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Statistical Inference For Randomly
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presented by

Wei-Hong Wang

has been accepted towards fulfillment
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STATISTICAL INFERENCE
FORRANEXIVILY CENSORED

IMARREBRESSICNMGDEL

By
Wei-ng Wang

A DISSERTATION]
Suhnitted to
Michigan State University
in partial fulfillment of the requiremnts
for the degree of

DWIOROFPHIILBCPHY

Department of Statistics and Probability
1983

/3A/- /f///)

ABSTRACT

STATISTICAL INFERENCE FOR RANDOMLY
CENSORED LINEAR REGRESSION MODEL

By
Nei-Hong Wang

In this theSis we construct an estimator of the slope parameter
in a simple linear regression model with randomly censored survival
data. The estimator is obtained by the minimum distance method and is

shown to be asymptotically normal under appropriate assumptions.

We also discuss the sufficient conditions for a randomly censored

multiple regression model to be locally asymptotically normal (LAN).

WIS

I wish to express my sincere gratitude to Professor Hira L. Koul
for the patience he accorded me in the preparation of this thesis . His
guidance and cements greatly inproved and simplified the results in this
work. Deep appreciation to Professor James F. Harman, Professor Roy
Erickson and Professor John Masterson for serving on my Guidance Calmittee.

I owe my special thanks to Professor Hannan for teaching me on many
aspects of mathematical statistics, and for his encouragenent and consis-
tent help during Professor Koul ' s sabbatical leave. His suggestion of the
inequality (2.1.13) also greatly improved the result in Chapter 2.

I am much indebted to Mrs. Vera Stephan and Miss Carol Jeane Case
for their excellent typing. I also appreciate the warm friendship frun
graduate assistants in Michigan State University, especially Mr. Rotana
Kanmammi.

Finally, I am most grateful to my parents and my wife for their

understanding through the years of this work.

TABLE OF CONTENTS

Page
INTRODUCTION AND SUMMARY .................... 1
Chapter
1 MINIMUM DISTANCE ESTIMATOR FOR REGRESSION WITH
RANDOMLY CENSORED DATA ............... 3
1. Introduction and Preliminaries ......... 3
2. Asymptotic Normality of 31 ........... 5
3. Asymptotic Normality of Q ........... 11
2 LOCAL ASYMPTOTIC NORMALITY OF RANDOMLY CENSORED
LINEAR REGRESSION MODEL .............. 24
1. Introduction and Preliminaries ......... 24
2. Lemmas and the Main Theorem .......... 28
APPENDICES 1. Proof of Theorem 2.1 ............. 34

2. Verification of (1.3.35) ........... 36
3. Verification of (1.3.36) ........... 38
4. Verification of (1.3.37) ........... 4O
5. E§(t,8) = 0 .................. 41
6. Upper bound of Var {§(t,8)} .......... 41
7. Verification of (1.3.42) ............ 43
8. Calculation of Var (0d(2 - 8)) ......... 44
9. Calculation of g ............... 46

BIBLIOGRAPHY .......................... 52

INTRODUCTION AND SUMMARY

Of interest are mathematical models for life-testing problems,
for example, cancer or heart-transplant data. Usually, at the end of
a clinical study, a patient may be still alive or lost-to-follow-up
and thus yields censored survival data. The problem of handling
censored survival data has received a great deal of study in recent years.
See, for example, Kalbfleisch and Prentice (1981), or R. Miller (1981).
Mathematically, we define survival variables {Xj} and censoring

variables {Yj} as sets of measurable functions and observe

.=‘ , , 5.: . .
ZJ min (XJ, Y3) and J I(XJ 5Y3) 9

where I(A) is the indicator of the set A. If 6j = 0, then Xj is said

to be randomly censored on the right by Yj; 6j = 1 means it is not

censored. The dataiXZj, 6j): 1 n} is said to be randomly censored

I/\

j

IA

data. Note that if Yj = <n, 1 5 j s n then zj = Xj, 6j = 1, 1 s j s n
and hence one has complete or no-censoring model.

In this thesis we assume the (log) survival time is a linear function
of certain variables and discuss the statistical inference for a linear re-
gression model under the above random censorship. '

Koul, Susarla, and Van Ryzin (1981) considered an estimator of
the regression parameters with the randomly censored data {(Zj, Gj): 1.: j s n}.

They proved the asymptotic normality of their estimator which are centered by

a truncated form of the regression parameters. In Chapter 1 we construct a

minimum distance estimator of the slope parameter in a randomly censored
linear regression model, and prove the asymptotic normality of this
estimator with the right centering factor.

-In Chapter 2 we give sufficient conditions for a censored
multiple regression model such that the log likelihood of {(Zj, 63):
1 s j s n} is locally asymptotically normal (LAN) in certain neighborhoods
of the true parameters. The importance of LAN property is that it provides
broad background to develop a unified asymptotic theory for the statistical

inference. For more detailed explanation see Fabian and Hannan (1982), or

Ibragimov and Has'minskii (1981) and the references therein.

CHAPTER 1
MINIMUM DISTANCE ESTIMATOR FOR
REGRESSIDN WITH RANDOMLY CENSORED DATA

1. Introduction and Preliminaries

 

Consider the following linear regression model:
(1-1) X1 = a + BCi + 81, i=1,2,. . .,n;

where {Ci} are known input (design) variables, a are independent

i
identically distributed (i.i.d.) non-degenerate random variables with
an unknown right tail distribution F, and a, B are the parameters of
interest.
Under the model (1.1), we observe not {Xi}’ but
(1.2) Zi = min {xi’Yi}’ and 6i = I(Xi 5 Vi) _
where I(-) is the indicator function and {Yi} are random variables
which are independent of {61}. The above model is known as randomly
censored linear regression model. Miller (1976), Buckley and James (1979),
and Koul, Susarla and Van Ryzin (1981) proposed various estimators of (a,8)
for the above model.
Miller's estimator is obtained by minimizing the weighted sum of
squares of the residuals with the weights determined by the Kaplan-Meier (1958)
estimator of the error distribution based on the residuals. Buckley and James
(BJ) estimator is based on an ekpectation identity. Both of the estimators
are computed using iteration methods, and the iterations may settle down to
oscillation between two values. No formal mathematical proofs of consistency or

asymptotic normality of their estimates are available at this time. Koul-

Susarla-Van Ryzin (KSV) considered another estimator of (a,B) and proved mean

4
square consistency and the asymptotic normality of their estimator.

In this paper, we consider the minimum distance Cramer-Von Mises
type estimator of the slope parameter under model (1.1) and (1.2). For
the uncensored case with F unknown Williamson (1979) defined minimizer
of f{X(c1--T)I(X1 f t + Aci)}2dt with respect to A as an estimator of 8
and proved the asymptotic normality of this estimator. For the uncensored
case with F known, Koul-Dewet (1983) proposed another family of minimum
distance estimators of B. Millar (1982) considered a general model and
proved that his estimator of the regression function is asymptotically
normal, efficient and robust. For a more detailed references of the minimum
distance method see the bibliography by Parr (1981).

In this paper we follow the Williamson's approach. In the case {Gil
are known, F unknown, this approaCh is modified by inserting the random
weights {5iG;1(Zi)}, by changing Aci to be A(ci-E) and by replacing dt to
be durfit), where {un} are o-finite measures on (R,B). These modifications are
crucial in the proofs of the following sections.

