.
J l+N n(v.) J J P
J VJ
Lemma 3.3: If conditions (2.9) and (2.13) hold then
VJ. M
‘32 H§2(vj)yj(f chp)2 (f Fd<p)2 EV. 5 M] + a: .
j 0 VJ. 3 P
Proof° Since 02 < m we have that o2 + 02 where
——4 2 . N,2 2
2 M _] M t 2 z
(3.5) ON,2 6 {H2 (t) { F(s)d¢(s) f F(s)d¢Js)} d H2(t)

Thus we need only show that

27

V.

- J M
(3.6) -l X H 2(V.)y. (f Fd¢)2 (f chp)2 EV. < M] - 02 + O.
n - 2 J J J - N,2
J 0 V. P
J
But
2 -l M 7 -4 M
E{L.H.S. (3.6)} 5 (j Fdo) E (H2 (vj)(f Fd¢)yj[Vj 5 M1)
0 v.
J
_] M )7 M W4 M ._
= n (f Fdo) £2 H m({ Fdo) G2 sz

7M
(f Fdw)7 IH 4do + 0

IA

by conditions (2.9) and (2.l3). Therefore, statement (3.6) holds

which completes the proof. D

Theroem 3.]: If conditions (2.9), (2.11), (2.12) and (2.13) hold

 

«2 2
then ON,2 3 d2 .

Proof: We have that

1'X Y {_ 1+" fj f dm M f d? - H'](v ) Mj F do M F doiztv < M]
nJ‘lliN:j(V)0 n an 2 JO 2 sz J-
v.
l l+n . M .
=32“ 1V)6‘1031 F)dcpf (Fn-F)d<p
+ n
J Nn(V J VJ
l+n Vj ~ M
+ + f (F d? f Fd¢
n
1+N (V ) 0 vi
l+n VJ M .
+ + f Fdo f (Fn - F)do
l+Nn(V.) 0 Vj
l+n -l Vj M 2
+ H2 ( f de f FdP} Ev. 5 M] + 0
l+Nn(Vj) 0 vi 3 P

by Lemma 3.l and Lemma 3.2, and

28
V.

- J M
12 r. H 2(v.)(I chp)2 (I Fdo)2 EV. < M] +02
n . J 2 J J - 2
J 0 Vj P
by Lemma 3.3. Thus as 2 + US by Lemma 1 of Sen (l960). D
, P
Now define 8: 1 as follows,
M M t
.2 m+l “ ‘ 2 z
(3.7) o = f {-——-———- f F dm f F d@} d H (t)
M" o 1+N;(t) t m o m m

where fim(t) i = 1]. Then proceeding as above

i-‘M
F‘1
C
A
f'.‘
to
O)

we have the follo ing result.

Theorem 3.2: If conditions (2.9), (2.l0), (2.l2) and (2.l3) hold

 

62 2
then 0N,] E 01.
Combining the results of Theorem 3.l, Theorem 3.2 and Corollory

2.1, we arrive at the following theorem.

Theorem 3.3: Under H0: F1 = F2

82 82 D
N,] + N,2
m n

provided a? < e, o; < e and conditions (2.9) - (2.14) hold.

 

 

 

 

 

4. Computational Aspects:

Let V(]) f ... f V(n) be the ordered V1,...,Vn and

denote by dl""’dn the corresponding Y],...,yn. Thus dj = Yi

i.f.f. V(j) = Vi'

(4.2)

where

(4.3)

and

(4.5)

(4.6)

29

Observe that for V( 1) 5 x < V(j) one has

N:(x) = n-j+l

 

 

é;1(X) = n {(2 + n-i)/(l + n-i)} 1 . J = l. ,n
i=0
V(O)=0 and d0=].
Observe that Fn jumps only at V(]),...,V(n), and
F(v )_-1 .5 . .[di=0]._
. - n (n-J) H {(2 + n-1)/(l + n-1)} , J - l,...,n.
n (J) i=0
Now
v(i) . i-i .
g F (S)d¢($) ‘ jZO Fn (V(j))(¢(V(j+1)) ' W (V(j))
'-l J [d =0]
= -l 1 _. 2+n-2 2 _
n jZO (n J) £20 {1+n_£} (¢(V(j+])) ¢(V(j)))
M n-l k [d =0]
_ -l 2+n-2 2
6 . Fn(S)d¢(S) — n kéi (n-k) £30 {T:E:EM {¢(MAv(k+]))
(1)
.- P (v(k))}[v(k) 5 M].
Thus
M t n
«2 l+n “ * 2 z
o f j F do f F do} d H (t)
"’2 o l+N:(t) o " t " "
2 n i-l n-l
$1131- 2 d <i+n-i) 2 { z z (n-J)(n-k)
n 1‘] j=0 k=1
x i (2*" “)[di=0] E 2*” litdi=01
2=0 l+n-2 2=0 l+n-2