Formally, under model (1.1) and (1.2), we assume {Yi} are independent,
P(Y1. > t) = Gi(t)' Gi is continuous and Gi(t) > O, for all t. Let
(1.3) S(t,A) = 2d.5.G71(Zi) IEZi 5 t + Adi], di = c

l l l
(1.4) 0(a) = Isz(t.A) du(t);

where u is a fixed o-finite measure on (R,B). Note that S(t,A) is a

weighted empirical process and

ES(t,A) Edi P(Xi 5 t + Adi)

‘—2di F(t - a - BE'+ di(A - 8)).
It is then clear that ES(t,B) = O; ES(t,A) 2 O, for A‘> B and ES(t,A) 5 O.
for A < 8. Hence, similar to a rank test in Hajek and Sidak (1967, p. 103)

for testing H0: 8 = 80, H0 is rejected for large values of 0(80). Therefore

A
if {Gj} are known, it is reasonable to define an estimator 3 for 3 by the

relation

(1.5) 0(31) = inf 0(3).
A

For the case Gj = G and G is unknown, we estimate it by a product-limit
estimator of G, same as KSV (1981), and then obtain an estimator g for B.
In section 2 we prove the asymptotic normality of $1 when {Gj} are
supposed to be known. In section 3 we prove the asymptotic normality of g.
The numerical calculation of A will be shown after we formally introduce

an estimator for the unknown censoring distribution G.

2. Asymptotic Normality of 31. /’/

 

Under model (1.1) and (1.2), we define

“3

assume Giis continuous and F has a- density f. The index i in the summation

(2.1) = Edi, F1.(t) = P(X1. > t) = F(t - a - 861.);

and maximization runs from 1 through n. The assumptions needed in proving
' A
the asymptotic normality of 81 are as follows.
(A1) 051- max Idil'T O , as n +-~.
(A2) ff2(t + a) du(t) < a, for all a; ffdu < w, and f is continuous a.s. [u].
_ (A3) ff2(t + a + s) du(t)l+ ff2(t + a) du(t), as s‘+ O, for all a.

.'. -1 '
(A4) of 2d? Var {61.61. (21.) 1521. s t + Bdilldu(t) <. on. n > 1

(A5) 0’2 2d? fit + Bdi + an -GTl(x)dF.(x)du(t) + 0 , where a = adalomax Id.l+ O .
d 1 t + 8d. - a 1 1 n 1
1 n

(A6) max Iiimu - 8d,, a) G€1Tn(x)d(-Fi)(x) < ..
where‘a(A) = fAf(t - a - BE)du(t), and

n is O or some small positive number.

Remarks on Assumptions

 

1. For the uncensored case, (A4) reduces to
(A4.1) fF(t + a)(1 - F(t + a))du(t) < w, for all a;
which is similar to (3.8) of Koul-Dewet(1983). For the censored case,
(A4) holds under
(A4.2a) u is a finite measure, and
(A4.2b) max Var(6iG;1(Zi)) = max I -G;1dFi < m.

Note that to verify (A4.2b), it is enough to check that
4.11;,

I; -G;1dF1. < 0°, for some b > 0. Assume for t > b, F(t) = Cle

Gi(t) = G(t) = Cze’AZt, for all i, A1 > 12 and {di}I are bounded. Then

‘r: - G'1(t)dFi(t)

f I: -G'1(t + a + BE'+ maxlei|)dF(t)

(m.

)t

0(1)e dt

l2(a + BE'+ maxlei|)fl;ne-(>.1 - A2
It is important to realize that for this case, 11 > A2 means F(t) < C(t) or
P(ei > t) < P(Y > t). For the case Xi = ei’ this implies that at time t the
probability of getting complete data is larger than the probability of getting
‘incomplete data.

Another example for (A4.2b) is that F is N(u,02) and G,{di} are as

before. Then

f; -e‘1(t)dF,(t)

= 0(1) f; expi-(t - u - a - BC, - *2.°2)2/202}dt < .,
Again, the tail of a normal random variable is smaller than the tail

of any exponential random variable.

7

2. For the censored case, (A5) holds under (A4.2) and the
fact that F and G are continuous functions. For the uncensored case,
(A5) holds under
(A5.1) lim f f(t + a + s)du(t) = f f(t + a)du(t) < m, for all a.
5+0
3. (A6) holds under (A5.1) for the uncensored case or (A2)and

the following assumption (A4.2+).

(A4.2+) u is finite and max I -G.'1'ndF. < a,
is" 1 1

for some small positive number n.

4. (A4.1) can be simplified to be I F(1-F)dUo < °.

1T u < < u. (Lebesgue measure) and du is bounded. So is (A5.1).

V
O

5. One of the crucial assumptions in KSV (1981) is that E(ai) = 0.

Note that this assumption is not required here.

For the finite sample property, we have
Lemma 2.1. Let
(2.2) L(A) = f S(t,A)/f(t)du(t),
where S(t,A) is defined in (1.3). Then
(2.3) S(t,A) and L(A) are non-decreasing in A,
(2.4) L2(A) s Q(A)rf(t)du(t).and

(2.5) 0(a) achieves its minimum in a compact set of R.

Proof. First consider the case A < A1.

If di 2 0 , then

1 w
di{I(Zi 5 t + A di) I(Zi s t 1-Ad,);

= diI(t + Adi < Zi

IA

t + Aldi) 3 0 ;

and if di < 0, then

I.
dl{I(Zl S t + A d1) I(Zl S t + Adl)}

= _ 1
di( 1)I(t + A d, < z, s t + Adi) 3 o.

1(2,.)

IV

Note 616; O , for all 1, hence

5(t,A1) - $(t,A) z o .

The same conclusion holds for the case A > A1. This proves (2.3).
The Cauchy-Schwartz inequality yields (2.4), whereas (2.5) holds by'(A21(2-3),
(2.4) and the fact that L(A) is not constant in A.. D

To prove the asymptotic normality of ’31 we need the asymptotic '

quadratic approximation of O(A) by the quadratic fUnction'

(2.6) 01(A) = 01(B;A) = f {S(t,8) + (A - B)°§f(t - a -ABE)}2du(t).
Theorem 2.1

Under assumptions (A1) - (A5), we have

(2.7) E{ 'SUP 0&2 IQlA) - 01(A)l} +~0 . as n +-~ ;
OdlA ' 8| S a

where a is a fixed number.

Proof. See Appendix 1. D

Denote
(2.8) t = t - a - BE', and
(2.9) (m2 = r f2(i)du(t) .

Note that Ql(A) has unique minimum at Al with 21 defined by
(2.10) o (3' - s) = «'1 JUII’2° J'S(t B)f(ft\:l)du(t)
° d 1 d ’

= -oc—iIML-Z 2d151‘3iluifiui - Bdi. 1»)

’b
where u is defined in (A6).
Note that 05(31 - B) is a sum of independent random variables and

A
E0d(A1 - 8) = O. Straight-forward calculation yields
A
(2.11) Var {Od(A1 - 8)}
= ufnf4052 Edi rri LEIAti-e;1<x>dri(x) - (1 - F(§))(1 - F<¥))}dt(t)di(s) .

where

(2.12) ti = t + 8d,.

Theorem 2.2
Under (A1) and (A6), ord(’A‘1 - B) is asymptotically normal with mean 0

and the asymptotic variance as in (2.11).

 

10

Proof. Denote

 

 

= l ”’20"1 ‘ "‘1 ” \P 7 _ a m
(2.13) AnI if; d d] 310i (Lila-(hi "div )
\ 2 _ .1 2 = Y 2
(2'14’ cni - Var (Ani)’ On ”Uni
By (Al) and (A6), we have
2 : . 2 : l ('7 A _ =
(2.15) max Uni 0(I), and on lar(..d(_.1 3)) 0(1).
Note
-2 _ 2 _
(2.16) “n 2 E(Ani EAni) IIIAni EAnil > e an}
o - A o A o - EA 0 e a
2 n1 n1 n
5 max E( n1 "1) III ' > -—-———— } .
“n1 “n1 ‘ ”ax “n1

By_(2.15), (2.16) plus Holder and Markov inequalities, to Verify

the Lindeberg's condition, it suffices to prove

(2.17) Max E cag'”lAnil2 T n < w , for same n > O .

, Note, by (A6),

aim amazi”=MU-(QI NAM.
'while
(2.19) cg, = 0(1)agzd§(r-tz(x-adi,~)e;1(x)dri(x)

-{rfi(x+a+a3, a )dF(x)}2)

'3 0(1)e;2df- Vari§(ei + a + SE; an)1

By (2.18) and (2.19), (2.17) holds since the errors {5i} are non-degenerate.

We therefore complete the proof of the theorem. 0

11

The asymptotic normality of B1 now follows by Theorem 2.2 and
A A
by a standard argument (see Williamson, 1979) that od(B1 - A1) = op(1)
Note we assume the measurability of $1 without further discussion.

3. Asymptotic Normality of g

In the previous section, we proved the asymptotic normality of the
minimum distance estimator for the regression parameter 8 with censored
survival data when the censoring distributions are known. In this section,,
we extend the result to the case where the censoring distributions are the
same but the common distribution is unknown.

To estimate the unknown censoring distribution, following KSV (1981),

 

IQt +
A 1+ N (23.) 1 a
(3'1) S(t) ‘2 Et{ + 1 ‘ i , where
j- 2 + N (Zj)
n
(3.2) N+(t) = 2 1rzk > t] .
k=1
Consider
A A_1
(3.3) S(t,A) = 2di6iG (Zi)I(Zi s t + Adi) ,
(3.4) 6(a) = f§2(t,A)duh(t) .
(3.5) 61(B;A) = mime) + (A - B)o§f(t)}2dun(t) ,
(3.5) I(A) = fS(t,A)f1/2(t)dun(t) .
We now define
A A A A
(3.7) B to be such that Q(B) = inf Q(A) .

A
Note that the finite sample properties in Lemma 2.1 hold again. That is

A
(3.8) S(t,A) and L(A) are non-decreasing functions in A;

A A
L2(A) < Q(A)flffl, and Q(A) achieves its infimum in a compact set of R.

12

He further denote

(3.9) F = nTIZFi, A = is .
The assumptions needed in this section are listed in below.

(81) maxldil < w , for all n; and n'loj = 0(1).

-1-n

(82.a) maxf-G dFi < 0° ,

(BZ.b) max/:G'3'anidG <<» .
where n is a fixed small positive number.

-2 2

(82.c) max I-F'le FidG < .. .

(B3) un< < vo(Lebesque measure). dun/dvo is bounded and ff2 < w .

(84) “n is a finite measure on (-m,£n), where {an} satisfy (85).

(B5.a) max n'lfinR-BGdFi + O .

isn
(85.b) n‘lfi'4(an) + o .

We denote in = E for the sake of convenience.

Remarks on Assumptions

1. In the proofs, (82.b) can be replaced by weaker assumptions:
(82.b1) maxf-G’z'IiFidG < w, and
(82.b2) maxf([:-szG)2 + nd(-FiG) < a .
Then (82) family holds for the special cases discussed in Remark 2.1 with
A1 > (1 + n)l2 when both F and G have exponential tails. Note n is

arbitrarily small and is negligible in application.

13

2. For the case Fi(t) = F(t) = C e 1

(B5) reduces to

-1e(711-+712)a

cn n + O .

A
In order to prove the asymptotic normality of B, we have to
A
prove the asymptotic quadratic approximation of Q(A), i.e.
(3°13) E{ SUP 0d 2|Q(A) " 01(8 AH} + 0 °
<
A-8|-a
In view of the proof of Theorem 2.1, it suffices to verify

A A
. -2 ' 2_
(3.14) l1mnsup 0d EJLSA - SB - SAfl_ - o , and

(3.15) lim sup 03256(3) < m .
n

(3.14) will be proved in Lemma 3.2 and (3.15) will be proved after Lemma 3.6.
We first state Lemma 3.1 which will ease up a lot of calculations.

Lemma 3.1 Under (B5.b), for large n, we have,(r1{Zi 5 an},

 

(3.16) 3,5,18'1(zi) - G‘1(z1.)}2 5 Ch 5 H (2.) ,
A 4 . n + 1 -1 2 -1 2
3.17 6.E.{lnG(Z.) - lnG(Z.)} i— . + G (2.)) 5 G (2.))
( ) I 1 I 1 1 + N+(Z.i ) 1 T
5 Ch 23.1l'8(z ).

l

A A A _
(3.18) ((6'1- 6‘1) - G'1(1ne - lnG)|s%ilnG - 1n(;)21%—$—.§_.+ + e 1}. for all nzzl.

Wmere Ei is the conditional expectation given (61,2i), and C is a constant.

14

Proof. Note (BS.b) implies

(3.19) lim inf nH(an) 3 b , for some b > 2.

Hence in view of (7.4), (7.5), (7.7) of KSV(1981) as wellas an argument

in (7.15), we have
--3 -1 z, ---8 %
LHS(3.16) s CGiH (2,)n {-54, FH d6} 5 RHS(3.16).

Similarly,

_ z- --8
Zr 1-FH dB 5 RHS(3.17) .

-oo

LHS(3.17) s Céin

Finally, (3.18) holds by a similar argument as in (7.4) of KSV.

Lemma 3.2
(3.14) holds, under (Bl), (BZ.a), (B3), (B4) and (BS)_
2599:, By'Appendix 1-(ii), we have

LHS(3.14)

5 lim sup 2052 E fl_§A - g - SA + 58 fl_ + 0(1) .

B
A A _ - .
Let SA - SB - SA + SB -. D - D1 + 02 , where D1 15 the

sum of all terms with Ed, 3 Adi , while D2 picks up the rest.

-2
Edd flplfl?

IA

where an is defined in (A5). By using (3.16) and (BS.b) we have

-2 2
Cd E flp1fl_t

d2 zd§8d1+ an -H'6(x) F—1(x)dFi(x)dun(t)
an

IA

ft + 8d; -

0'2 2d§1i3(:-:3;-:2 d“n(t) H'7(X)d(‘Ffi) x) °

IA

_ 2 co A_ _1
nodz Edi L. 531(3 1(21) - e (Zi))zI(t + Bdi — an < z,

D

Note

m A- -
032' 51. {2d151(G 1(21.) - G 1(21.))I(t + ad, < 21 5 t + Bdi)}2d"n(t)

< t + Bdi + an)}dun(t).

15

By (B3), (BS.a), (Bl), we conclude

-2
0d Eflp1fl_

. . -1. -2 2 E "7 +
s C bO a Gd maxldil 0d Edi f n H (x)dFi(x) 0 .

Similarly, 0&2 EHDZH2 + 0 . This completes the proof of the Lemma. D

Next to verify (3.15). Note by Appendix 1-(i),
(3.20) LHS(3.15)

5 lim sup 2xoazs{§(t,s) - S(t,B)}2dun(t) + 0(1) ,
where
A A_1 _1
(3.21) S(t,B) - S(t,8) = 2di6i{e (2i) - G (Zi)}I(Zi s t + Bdi) .
Note (3.16) is not enough to find a finite upper bound for the RHS(3.20).
We will use (3.17) and (3.18) for this. The following Lemmas 3.3 - 3.6 will

accomplish the proof of (3.15) and serve for further purpose.
Lemma 3.3. Under (BS), we have

.. A
(3.22) 052 Ef{(§(t,8) - S(t,8)) - ZdidiG I(Zi)(lnG(Zi) - lnG(Zi)) .

I(Z.

2
1 s t + Bdi)} dun(t) + 0-

Proof. By (3.21), (3.18) and CauanSchwartz inequality,
LHS(3.22)
.. 2 2 , A 1+
5 nod 2dif EéiEi{1nG(Zi) - 1nG(Zi)} .

{ 1 I " + G'1(Zi)}ZI(Zi s t + Bdi)dun(t) .
1 + N (Zi)

 

16

Hence by using (3.17) and (85), we have
LHS(3.22)

-2 2 -2-—8
s cnod lair E61.n H (2))I(Zi s t + Bdi)dun(t)

f cn'loazzdgffi fi-8(X)G(x)dFi(x)dun(t) + o , 0

Lemma 3.4 Under (B4) and (BS),

-2 -1 A
(3.23) 0d Er(§diaie (Zi)I(Zi s t + 6di){lnG(Zi) +

_ + 2
End]. - 0. ZJ- s Z1.)/(2 + N (z.)}) dun(t) 3 o .