30

X (¢(V(j+l) “ ¢( V('

J)))(¢(MAV(k+1))

Similarly, let U(]) f ...

and denote by b1,...,bm the corresponding 5],...,6m. Thus

i 1.f.f. U(j) = U.. Then

1
2
«2 _ l+m)
°N,1 ' l'E‘ '

m i

b. = 5
J

i-l m-l
bi (l + m-i 2{ f X
l j= -0 k=i

x % (2+m-z [b2=0] k [b2=0]
_ l+m-2 _
2-0 2—0

(4-7) (m-J)(m-k)

"Ma

2+m-2
l+m-2

 

X (¢(U(j+l)) ' @(U(«

J)))(¢(MAU(k+1))

where U(0) = 0 and b0 = 1.

Now observe that

M . t .
(4.8) 5 Fn(t) g Fm(s) dP(s)d¢(t)
_] m-l j

. 2+m-2
m 2 (m-J) H ( _ )
i=0 i=0 1+m i

[b =01 M
(I Fn

 

M .
‘ ¢QJ(j)) é Fn(t)flJ(j) < t] d¢(t))

j [b =0]
(m-J)(n-i) H (§:+_ :) M
i=0

§|

=0] 1

2+m- :2)[ d2 {2(WZ (U U(j+])AV(T+])AM)

f U(m) be the ordered U1,...

U(J’)))[V(i) V ”(ii 5 M“”<i+1>3”(i+1> > Um]

- P(V(k)))£v(k) 5 M112

-wwmnwk<m¥

M( ) < t]¢(tAU(j+]))dW(t)

3l

+ ¢ (U(j+l))(¢(v(l+])AM) ' ¢(V(l) V U(j+])))

V E”(am V V<i> f M3:V(i+1> > ”(amJ

 

 

‘ W (U(j))(¢(V(i+1)AM) ‘ 9 (V(1) V U(j)))
X [”m V V<i> f M35V<i+1> ’ U(J)]-
Therefore,
~ M A M . M A t .
(4.9) AN(¢) = 6 Fm dw é Fn d¢ - 2 g Fn(t) é Fm(s)d¢(s) d¢(t)
- 1 m-l n-l . . J 2+ _£ [b =0]
- fifi-jgo 120 (m-J)(n-1) £30 (1+$_£) V
i 2+ _1 [d =0]

X £20 (1+$_£) E {(¢(MAU(j+])) ‘ ¢(U(l)))[U(j) < M]

X (¢(M A V(i+])) ' ¢ (V(i)))tv(i) < M] + 2 W (U(j))

X

(¢(V(i+]) A M) ' ¢ (V(i) V U(j)))[U(j) V V(l) : M]

EV<i+1> ’ ”(3')J

 

2 ¢ (U(j+]))(@(v(l+l) A M) ‘ w (V(l) V U(j+])))

X

E”(am V V(i) f M3EV<1+I> ’ ”(:M)3

2

2
-W<WfinAVnn)Am'¢ ”o)VWn”

X “(1) V ”m f ““Uonflfvom > ”(j)“ -

CHAPTER III
THE ESTIMATOR EN.

1. Notation and Preliminaries. Let

Mt.. .. ... = ...
(1.1) EN = 6 6 {Fm(t)Fn(s) - Fm(s)Fn(t)}d(ANFm(s) + (l-AN)Fn(s))d(ANFm(t) +

(l-AN)Fn(t))
and

M t
(1.2) 5N = 6 5 {F](t)F2(s) - F](s)F2(t)}d(ANT,(s) = (T-AN)Té(s))d(xNT,(t) +

(l-AN)Fé(t)).

In this Chapter, we obtain the asymptotic distribution of /N (3N - 5N)

and obtain a consistent estimator for its asymptotic null variance.

2. Asymptotic normality of /fi'(3N - 5 Let F M, G , G

N)' 1’ F2’ l
satisfy the following conditions as N + m, and for i = l, 2.