J

Proof. Argue as (7.21) of KSV(1981), and use Canchy-Schwartz inequality,
LHS(3.23)

-2 2 -2 _ + -2 2
nod Edif E5iG (Zi)Ei(j;iI(55—0, Zj s Zi){1 + N (Zj)} ) .

I(Zi S t

"A

+

Bdi)dun(t)

4' -1+

“2052 Zdfir eaie‘z(zi)ei 2 1(6.
1' 3H 3

IA

I(Zi

A

t + 8d,)dun(t).

where E1.

J is the conditional expectation given (6i,Zi) and (aj,Zj).

By using (7.6) of KSV(1981) and (3.19),
+ -4 -4 ' 2 -4
Eij{1+N(Zj)} an {H(ZJ.) - F}
Sen'4b4(b - 2)'4H'4(zj) .

4

Using (B4), (BS.a) and the fact that B- is non-decreasing,

17

LHS(3.23)
-2(

IA

-2 -2 2 _ "4 .
cn 0d Edi! EéiG Zi)E1 Z I(é. - o, z. < Z.)H (2.) I(Zi < t + Bdi)dun(t)

2( '4(

IA

-2 —2 2 - "
cn ad {dif EdiG Zi)H Zi)(n-1)I(Zi 5 t + Bdi)dun(t)

IA

cn'loazxdfini H‘4(x)e‘1(x)dn=1.(x)dun(t) + o . (3

Lemma 3.5. Under (B4) and (BS),

(3.24) 05251 (zd.a.e'1(z.)1(z. < t + 3d.)21(5. = o, z. s z.) .
i 1 1 1 1 1 j J

{(2 + (1“(zj))‘1 - n‘1(2H'1(zj) - H(zj)H'2(zj))})2dun(t) + o ,

A
where H(x) = n-IZHZk > x) .
Proof. Argue as Lemma 7.5 of KSV(1981). D

By Lemmas 3.3 - 3.5, we have
(3.25) A 'b 2
-2 _ _
odews sB 5J530.
where ’
W _ ‘1 o ’1 = o
(3.26) S(t,B) - édidiG (Zi)I(Zi 5 t + Bdi) {lnG(Zi) + n §I(6j o, 23 s 2,)
--1 A --2
(2H (Zj) H(Zj)H (Zj))}
To prove (3.15), it now suffices to prove
’b‘
. - 2
(3.27) l1m sup odzEfl§Bfl_ < «=.

This will be done in Lemma 3.6.

18

Denote
"t
(3.28) S(t,8) = B1 + 82 - B3 . 1 e
n _1 -
(3.29) 31 = iéldiaic (zi)1(zi g t + Bdi)lnG(Zi) = :zui, say
_ 2 -1 _ --1‘
(3.30) 32 — 5% E (11.311; (21.)1(z1. s t + 3d,)(1 aj)1(zj g z.)H (23.) ,
2
= : - U..
n E E 13

(3.31) 83 = 1- Z 2 E d-5.G-1(Z-)I(Zi s t + ediXI-éj)I(zj < z.)H'2(zj)I(zk>zi)
1 J .

n2 1 1 1
= -2
"H153
Note
_ n -1 . _
(3.32) B2 - (2) 1Ean-IHUU. + Ujk} , ($1nce Uii — 0) ,
and
{U.. + U. . + U..
(3 33) B3 = (3)-1 g {12:l%%fl:é) 13k 1k1 31k + °p<sa) ,
‘<3<k + Ujki T Ukij + Ukji}

where, by using (BS.a),

0&2E(2nd term)20f (3.33) = 052E{n'2 Z Z Uiji} s n‘lo;2 2d? ft + Bdi H-1(y)G'1(Y)dFi(x)+C
1 J -w
i.e. 32’ B3 are approximately U-statistics.
’L'.
We next approximate S(t,B) by a sum of independent r.v.'s.
It is well-known that .

(3.34) E = : 2 E B = 31 + (n—1)EB1 ,

We further have

19

n
’b
(3.35) 32 = :£§15£32 = -231 + (n-2)EB2 + q)(cd) +

+ 2-2 Z d
1

n 2 .a,3£)H'1(z£){Fi(z£) - F(t - a — BE)}I(Z

Z

the detailed calculation is shown in Appendix 2. By Appendix 3,
q, n

(3.36) B =:ZEB
3 £=1 2 3

-B1 + (n-3)EB3 + °p(°d)

-1 --1 ._
+ n E Edi(1-6£)H (Z£){Fi(Z£) - F(t - a - Bc)}I(Z£ < t + Bdi)
+ n'1 E 2 di 122At+3d1 -H—1(y)G'1(y){Fi(y) — F(t - a - 3E1do(y) .
1 -¢)
By Appendix 4,
(3.37) (n-1)EB1 + (n-2)EB2 - (n-3)EB3 = 0(qd) .
Combining (3.34) - (3.37),
8 _ m m m m

(3.38) S(t,B) — . 31 + 32 - B3 - EE£S(t,B)

_ -1 --1 -
- n E Edi(1-6£)H (Z£){Fi(Z£) - F(t - a - Bc}I(zz 5 t + 8di)

+ l{-11% id) ,_Z£At+8d1 fi-1(y)G'1(y){F1.(y) - F(indem + op(od)
1

Note, see Appendix 5,

(3.39) B§(t,8) = 0 .

13mm 36
(i) Under (BZ.c),

lim sup 0&2 Ef §2(t,8)dun(t) < «>,
(ii) Under (B4) and (BS),

’1:

__ r1. r1, '
lim sup odz Ef{S(t,B) - S(t,B)}2dun(t) + 0 .

20

Proof. (i) See Appendix 6.
(ii) Let

’1)
(3.40) Ut:= n'P0515(t,s) ,

(3 41) w - n‘Pc-1§(t 8)
° t d ’ ’

where p is a fixed number and .5 < p < 1 . By Appendix 7, we have

(3.42) sup not - Ht)2 = 0(n‘2) .
tsE
Hence
2 m k
(3.43) sup 0; E{S(t,B) — S(t,s)2
tsE
n
= sup E{nfijt - npwt}2
tsE
n
= n2P ° 0(n-2) + 0 .
This together with the fact that {un} are finite measures yield the
desired result. U

Note (3.15) holds now by (3.20), (3.25% (3.27). and the above Lemma.

Theorem 3.1 Under (Bl) - (BS),

 

(3.44) E{ sup 052|6(A)- 6*(B;A|} + 0 , where

(filA’Blfa

(3.45) 6*(1) f{S(t,B) + 's’(t,s) + (1.- 8).o§f(rt)}2dun(t) .
Proof. Let
(3.46) 61(1) = x_:{§(t,e) + (1-3)o§f(1’)}zdun(t) .

By the discussion before (3.14) and (3.15), we have

 

21

(3.47) E{ sup 452l3(3) -81(A)I} + 0 .
odIA—Blsa

Using (3.25), (3.27) and Appendix 1 - (i),

E{ sup 082l61(A) - 0*(A)I}
odlA-Blsa

IA

-2 A ’b' 2 - A ’b N
°d Eflfis - SB - SBH— + 2E0d fiss- SB - Ssflgo lzugl<afl58 + $8 + (A-B)ijl
d

|A

l, 'b
0(1) + 0(1)031E2{O sup flfis + (A—B)odffl? + flfifiugl-

0(1) + 0(1){0(1) + 0(1)} = 0(1) .

This completes the proof of the Theorem. D
.A o o o A 0
Note Q*(A) has unique m1n1mum at A, w1th

8 - B = -flffi:2 0‘2 [{S(t B) + §(t B)}f(t)d11(t)
d 9 9 n o
By (3.43) and the fact that ff(t)d1%(t) < c», we have

A ,- _ . % m I
(3.48) 0d(A-B) = -flffl_20d?f{s(t,8) + S(t,8)}f(t)d1%(t) + op(1) .
’1:
Comparing to (2.10), the only difference between_od(3-B) and 06(A1-B)is the

m %
S(t,B) part. Note if B=0 or no-censoring, then S(t,B) = 0 . Without loss of

generality, we will drop 0p(1) in (3.48).

Note qu(2-B) = o , and
(3.49) lim sup Var {0d(2-B)}-< w.
Indeed, by (2.15), (B3), (B4) and Appendix 6,

22

LHS(4. 38)
-2” W

0(1) + flffl_: (_ “d E52(t, B)f(t)d1h(t)
+ flfflf4 - 0(1) - 0(1)

IA

Theorem 3.2 Under (81) and (82. b), Ud(A- 8) is asymptotically normal
with mean 0 and variance as in Appendix 8.
Proof. By (3.48), Theorem 2.2 and the inequality
(3.50) E(A + B)21([A + B|>e) 5 4EA21(|A|> 8/2) + 4EBZI()B( > e/z),
to prove the asymptotic normality of od(A-8), it suffices to verify the Lyapounov
condition for
"1% a” m - fwd
0d I (t,B) un(t), where un(A) - IA (t) un(t) < «>,
for all AcR and for all n.

Denote
%

(3.51) W; fS(t, 8)dun (t t)= Xan, and
(3.52) In = Var( (Ean) ,
with
(3.53) an ,

= a‘ln‘lzd.(1-a.)f°° "1 W "I

d i 1 J Zj-Bd H ( ZJ.){F1.(ZJ.) - F(t)}dun(t)
+ o-ln-lid rrZJAt*Bdi e W<y> H( ){F (y) - F(t )}dG(y)du (t )
l

= : Bnlj + anj , say.

Then,
2+0

(3.54) EIBnljl

-2-n -2-n 2+n --1 m _ W 2+n
5 0d n max Idil E((1-6j)H (Zj)§ ij-Bd.{Fi(Zj) F(t)}dun(t))

23

5 032-” . 0(1)E((1-6j)fi‘1(zj)n‘1§Fi(zj))2+n

_ - - -2-
- od2 ” - O(1)E(1-6j)G n(23.)

aaz-h - 0(1) - f-e‘z'”(y)F.(y)dG(y)

J
= 0(1)oaz'n
for all jsn, by using (BZ.b1); and
2+n
(3.55) EIanjl

z. __ m m
< 0&2-n 0 max Idi|2+n . E(f[mJ_G‘1(y)H I(Y)n-IZ(Fi(Y) _ F(t))dG(Y)dun(t))2+n
1

052-" - 0(1)E([£§ - G‘Zde)2+”

IA

032‘” - 0(1) .

for all jsn, by (BZ.b2).
Using (3.54) and (3.55), the Lyapounov condition reduces to

(3.56) noaz'n 132‘” + O .

But (3.56) holds obviously by using (BI) and the fact that Tn2+n > 0.

This completes the proof of the Theorem. D

CHAPTERZ
ImALASYMPI‘OI‘ICNOEMALITYCF

RANDCMIX CENSORED LINEAR MERESSION MODEL

1. Introduction and Preliminaries

 

For each A a RP, let (Q , F ’PA) be a probability space and
X1, X2, ..., ane random variables on ($2 , F ) whichare independent

underPAandaresuchthat

_. = _ T .
(1.1) PA(xj>t) -. FAj (t) F(t A Cj), l i j in,

whereF is adistribution functionandC§= (le’ ..., ij) isavector

of known constants, l f_j f. n. Furthennore, let Y1

setof independent randanvariableson (S2 , F , PA) suchthat

, ..., Yn be another

(1.2) {Xj} and {Yj} are independent, and

PA(Yj>T) = Gj(t), for all A 6 12", 1 _<_j in;
i . e. the censoring variables are irrelevant to the regression parameters .
In survival analysis, what one gets to observe is not the Xj's or Yj's, but
(1.3) zj = min (Xj, Y3.) and 63. = I(Xj :Yj).
Note that (Zj, 5].) is a measurable transfomation from ($2 , F ) to

(Rx {0, 1}. B x {0, 1}). Let

(1.4) unAj be the induced treasure of (Zj' 63.) on B x {0, l},

and

(1.5) fAj be the density of FAj with respect to the Lebesgue measure.

'Ihen

(1.6) “mj((‘°°' t] x {1}) = PA(Zj _<_ t, 53. = 1) = fiGj(x)fAj(x)dx,
uij-oo, t] x {0}) = pA<zj _<_ t, 5]. = 0) = 41% 0;)de (x).

24

25

Consider a measure an on B x {0, l} by the following relationship:

' z
an((Y' 2] x {1}) fy Gj(x)dx. y < z,

(1.7)
vnj((y. 2] x {0}) = Gj(y) - Gj(2), y < 2.

Then “nAj is absolutely continuous with respect to on]. and

c111 .
nA _ d
(1.8) ——l (x. d) — (fAj(x)) ~(FAj(x)

dvnj

l-d

I

wherexeR, d=Oorl.

In this chapter, we will use the Radon-Nikcdym derivative in (1.8)
to prove the important LAN (locally Asynptotically Ncmal) property for
the above censored regression model, which provides broad background for
statistical inference.

Definition

 

For all n= 1, 2, let @ncRP, Fnbe aU-algebra, 1:nA

an expectation on Fn’ for all A a On. We say that conditim LAN
<6, Mn' Yn> holds, if

(i) For all n, Mn is a p x p positive-definite matrix, and Yn is a
p-din‘ensicnal randcm vector on Fn' such that

(ii) Yn =>N(O,I) under En where I is the identity matrix, and

6!
(iii) For any given bomded sequence <tn> in RP, if

1/2

A =6+M- t iseventuallyin9,thenwehave
n n n n

as
n'An T 1 2 .
(1.9) log as - (tnyn - Elfin“ )+0 in <1:n e> prob.
n,9 ’

 

26

Remark 1.1 If tn+ t, then the log likelihood ratio

dE‘r

 

n'An dNt l. .2 2.
(1.10) log (33 =>log arr“ N(- 511th , {It}: ).
n,6

where Nt is the expectation of p—dimensional normal random vector
with mean t and covariance I.
‘Ib prove the IAN property for the model (1.1) - (1.3), we need
the following assumptions:
(Al) F has density f with respect to the Lebesgue measure; f is
absolutely continuous and the Fisher informatial amount
I(f) = -f (1%) 2d}? is finite, where f' (x) is the a.e. derivative of
f(x) .

(A2) (Noether cmditon) max [C?(CTC)-1Cj] + 0, where C is the design

3531
matrix with raw vectors {Cg}.

(A3) Mne is positive definite,

 

 

T -l __
where
n T
(1.11) M =: 4 Z R .c.c.,
n6 ._ n63 j 3
j-l
(l 12) R =- 1'6 (x + eTc “(9)2 (x)dx - I—f-i(x)dG (x + eTc )
' nej ° 3' a" f F j 3' °
Remark 1.2
(i) For the uncensored case, Rnej = I(f) and Mne = I(f)CTC is

positive-definite whenever C has full column rank.

27

(ii) Since I(f) < 0°, f(x) +0 as x+j~_ 0° (see lemma I.2.4.a of
Hajek and Sidak (1967)). Hence

I 00 i 2
(1.13) f2(x) = (f°° L(x) ./f(x) ax)2 < f EJ-mdy-Nx)
X /?- — X f
an important inequality which will be used to prove the main Theorem.
We first apply (1.13) to claim

(1.14) Rnej _<_ I(f), for all 3 i n.

Indeed, by using (1.13) and 'Ibnelli's Theorem, we have

Rne j

2 2
T (f') 0° (f') T
_<_ ij (x + 6 Cj) -—f——(x)dx - ffx -?——(y)ddej(x + 6 Cj)

2

 

, 2
= ij (x + eTCj) g—L—(XMx + I(l - Gj) (y + STCj) (f ) (Y)dY

I

f
= I(f).