2

 

M
(2.1) M‘2 1 F1 H§8 do, + o
o
M __
(2.2) N'] f H;]2 G;2 dF1 o
o
M
(2.3) N" j H;5 H55 dF} +.o
o
M M
-l -6 -l .— -6 -l -—
(2.4) N 6H] (31 dF] 6H2 G1 dF1+0
M M
N‘1 I Hi6 6;] dTé / H56 651 dig + o
o o “

33

and
M t
-l -l -12 -— -—
(2.5) N é F2 62 6 H1 G1 dF1dF2 + o
M t
-l -l -12 -—
N g F1 G1 5 H2 G2 sz dF1 + 0
Observe that
N 2 M t . . . . x =
(5N - 5N) = AN(£ é {Fm(t)Fn(s) - Fm(s)Fn(t)}dFm(s)dFm(t)
M t
(2.6) - I f D(s,t)d —F'](s) dl—'1(t))
o o
M t ,. ,. ,. A : 7.
+ AN(l-le(g é {Fm(t)Fn(s) - Fm(s)Fn(t)}dFm(S)an(t)
M t
- 6 A D(s,t)d F](s) d F2(t))
Mt ,. ,. ,. .. z 7:
+ AN(1-AN)(6 é {Fm(t)Fn(s) - Fm(s)Fn(t)} d Fn(s)d Fm(t)
M t
- 6 6 D(s,t)d F2(s)d F](t))
2 M t A A A A 1' I
+ (l-AN) (6 g {Fm(t)Fn(s) - Fm(s)Fn(t)} d Fn(S)d Fn(t)
M t
- f f D(s,t) d F2(s) d F2(t)).
o o

= i§:§ g {E (ui)fin(u.) - ?m(u.)% (U,)}{aj é;1(U )

m J J n J
(an +<Lam4wim+NWun4naéJwi
J m J m J l m 1

“-l -l
+ (T - a.)em (u,)(2 + Nm<”1)> TTU. g u, 5 M].

34

Expanding the right hand side in the usual fashion and using Lemmas

2.2, 2.3, and 2.4, of Chapter 2, we get the following Theorem.

Theorem 2.l: If conditions (2.1) - (2.4) hold then

 

M t . . .

. M t
N E{(é éTFm(t)Fn(s) - Fm(s)Fn(t)} dF} dF} - g g D(s,t)dF} at})

(l -l “-l

m l

g g {(fi Tu.) - F,(U,))F2<uj) - (En(u.) - F2(Ui))F1(Uj)}5i 5

+
anJ“

-l -l
x G] (U1)G1 (Uj)([Uj g u, 5 M] - [U1 5 Uj 5 MJ)

1 -l -l
-,ffD(s t)dF 1dF1)} +0.
0 0

NOw proceeding as in section 2 of Chapter 2, we have the

following.

Theorem 2.2: If conditions (2.1) - (2.4) hold then

 

-
A

M t ,. A .. ., a: M 1'.
AT (6 g {Fm(t)Fn(s) - Fm(s)Fn(t)} dFm(s)dFm(t) - 6 A D(s,t)dT, dT,)
U.AM
_. 1 1 -1 ._
1 3

...;
Osz
‘1']

d
m
N
3
0..
T!
—J
H4

M
l -l .—
+ 5-; (WU-69H1 (v.) 5 (Q1 M + F1 RZ’M)dF][Ui 5 M]

35

U.AM
T _2 M __ ‘_
' g Hl FT 1 (Ql,M + FT R2,M)dFl dGl}
X
1 I "U 6 U Mx = Z
+ HER-MG](i)iQ],M(U1-)[1§MJ- 2 6 g D(y. x)dT=l dF1}
--1 2 (>){(T- )H-](V ) ? F R dF [v < M]
n J 2 VJ 2 3 v. 2 l,M T j -
V.AM J
J -2 M _ _
- 6 H2 F2 j F2 R1,MdF] dGzl
X
V.AM
- 1- {fJ G 1 R d? - W F R dF })
n g 0 2 T,M T 6 2 l,M T

X M
i = l, 2, and 0] M(x) = 6 D(y’x)dFl ' I D(y,x)d?,
: X

Proceeding as above with each term in (2.6), we get the

following theorem.

Theorem 2.3: If (2.1) - (2.5) hold then

 

M t
AV (EN - é 6 0(5, t)d(ANFM + (l-AN)?é)d(AN?, + (l-AN)Fé))

- AV (Afi, (l-AN)2, AN(l - AN)) - (5&1) - S§2))

converges in L2 -to zero, where S§]) and Séz) are defined in

appendix B.

With the notation as defined in appendix B, we have the

following two theorems.

(l) (2)
Theorem 2.4: If I < m and X < m, then

X“) + Z<2)

’ A l-A

 

JM (5&1) - s§2)) E N

20

3 (

36

(1°) (1) 1)
Proof: Since E(SN ) = Q for i = l, 2, IN + Z < m for
‘ - fl+ E J— 1 1 1 1
1 - l, 2, m l/A, n + ]_A , and the U, s, 61 5, vi 5, and vi 5

are all independent, we have that

 

(l) (2)
fir-(5,81) - 5,3“) + N(0,a___1:__a_ + EH) for all a 6 R3.
D
Thus by the Wold-Carmer device, we may conclude that
(l) (2)
Jfi'(s§1) - s§2)) 6 N3 (9, I—:—- + 4:35) . D
(l) (2)
Theorem 2.5: If conditions (2.1) - (2.5) hold and I < m, I < m
then
xfi (aN- g 5 01s. m1 (TNF11s s) + (l-AN)F2(s))d (TNF111 ) + (T-TNTFZTtTTT

converges in distribution to a normal random variable with mean

zero and variance

2

(T . (l-A)2. A( (T A) MIL—-+ légr-Mfi , (l-A)2. TTT-TTT
Proof: This is a result of theorems 2.3 and 2.4. D

Corollary 2. 1: Under HO: F‘3° = F2 = F, if conditions (2.l) - (2.5)

hold, I Gfl F2 dF < e, and j 6'2 F3 as, < e for 1 = T, 2, then

 

/N3 + N(0, 08) where

ND

2_T_-1°°-T 2_ 4 6- -T°°-1 2

co - 64 {T 6 G1 (7 F 18 F + 11 F )dF + (T-T) g 62 (7 F
-l8 F4 + 11 F5TdF‘}.

37

Proof: Under H

0,
(1) 1 1 2 °° _ _ oo
1 = 1 T 2 (%-j G1‘(F2 - 3F4 + 2F6)dF - g%-f Biz (F3 - 2F5 + F7)d§,)
2 2 4 o 0
But ] e;2 (F3 - 2FS + F7)dfi, = j 6;] (3F2 - 10F4 + 7F5TdF, so
0 0
we have that
(1) 1 l 2 w ._
I = T T 2 (fig) f 6;] (7F2 - 18F4 + llF6)dF ,
2 2 4 o
and
(2) T T 2 m _ ._
I = l 1 2 (fig) 1 62‘ (7F2 - 18F4 + llF6)dF .
2 2 4 0
Therefore, by Theorem 2.5, /N EN + N(O,og). D

0

Remark: Notice that if (31 and G2 put all their mass at m,

. . . 2 _ l . .
i.e., there 15 no censorTng, then 00 - 210 A(l-X) , thch 15 the

 

asymptotic null variance of Kochar's statistic VN'EN .

3. A Cohsistent Estimator of °3 . Let

 

(3.1) 03 1 = j 6;] (7F2 - 18F4 + 11F6)dF' for 1 = l, 2
’ o
(3 2) “2 = ) é‘] (7F2 - 18F4 + 11F )d;
' O0,m 0 m m m m m
and

-
A

M - x - -
(3.3) 52 = f G“ (7Fn - 18F: + lng)dF
o

n

 

Theorem 3.l: If 03’] < m, 03,2 < w and conditions (2.l), (2.2)
hold then 83,m P 03,] and 83,” 3 08,2 under H0.
Progf;_ Consider
(3.4) 2 6;] F: dFm = %.igl 6&2 (U,TF; (U1) a, [U1 < M]
+ %-§ é$2(ui)F; (Ui)(2 + M; (U1)) 1 [51 = 0, U1 5 MT

But

2 “-l -l 2 *2 + -l _
5512mm (Up-G1 (11.)) Fm(Ui)(2+Nm(Ui)) [61-0, U15M]
+2.;e2wT1? (U)(2+N (11))“ Ta =0 U <MJ

m 1 i 1 2 i -

However,
~-1 T 2 2 + T _
E{(Gm (U1) - G] (U,)) F (U,)(2 + Nm (U )) [41 - 0. U, 5 M3}

< n‘2 ? H‘7 F d6' 4 o b (2 T)

- o 1 1 y '

and
-2 *2 + -l =
E161 (ui) Fm (U,)(2 + Nm (U,)) (a, 0. U. 5 MT}
<n-17’H-3Fd-G—+O b (21)
_ 0] 1 y . .

Therefore, the second term on the R.H.S. of (3.4) converges in

probability to zero.

39

Now observe that

(3 5) 311N613] (”1) gm (U1) - GI] (U,)F (U1)}2 51. [u1 < M]
l
-1 A] -] " _‘| .. 2
" ,‘n'Z {(Gm (U1) - G (U1))F (U1) + G1(U1.)(F(U)- F(Uim 5
x [U f M]

But

A

-l -1 2 “2
E((Gm(U1. - (31 (u.)) F

by (2.2) and
E {G'2(U.)(F (U.) - F(u.))2 MU.

l 1 m 1 T 1 1

by (2.2). Therefore,

1 “-l ~ -1 2
(3.6) a; {Gm(U1.)Fm(U1.) - (a1 (”1” 51w, 5 M] T; 0.