‘Iherefore, for the case p = 1, O < Mne <°° and (A3) is always true.
2. The Main Theorem

Denote

(2.1) s =: 131/2, 3 =: 5'1/2.
du .
_. n11] 1/2
(2.2) hnAj -- (dvnj ) ,

' I

- _._ ,_T d., _T 1—d.
(2.3) hm]. —. (l){s (x ch)} {S(x chn cJ

where s' and S' are a.e. derivatives of s and S, respectively (see e.g.

Hajek and Sidak, 1967, p. 212). Note that if f' were everywhere derivative

of fthenhnAj 15%5hnAj' We furtherdenote
, fine.
(2'4) qnej =‘ 5—1. I(hnej > 0"

n6]

28

and
_:-1/2 -
(2.5) Unej ZMn qnej o (Zj , (S. 3.)

Note that straight— forward computation yields

. T
(2.6) Nine: 4jzlff(hne j)(hnej) dvnj

We new state the main Theorem.
Theorem

Under (Al) - (A4), the condition lAN <e,M Zune 3> holds for

ne'
the model (1.1) - (1.3).

To prove the main Theorem, we need the following lemma.
lemma (Fabian and Hannah)

OonditicmlAN <9, Mn, XUne j> holds, if

(i) ”h nGjHELZan '
(ii) ffqnejdunej = o,

n
(iii) j=1 Enzeuunejufluu nje “_>_a}+Oasn+°°, foralle>0, and

(iv) For all bounded sequence <tn> in RP, let

A = e +M’1/2t , wehave
n n n
T‘ 2

Proof. See Fabian-Hannah (1980), Theorem 4.5.

Proof of the Theorem

 

We will verify (i) - (iv) in the above lemma.

(i) By a similar carputaticn for M

n6 in (1.11), (1.12) and

(2.6), we have ff11hnej11 dvnj - HC]. )1 RnGj' Therefore (l) holds by (1.14)
and (A1).
(ii) Denote

_ l d. f' _ T d. g. _ T l-d
(2-7) Onej --2'('1) {(—f-) (X 6 Cj)} {F(X 9 Cj)}

tren onej = Qnej‘cj. Note that

”onej'dunej = 0'
bythefactthat f(x) +0asx+:m, andthe integrationbyparts
(see, e.g. Hewitt and Strarberg, Theorem 21.67) . This catpletes the
proof of (ii).

(iii) LHS

 

-l/2 2:2
_<_ j: lne411M c.1123n6Qnej(zj, 6 j)I{411Qnej(Zj , cs )11>max W“ M.l/2C “2}

i<n

1/2 2 £2 _ T _f_'_2 _ T
3.211le €3.11ij(x)(f) (x ecj)I{(f) (x ecj)

 

"131‘le C- 12“

n
-l/2 2 f 2 T f 2 T
o u - . I — - .
+ jgluMne C311 fFejbc) (—F) (x 9 C3) {(15) (x 9 C3)

 

2
> WINE/2 Ciszhiu - G]. )(x)
= (I) + (II), say.
Note
(2.8) xmax11M;19 1/2c. 112 =max CT (cT C) 1[(CTC)M;1]Ci + o,

i<n i: n

30

by (A4) and the Noether condition. And

 

 

n n
-l/2 2 __ _ T -1

(2.9) 11; 11Mme c111 — ECiMneCi - E tr(ciciMne)

1.11—1 1—1

= tr(CT CM me) = 0(1).
where tr means nthe trace of a matrix. Now

2
2 f' 2 e
u)<2fluM Heur()(mn()%()> Zmu-an
j=l ne f f maXHMne -I|./2C €12"
= 0(1) ° °(1)
Next to claim
(2.10) (II) ~> 0.
To see this, note that by (2.9)
(II)
f2 f2 T
(2.11) _<_ 0(1) max f—- --(x)I[-—2-(x) > K]d(1 - G.) (x + 6 C.),
if?! F J 3
where
E:2
(2.12) K = + °°o as n + °°r
”1114,19”chle
i<n

by (2.8). We further have, by usm' g (1.13) and Tbnelli's Theorem,
2 2
f f T
f-F'(X)I[;'-Z(X) > K]d(l " Gj) (X + B Cj)

f2
.gfifi‘fi’2(w@1§gm>Kmu-G”)m+eC)

Yumm<$§hm1-GWHx+ech’2(we

-00

n
Fe

-1 I(f)

( )<mm1-GWMx+ech’2(w®

 

1 A
5‘8
3‘8

I(x< y)I[F

. 2
%§‘§’me

1 A
2‘8
5‘8

 

nfh%fl)<wM1-%Hx+e

31

(2.13) i [A Egg—Why, for all j _<_ n;
where A=An={Y:Y>F- ].(_I___(f))}’an dF-l(£1(<fl)'*F1-(O)=°°IaSK+°°°
Hence by (2.11) - 2.13), we have (II) + 0 This concludes the proof of
(iii).
(iv) In View of (1.7), (1.8), (2.1) - (2.3), we have
'1" 2
”{hnAnj - hnej - (An -6) hnej} dun].

_ _ T _ _ T _ T T 2
f{s(x Ancj) Sb; e (3].) + (An 6) c. S(x 6 c3.)} Gj(x)dx
T T T 2
+ f{S(x - AT nj—C) S(x - e cj) + (An - e) CjS(x - e cj)} d(l - G3.) (x)
(2.14) = Bnlj + Ban’ say, l_<_j in.

By using Ietrma VI.2.l.b of Hajek-Sidak (1967), we have

(2.15) {.ij = 0(1).

‘lb prove 23an = 0(1) , we first claim
3'

(2.16) (s' (y))2 =

.h-IH
uanIN

(y) +0, asy+i°°, and
(2.17) 71(2) =
Y

Indeed, by (1.13), we have

(S'(y) - S' (y - 2))2 is continuous at 0.

(s on)2 _<_4 fy f°°"—E——’2 (x)dx+o. asy +00.
since I(f) < 00.

By Cathy-Schwartz inequality, we also have
(s' (y))2 5&1: -‘—E—'l(x)dx

Tnis concludes (2.16) .

(1-F) (x) + ., ...,
F(y) 0, as y .

(2 17) now follows by using (2.16) for large y
and by usmg the fact that S' is continuous and hence uniformly continuous

32

on compact sets.
Next, let
— _ T.
(2.18) bnj - (An 6) le
+ + . .
and denote 2‘3an = 2 Ban +2En2j’ where Z 15 the sum over those 3 for
whichbnj > OandZ" isthesumoverthosejforwhichbnj < o. Wethen
have, by using (2.9), (2.14), (2.17), (2.18), Cauchy-Schwartz and Tonelli,
+
Ban
= Z+f{S(X - b -) - S(X) + b .S'(x)}2d(1 - G.)(x + 6Tb.)
n3 n3 3 J

= Z+f{-Ibnj S'(x - u)du + b .S'(x)}2d(l - G.)(x + 01C.)
0 n] J J

b .
_ + 2 1 n] . _ _ . 2 _ T
-2 bnd{ 3; I0 (8 (x u) s (x)du} d(1 Gj)(x + e cj)
b .
0(1) max+ b'l. I njn(u)du.

2.19
( ) jm n] 0

IA

By (2.17), for any given a > 0, there exists 6 > 0 such that |n(u)| < e
for all |u| < (S. For this (3, by (2.8), there exists no such that
max+ b . < 6. Therefore

“3 b .
RHS(2.19) _<_ 0(1) °max+ til; Ion] e du = 0(l)°e, for all n 3 no. By

j_<_n

a similar argument for z- B n2j we conclude
(2.20) 23an = 0(1).
(2.20), (2.15) and (2.14) then yield (iv) of the Lemma. ‘Ihis also canpletes

the proof of the main Theorem.