We will now show that

1'" -2 2 °° -12 —
(3.7) a.) G] (1)1.)F (U145, [U]. fM]->fG] F dF.
T-l P 0
“-12- "-12—”42-
Since I G1 F dF < m, then f G1 F dF + f G F dF1 . Thus
0 o o
to prove (3.7) we need only show that
M
1'" -2 2 -l 2 —
(3.8) T117; {(31 (U1.)F(U1.)<51.[UifMJ-j'G1 F dFl-rO.
1-l 0 P
-2 2 M -12 —
But E(G1 (U )F (U.) 6. EU. < M3} = f G F dF , and
1 1 1 1 - O l

1 G;3 F4 dF'+ 0 by (2.2). Hence statement

(3.8) is true which implies that statement (3.7) is true if condition

Var(L.H.S. of 3.8) 5 n‘

0‘53

(2.2) hold.

40

Therefore, by (3.6) and (3.7) we have that the first term

@

on the R.H.S. of (3.4) converges in probability to f 6;] F2 dF}.
0
By similar arguements, we have that
? G.1 F4 d; + ? G'1 F4 d? and ? G'1 F6 d; + ? G.1 F6 d? Therefore
0 m m m P 0 l 0 m m m P 0 l ' ’

2 . . . 2
00,m ; CO 1 prov1ded conditions (2.l), (2.2) hold and 00,1 < m. D

Combining this result with Corollory 2.1, we have the following

theorem.

 

 

 

Theorem 3.2: Under H0: F1 = F2 = F, if 0811 < e, 0312 < e and
conditions (2.1) - (2.5) hold, then
8 5N
f4. :3. T’ ”‘°’ "
__4_. + 2
m n

. «2
4. ComputatTOnal form of 00,m and 00,” .

 

Using the same notation as in Section 4 of Chapter 2, we

have that
*2 _ l 2 .4 .
(4.l) 00,m - fi-§% 2(U(1.))(7Fm (U (. )) - ll Fm(U(1))+- 18 Fm(U(1)))

x {1.1. + (T-b,T(2 +Nm(U(1)))'1}[U(1.) 5 M1

m

l + m-1

i [b2=01
- , + -
= m 3 Z (m-T)2 H ($ 1 $_})4 (7
1 2=l
. i . 2Tb = O]
_ 11 (m-1)2 (2 + m-T) 2

and

(4.2)

4l

. . i . 4Eb = O]

 

)

x (b1. + (l-b1.)(2 + m-1)")[u(1.) < M1.

- J _. 4Ed = O]
= n 3201-1)2 11 ($3,243) 1 (7
j 2=l
3:112 2 + n-J thz = O]
- ll ( ) n (1 + n-J)
1 2:1
. j . 4Ed = 0]
n;;_4 2 + n-g 2

2=l

x (41. + (1-111.)(2 + n-TT“TTv(1.)5MT.

APPENDIX A

 

In this appendix, we want to obtain (2.2l) where 2 Mn 1

is defined by (2.20). Observe that

v AM
4 Var(wn11) = Var(é1 911M 6;] d¢ + “n,M(vl)H21(Vl)[Yl = 0, V1 5 M]
V‘AM -2 ~ 1 -l
' g u2,M”2 dH2 T fi'{“2,M(V1)M2 (V1)CV1 ‘ 0’ V: f M]
v AM

1
-2 ~
+ 6 “2.MM2 dM2})

= Var(A + B - C + D).

Now we see that Var(D) = O(n"2 H£2(M)) since “2 M is bounded and
that

V AM

I] H'2 dfi < F‘1(M)G"(M) = H“(M)

0 2 2 - 2 2 2

where we have used the equality d M2 = F2 dGé. Consequently,

-2(

2 MT) + O(n“ Hg‘UTTT

4 Var(w ) = Var(A + B - C) + O(n'2 H

n,l
provided Var(A + B - C) is bounded for large N. Thus a; < m
will imply

M -2 M 2 z -1 -2
4 Var(4n11) = 6 H2 (t)({ 911M {2 d4) dH2(t) + 0(n H2 (M))
where fi2(t) = P(y1 = l, v1 5 t) if

M 2 M

Var(A + B - C) = g H; (t) ({ gl,M F2 d¢)2

dH2(t).

42

43

Since
00 mm -2 ~
E(C) = g £ “2 M(S) H2 (S)dH2(S)dH2(t)
M m _2 __ N
= 6 { p2,M(S)H2 (s)dH2(t)dH2(s)
M _] ~
= g “2,M(S)H2 (s)dH2(s)
= 5(3),
we have
Var(A + B - C) = Var(A) + E(BZ) + E(CZ) + 2E(AB) - 2E(AC)
- 2E(BC).
But
” t“” -1 2 —- ” t“” -1
(Al) Var(A) = f (f g.I M62 dw) dH2(t) - (f f 91 M62 dQ dH 2(t))2
0 0 ’ 0 0
on UN _
= 2 é {6 91 M(u f g] M WV)G (V)d¢(v)d¢(u U)}dH2(t)

M
(g 91,M(u)eg‘(u)27 dfiznwwunz
U

M u
= 2 g g1,M<u>F2(u) g g1,M(v)eg‘<v>d¢(v)d¢<u>

M u
2 g 91,M(”) 6 g] M(V)F2(V)d¢(V)dw(U)

2 -2 - _ -2 ~
“2,M H2 F2 dGz - H dH

(A2) E(B)- niM 2 2

N
I
0%:
0%:

(AS)

IT!
A
(.3
N
v
II
0‘5 8

N

0‘53 0‘38

2E(AB)

-2E(BC)

-2E(AC)

II
N

N N
0%.: 0“: 0‘“:

N

(f
>
3

A
O‘fi
T.’
N
U
2
A
C
V
I
N
A
C
V
D.
I
N
A
C
v
V
N
{l
I
N
A
ff
V

>
3

2

u N N _
(u)H§ (u) g uz,M(v)H52(v)dH2(v)dH2(u)dH2(t)

O‘Hf”
1:
N
3

9

l -2 ~

u N
u2,n(U)H£ (U) 6 u2,M(V)H2 (V)dH2(V)dH2(U)

-1 UN 4 ~
,M(t)H2 (t) g 91,M(”)Gz (U)d¢(U)d H2(t)

‘C

A
(+%3 N

tAM ~
g1,M(v)F2(v)d¢KV)) g g],M<u)eg‘(u>d¢<u>Hg‘(t)de(t)
M v

91,M(u)eg‘<u)<£ 9,,M(v)rz(v) i H“

2 (t)dfi2(t)d¢(V)d¢(U)

M UAM _]

-2 g (6 u2,M(V)Héz(V)dfi2(V))u2,M(U)H2 <u)dfi2<u)

M _1 u _2 ~ ~
-2 g u2,M(U)H2 (u) 5 u2,M(V)H2 (v)dH2(v)dH2(u>

0° SAM _.' SAM -2 .... _

'2 g (6 91,M(U)Gz (u)d¢(u))(é “2,M(t)H2 (t)dH2(t))dH2(s)
M _2 t _] m __

-2 6 uz’M(t)H2 (t){é g],M(u)GZ (u) { dH2(s)d¢(u)

M ~
+ { g],M(u)eg‘<u) dfié(5)d¢(U)}dH2(t)

CIR—.8

M

t ~
-2 5 uz’M(t)Hé1(t) é g],M<u>eg‘(u)d¢<u)de(t)

M M ~
-2 I uz M<t>ng(t) { g1 M(u)F2(u)dw(u)dH2(t)
0 ’ 3

M M t _ N
-2 I (I 91 M(v)F2(v)dcp(v))Hg‘(t) 6 91 M(U)Gz](u)d¢(u)dH2(t)
o t ’ ’
M

-2 6 ug’M(t)H§2(t)dM2(t)

45

M M v

= -2 g g]’M(u)G£1(u) j 91 M(v)F2(v)(f H2](t)dfi2(t))d¢(V)d¢(U)
u ’ u

2 M 2 -2 ~
- é p2,M(t)H2 (t)dH2(t).

By adding (Al) through (A6), we obtain that

M u
Var(A + B - C) = 2 f 91,M(U)F2(U) é 9],M(V)62](V)d¢(V)d¢(U)

U
-2 é 91,M(U)F2(U) 6 91,M(V)F2(V)d¢(V)d@(U)

2.
However,
2 2 H-2 dH = M M F 2 22 ~
‘ H2 M H2 2 ' é ({ g],M(u) 2(U)d¢(U)) H2 (t)dH2(t)
M u _2 u ~
= -2 g g],M(U)F2(U){l H2 (t) f g1’M(V)F2(V)d¢(V)dH2(t)}d¢(U)
M u v H_2 ~
= 2 6 g,,M(u)F2(u u) g 91 M(v)2 )F (v) é<- H2 (t))dH2(t)d¢(v)d¢(u)
M u
=25MMwMgo591ngnMM)MM)
M u _]
-2 g 1 (U)F2(u U) A 9] M(v)G2 (V)d¢(V)d¢(U)
M u dz
+ 2 g 9] M( F2(U) 6 g1,M(v WF ”(v )(I H2 2(t) )dH 2(t ))d¢(V)d¢(U).
Thus

V
Var(A + B - c) = 2 j g1,M(u)F2(u){é g1 M(v)F2(v) 6 H;

T ()()'f‘2 u
2 F { H
91 M u 2 u o 2 t ,
M_2 M u
= 2 6 H2 (t)({ 91,M(u)F2(u) { 91,M(V)F2(v)d¢(V)d¢(u))dH2(t)
M _2 M 2 z
= 6 H2 (t)({ 9] M(U)F2(U)d¢(U)) dH2(t)

which comp1ets the proof of Theorem (2.2).