Remark

 

(a) An alternative proof of (2.10) is as follows. By (2.11), we have

(2.21) (II) 5 0(1) sup (S'(x))2I[/\2(x) > K],
X
where

33

(2.22) A2 =

hid H)”

By (2.16), S'(x) + 0 as x+j_-_°°. Hence for alle> 0, there exists x0

such that IS' (x)| <./E/2 for all |x| > x0. Note A2 is continuous and
hence has a maximumvalue on the compact set [-x0, x0]. Choose n such

that K: Kn?— | Tax A2(x), where K is defined in (2.12), then
x<x
-0

sup (5' (X))21[A2(x) > K] = o, and
lxl: x0

A
m
0

sup (8' (x))21[A2(x) > K]
le> x0

Thus by (2.21) (II) + 0.

 

. , _ 1 f . . .
(b) Since 8 - 2 FJ7§ is the everywhere derivative of S, we can
prove (2.19) by another approach:
ZBn2j
S(x-AnTC.) - S(x - eTc.) T 2
_<_maxf{ 3 T J +S'(x-0C.)}d(l-G.)(x)
jin (an - e) cj 3 3

, _ T 2
Emu e)cj}

T T
S(x-AnCi) - S(x-e Ci)

 

: 0(1) max sup{ T
jin x (An - 0) C3.

+ s' (x - ech)}2

= 0(1) max sup {8' (x) - S'(x - 5.)}2,
jin x j

bythemean-value Theorem; where |§.| < IATC. - GTC.| < maxlb .| =O(l).
3 -' n3 3 _jn n3

Hence )3an = 0(1) by (2.17).

34

APPENDIX
1. PROOF OF THEOREM 2.1

N
Recall t = t - a - BE'. Denote

(1.1) §(t,A)

ES(t,A) - ES(t,B)

’b ’b
-Xdi{F(t + di(A - B))- F(t)} .
To simplify the notations, we further denote
. - . 2 2
01(8,A) - .fl§B + (A - 8)odffl_ ,
‘ - 2
I {S(t,A)-S(t,8) - S(t,A)}Zdu(t) = :fl5A - SB - SAfl_ , and
f {S(t A) - (A-B)02f(t)}2du(t) = ° 3' 4 (A - B)02f 2
a d °.u.A dlL .

Hence

(1 2) lQlA) - 01(B;A)|
sij-sB-§QE+¢§E-(A-snfimf

+ 215A - SB - 31 111(le <A - ~8>o§fl +113 - (A - swim}
+ 21158 + (A- (3)031‘1- 15A - (A - (3)0311

By the fact that §(t,A) is monotone in A and by following the method in
Koul-DeWet (1983), it suffices to prove

35

(i) lim sup 0&2 50(3) < w ,
n
.. -2 — 2
(11) 0d EflfiA - SB - SA” + O , and
... -2 2
(111) 0d HSA - (A - B)o§ffl_ + 0 ,

Proof of (i). Since S(t,B) is a sum of independent random variables

 

variables and ES(t,B) = 0, we concluded, by (A4) and Tonelli's Theorem,

LHS(i) = 0&2 E 1;: Sz(t,B)du(t)

_ 2 co ..
= ad2 Edi f, Var {die 1(2,)I(zi 5 t + Bdi)du(t) < e .

Proof of (ii). Note
EHsA - SB - An? = f E{XdiRi(t)}2du(t)

 

where

_ . -1
Ri(t) — '5iG (Zi){I(Zi s t + Adi) - I(Zi < t + Bdi)}

Since {Ri(t)} are independent random variables and ERi(t) = 0, for all i,

we have

E{2diRi(t)}2 = Zd§Var{Ri(t)}

2 -2 2
s 2diE6iGi (Zi){I(Zi s t + Adi) - I(Zi g t + Bdi)}
2 t + 8d. + a -1 o 0
§'- Edi ft + 8d} _ a: Gi (x)dFi(x) , w1th an as 1n (A5) .

Therefore, by (A5), LHS(ii) + 0 .

36

 

Proof of (iii). By standard arguments, LHS (iii)
< ka2 1 f eil {f(% + S) f(%)}2du(t)ds
.nax " I
" l<n 2|eil'leil
where ei=Adi- Bdi, andmax |ei| ia-oC-l max lei} _<_.a-od ~max |di| +0.

Denote h(s) = :[:{f(t + s) - f(t)}2du(t) . Then by
(A2), (A3) and Theorem 4.4.16 of Royden (1970), h(s) is continuous at O.
This and (A1) entail (iii).

That completes the proof of Theorem 2.1. D

2. VERIFICATION or (1 3.35)

We first calculate Ezuij for i % j; where {Uij} are defined in (1.3.30).

(2.1) Efiuiz

-) ,

- _ “‘1 m
where ti = t + Bdi .

(2.2) E£U£j

= d£6£G'1(Z£)I(Z£ 5 t3) -'{1 - Fj(Z£)}H-l(y)dG(Y) .

(2.3) E = XE B = 2E (2 '1)22u..
2 2 2 2 2 " ij 13

37

W

W m
‘ B21 + B22 + B23 ’ say'

Next, by using (2.1)

(2.4) E = 2n'1 2 2di(1—5£)fi'1(z£){ri(z£) - F(¥)}I(z

21 t.)

£ 5 1

- 2n‘1£d£(1-§£)fi'1(z ){F

2 (Z

A)

m
K - F(t)}I(Z£ < t

z)
= (F1rst term) + °p(°d) ,
for all t s 6, since by (B4) and (BS.b),

-2

0d E(second term)2

IA

-2 -2 2 "2
4n 0d nZd£E(1-6£)H (z£)I(zz 5 ti)

4n‘1fi'2(g) = 0(1) .

IA

By the same argument and (2.2), (BS.a)

-1 -1 1
2n 2 Xd£6£G (Z£)I(Z£ g t

m Z
3 "1 -
(2.5) 322 K) Lm-FjF G dG + op(od)

-231 + op(od) ,

Straight-forward calculation yields
N

Hence (1.3.35)holdstu'combining (2.4), (2.5) and (2.6). U

38

3. VERIFICATION of (1.3.36)

 

We first calculate EZUij k’ for i#j#k, where {Uijk} are defined in (1.3.31).

(3.1) EZUiJZ

d.E

1 £IEZ

>Zj JE

. . - . 6. . . .
Z 2J61 1 J J’ 1‘ 1

d. WE£IEZZ>Z 1(1-6fl)2(z.)(Fi (2.) - F(¥))I(z.<t.)
6, 23£Ati Fj<y>fi'2 (y)(F,(y) - F(t))de(y) .

(3.2) EKUifik

= di(1-6£)H-2(Z E a.e‘1(zi)1(z <L <1. )E I(Z >2 )

£) £ 1 Z' i' Z k K

--2 m .
= di(1-6£)H (z£)(Fi(Z£) - F(t)) Fk(Zg)G(Z2) ° I(Zzs ti)

(3.3) E£U2jk
_ --2
_ d262 e 1(Z£)I(Z£ <t 2) H£(1 6 )IEZ. J.<z£JE£J .IEZ k<2le (Zj)
= d£6£ G I(ZZ )1 (Z£<t£) H£(1 6. )IEZ. WZKJFk( .)o(z. )H “(z )
- -1 23 "2
— d£6£G (Z£)I(Z£s fl) 2‘” - FijGH dG .
S1nce Uijk - 0 if i = j or j = k, for any fixed 2,
-2
(3.4) B = n ( 2 Z 2 U.. + z 2 U..
3 1+2 1+2 k+2 ‘Jk 1+2 1+2 ‘33
+ U. + Z - + I U )
1E2 ki2 ‘zk 112 kizu ilk 1+2 zi‘
Note

3.5 '2 ..
( ) n I E13(1E2 1E2 kifl U‘Jk

= E(n-3)B3 + n'zE 2 2 U

)

£j£

(n- -3)EB3 + 0(0 d)

39

And, by (3.1),

-2
3.6 2 E u
( ) n I 1:2 j+2 2 ”’3

n4: 2 2 (11,2216 F.<y)fi‘2(y>xt1dF,<x)de<y)
2 1+2 j+2 J Y

"-2

v..,
iéfl jifi 13

2 (say).
K
n26 2 2 d 226““ My )H 2y( ) 251' dF,.(x)dG(y)
L i j
V

— ‘v.2v),
II”: H6630"

where the last 3 terms are op(ad), for example, by (BS.b),

0&2 E(n 2 X E V. Hz

i i
s n-4 0&2 n2 E E d? E 1:12AM-dGFEl-l-4
-1-- _
H (E) - 0(1) .