APPENDIX B

In this appendix, we present the notation that is used in

Chapter 3.
~ x __ M __ x ._ M ._
Let R1,M(x) = g F1dF2 - i Fisz, Ri,M(x) é F1dF] - i F1dF],
Oi M(X) = f D(x,y)dFi(y) - I D(X.y)dFi(y). and
’ 0 x
x ._ M ._
Qi M(x) = 6 D(y,x)dFi(y) - f D(y,c)dFi(y) for i = 1, 2. Aiso
’ x
(1) - (1) (1) (1) ' (2) _ (2) (2) (2) '
1et SN - (SN1 , 5N2 , 5N3 ) and SN - (SN1 , 5N2 , 5N3 )
where
5(1) = 1-2 (»){(1-5 )H‘](u ) ? (Q + F R )d?‘ [u < M]
N1 m 1 2 i 1 i u. 1,M 1 2,M 1 i -
U.AM 1
1 _2 M __ __
- 6 H1 F({mhM + F1 R2,M)dF1)dG]}
U.AM
+ 1.2 {f 6'1 R df - ? F R df 1
m i 0 1 2,M 1 0 1 2,M 1
1 L -1 M X - -—
+ 5'1 2{G1 (”i)51 Q],M(Ui)[Ui 5 M3 - 2 g 6 D(y,x)dF]dF1},
5(‘) = 1.2 »{(1-5 )H"(u ) ? F R dF [u < M]
N,2 m i 2 i 1 i u. 1 2,M 2 i -
U.AM 1
1 _2 M __
- 5 H1 F1 i F1 2,Msz dG]}

1 1
Sé3) = 5'; (%){(1‘5

2)
N2

and

3'5”]

2)
N3

II
3 [—J
LJ-M
A
NV
V
r5
A
_;
I
.<
C,
I
N I
A
<
v
“fl
A
O
—J
2

+ F2 Rln)dF2 [vj 5 M]

<

>

z
(.1

(4M
a-‘H
0““. <ka < 0%

(D

N
:02

——J

3
Q
'"l

—-J

I

MN _ .—
H2 F2 i (01M + F2 Rm)dF2 dGz}

+
DI—J
Li.
I
—l

+
3|-l

(.2.M
rH

+
DI—d
c—‘M
A
to“
v
H-a
A
—I
I
.4
C—l
v
I
N I
...:
A
<
V
8-.
'71
N
:0
—J
U
3
Q.
d-T“

M x

_ M] -2 f f D(x,y)d‘F’1 dB}.
0 o

+
:|—1
(HM
fh‘
_<
(.1
G)
N
—J
A
<
V
O
—l
3
A
<
V
r_1
<
A

Z
O
z
_a
m
C‘I'
M
A
_a
V
I
A
Q
A
—a
V
V
.1

1,2,3 where
1 2,3

1

M

2 ..
F1 (i F.l R2,M dF1

)2 do

3/4 f H;
o

O
N
3

Q
'71
...-a
I
43
A
0‘s 3
OK» X

C:

A

‘<

X

v

Q.

11I
N

Q.

""|
—l

M

M
I

x

- 3/2 I H'2 F (? F R dF')(f Q dF')dE'

1 1 x 1 2,M 2 x 2,M 1 1

M .—

i F1 RZ’MdF2)(£ FIR 2 MdF )dG

- 3/2 6 Hi2 F1 (

M x_1 _ M x_1~ __ _
+ {[ F1 R2,“ 1 G1 RZ’Msz dF1 + f F1 RZ’M j G1 R2,MdF1 sz
o o o o
M ~ ._ M
- é F1 RZ’MdF1 6 F1 R2,MdF2}
1M x_1 _ M __Mx
+ §.{f Q1," j 61 R2,MdF2 dF1 - 2 1 F1 R2,MdF2 f f D(y,x)dF1 dF1},
o o o 0 o
(1) = - l-? H 2 F (? Q dF )(? Q dF )dG
ON,1,3 2 0 1 1 1,M 1 2 M 1 1
X X
M _2 M ._ M
- 3/4 6 H1 F1 (f F1 R2,MdF1)(j F1 R2,MdF1)dG1
X X
M _2 M M
- 3/4 é H1 F1 (f F1 R2,MdF1)(f Qz,MdF1)dG1
X X
1 M 2 M M
- §.é H1 F1 (1 Q1 MdF1)(j F1 R2,MdF1)dG1
X X
1 M _2 M __ M ._ __
-.7 6 H1 F1 (f Q1,MdF1)(f F1 Rz’MdF2)dG1
X X
M _2 M __ M ._
- 3/4 6 H1 F1 (f F1 R2,MdF2)(£ F1 Rz’MdF1)dG1
M x_1 _ M ~ x_1 _ _
+ {1 F1 122,M f G1 RZ’MdF1 dF1 + f F1 R2,M f G1 RZ’MdF1 dF1
o o o o
M __ M ~
‘ I F1 R2,MdF1 f F1 R2,MdF1}
o o
M x_1 _ M x_1 __ _
+ {f F1 R2,M f G1 R2,MdF2 dF1 + f F1 R2,M f G1 R2,MdF1 sz
o o o o
M ._ M
- 1 F1 RZMdF1 f F1 R2,MdF2}
o o
M x 1 M ._ M x ._ ._
+ {6 QZ’M 6 G1 RZ’MdF1 dF1 - 2 6 F1 R2,MdF1 6 g D(y,x)dF2 dF1}

+

+

N
r-‘H

N

MN x_1 __ _
+ {f Q1,M f G2 R1,M dF1 d 2
o 0
M __Mx _
- 2 1 F2 R1 M sz f [ D(x,y)dF1 dFZ},
0 ’ o 0
(2) 1M~ x_1~ __ _ Mx __ M
0N,2,3 = 7-1; QZ,M ] 62 R1,M dF1 sz - 2 j I D(x,y)dF2 sz ( F2 R1,M dF1)}
o o o 0 o
1M~ x_1 _ _ Mx M
+ §'{I 02,M j 62 R1’M dF2 dF2 - 2 f j D(x,y) sz dF2( F2 R1,MdF2);
0 0 0 0 0
1M _1~ ~ Mx 1 _ _ Mx _ _
+ §.{f 92 QZ,M Q1,M sz - 4 (j j D(x.y)dF1 dF2)(f f D(x,y)dF2 dF2)}
0 o o o o
M ~ x_1~ _ _ M ~ x_1~ _ _
+ {f F2 R1’M f 62 R1,M dF1 sz + f F2 R1,M f G2 R1,M sz d 1
o o o 0
M ~ __ M ~
- (1 F2 R1 M dF2)(f F2 R1 M dF1)}
o ’ 0 ’
M N x_1 _ _ M x1~ _ _
+ {f F2 R1’M f 62 R1,M dF2 sz + f F2 R1,M f 92 R1’M dF2 sz
o o o o
M N ._ M
- (6 F2 R1,M dF2) 6 F2 R1’M sz}
MN x1~ _ _ Mx _ __ M ~ __
+ {6 Q1," (6 G R1’M sz)dF2 - 2 g g D(x,y)dF1 sz (5 F2 R1,M dF2)}

NI—4
:1:
N 1

N|—I

I
N I

I
4:400
0%3 0‘3 0‘53
I
N I
N
'31
N
A
X‘Z X2113 X‘s: NI
.71
N
m
—‘
3
0..
'11
N
V
A
“a
O
N
3
Q.
11
N
v
C.
CD
N

56

2
_G
2 .0
_G \ll
d «I
) 2 __Ll
2 _G d
_F d
d \11 M
2 Q.
M _F 1|.
9 d ~R
1|.
~R M 2
9 F
2 2
F ~Q MIX
/.\
MIX MIX \1
( ( 2
) ) _F
2 .1: d
_c. Fr
.0 d M
M M .11
, 3 NR
1| 1'
R ~R 2
F
2 2
F F MIX
(
MIX MIX
( ( 2
F
2 2
C... F 2
. 2
2 2 H
. 2 . 2
H H MIO
Mfo MIO 4
/
.

=MN

Define [(1) and [(2) similarIy except replacing M

with + w.

BIBLIOGRAPHY

BIBLIOGRAPHY

Bar1ow, R.E., and Proschan, F. (1975). Statistica1 Theory of
Re1iabi1ity and Life Testing. Ho1t, Rinehart, and Winston,
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Chikkagoudar, M.S., and Schauster, 0.5. (1974). Comparison of
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Doksum, K. (1969). Star-Shaped Transformations and the Powers of
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Hoeffding, w. (1948). A C1ass of Statistics with Asymptotica11y
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Kochar, S.C. (1979). Distribution-Free Comparison of Two Probabi1ity
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Kou1, H.L., and Susar1a, V. (1978). Testing for New Better than
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Kou1, H., Susar1a, V., and Van Ryzin, 0., (1980). Regression
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Puri, M.L., and Sen, P.K., (1971). Nonparametric Methods in Multi-
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Sen, P.K. (1960). On Some Convergence Properties of U-Statistics.
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Susar1a, V., and Van Ryzin, J. (1978). Large Sampie Theory for a
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Susar1a, V., and Van Ryzin, J. (1980). Large Samp1e Theory for

an Estimator of the Mean Surviva1 Time from Censored Samp1e.
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57