Similarly, we have, by (3.2) and (3.3),

3.7 ‘2 . .
( ) n (22: 1:2 k§2w6166+§z j§2k$2E£UKJU

= {2% 12 Edi(1<SH)lI-2(Z) ti -dF1.(x)Fk(Z£)G(Z

fzz K) °I(Z (13')

fi’ 1

2 -1 z --2
EE d£6£G (Z£)I(Z£< t1.) 1,} -FJ.FkGH dG + op(od)
Z d

n-
“-1

1

Z
Z
3;. 1.(1-£117”1(z£){r1.(z£) - F(tH - I(Zzs, t1.) - 31 + op(od)

Now (3.5) + (3.6) + (3.7) yields (1.3.36). D

40

4. Verification of (1.3.37)

 

By direct calcuiation, we have

(4.1)
and by
(4.2)

= ti - , -
EB1 Edi f4 lnGdF1 ,
using (BS.a), we can prove

_ .-1 -1
E82 - 2n gdiEGiG (Zi)I(ZiE ti)

. ,--1
XE].(1-<sj)1(zj 2].) H (2.)

iii 3
- - . --1
= 2n12d.55.e 1(z.)1(z.<_t.)2 121 -F.H dG
. 1 'I 1 1 'I . . J
1 3+1 -w
= 22di 1'}; fiFfi-ldeF-1(X)
- 2n-12d1. IE: 13; Fifi-ldeFiU) ,

-2EBl + o(od)

For E83, note

E fi‘2(2)fi(2)
ij j n i

-.2 _1 1
H (Zj)n kgj EijIEZk:>st
' 2 -1

H (Zj)n k§j6(zj)Fk(Zj)

fi'zajnffizj) -n‘le(zj)Fj(zj)}

.. _1- -2
H (Zj) - n ii (Zj)G(Zj)Fj(Zj) .

Hence by using (BS.a) for the second term of above, we have

(4.3)

EB3 = .SEB2 + 0(od) = -E81 + 0(06)

Combining (4.1) - (4.3), we have

(4.4)

(5.1)

(5.2)

6.

(6.1)

Then,

(6.2)

(6.3)
then

41

(n-1)EB + (n—2)EB2 -(n-3)EB

= 0(0

1 3 d) ‘

%

ES(t,B) = O.

 

E (first term)

n4 2 2 d. fti - F£(y)H (mam - F(indeo)

1 “a

Edi I}; - e‘lonnm - F(indeo) .

”1

(second term)

"-1 z 2 d1. I“- Mi fi‘lom'lomo) - F(t) de<y)d<rze)(x)

- :3 '00

n4 2 2 d1. [33; fi"l<y>e‘1(y)u=i (y) -F<¥>}(F£e)(y) - dam

Zdi L2: G’l(y){Fi(y) -F(¥)}de(y)

-_E (first term) D
%

nger Bound of Var {S(t,8)}.

Denote

% -1
S(t,B) - n E EdiDiK

2

k a
052 ES(t,B) = 0&2 Var {S(t,8)}

[A

-2 2
r: %4 E E£§diDi£}

!A

-1.2 2 2
n 0d XZdiE(Di£) .

Let
Ki(.) = fi'1(-){Fi( ) - F(¥)},

By

42

(6.4) 01. = ”‘32 1- Z)I(z11) < t1) + if” 6—1111 )mymem .

and

(6.5) EDEK = E(1-6£)K§(z£)1(z£s t.)

+ E2(1- «8 £1)K (Z )I(Z£_ t1.) rm“ 6 (y)K1(y)dG(y)

‘00

+ E{ I_Z§At1 G 1(y)K1-(y)dG(y)}2

(6.6) n‘1 2 2 d? E(D.£ )2

= id? 111' - F'(y)K1-2y( may)

1 -co

1

+ 22d1. 4E: - E<y)1<1.<y) 4: G'1(u)Ki(u)dG(u)dG(y)
+ 2d? P463?“ G'1(y)K1. (y)de<y)}2dfi(x) .

, .00
1

but, by denoting h(y ) to be

(6.7) G 11(y)l<(y) (Z, 6‘1 w)1g(u))de(u ,
(6.8) the third term of (6.6)
= 2 dij-xj‘fti61K(y)1y()_{e‘1(u)x1.(u)de(u)de(y)di(x)

0.
do N
I'fi
d
—Jo

II II
N N
do M I... M do M —‘o M
0..
—lo N
('1'
—‘

Q.
'“N
d-

—l

8

- 1’: h(y)dfi(x)dG(y)

H(y)h(y)dG(y)

I
8 .

II
N

I
8 .

F(y)K1(y )fy G1 M)K (U)dG(U)dG(y)

- (second term) of (6.6).
(6.2), (6.6), (6.8) and (BZ.c),

43

%
(6.9) sup “621521

tsin

t,B)

<sup Udzzd§£t°1-F1{i2dG
tfgn

. --1
s sup 61-12 Ed? -/_t°1 - F (y)G
tst:n

‘2(y){F1(y) -F(?)12de(y) < ., o

7. Verification of (1.3.42)

 

Note
-‘ _ m -
(7.1) Ut = n podl$(t68) = I1F)oa1(81 + 82 - B3) is approximately an U-statistic.
*
Let 8m to be the projection of Bm’ m=1,2,3, (see Hajek, 1968). Note

*
(7.2) B1 = EEZB1 - (n-1)EB1 = B1 .

Denote, see (1.3.30) and (1.3.32),

- -2 -p '1 n -2
(7.3) n Pa 13 = (") 2(n-1)n o (u..+u..) =; ) ,2 6
p d 2 2 i<j d 'u J1 (2 i<j

then, by (81) and (BS.a), 3 A. no, and .5 < p < 1, such that for a11 n 3 no,

r119_(’c;1',J');

(7.4) sup max Eofiz (t;i,j)
tsin 1' <J°

2-2p 0'2 max d? {:"G 1H -1.2dF _<_- A < 0°.

1<_n

5. 2n
Hence, argue as (8.6) - (8.10) of Hoeffding (1948),
(7.5) sup E(n'PoflB -3*))2 = 0(n'2) .
t<€ p d 2 2
-n

Simi1ar1y, see(1. 3. 31)and(1..3. 33), by denoting

(7.6) n 1105133 =(3")‘1<§<|(n -1)(n 2) '1'1"poa1¢n3(t;i ,j,k) ,

and by'using (85.a), we conc1ude

(7.7) sup ' max E¢23(t;i,j,k)
15%" 'i<j<k n
- - -- --1
< sup n1 2'1 - 0(1) - 661.6 2(21.)I(Z1.5'c1.)H (Z1)F (Z-)

tsg1

44

1- m3

1'21) H F ldF. <A< co.

-0(1) 4:" G’

Argue as Hoeffding (1948) again, we have

(7.8) sup E(ngpoaHBB -B;))2 = 0(n'2) .

1:56"

Now (1.3.42) holds by (7.1), (7.2), (7.5) and (7.8). U

8. Calculation of Var (qj(2 - 8))

 

Let
(8.1) K1(y,t) = fi'1(y)r;1- dF1(x) ,

and

A _ -2 -1
(8.2) ‘16” - B) - -(Lf(L °d E£Lo£+ Lu + Lu} ,_

_ -1 m m
(8.3) 101, - dfiéeG (22) {111(2z 5 t£)du(t)
-1
(8.4) L1}, =(1-<5£)n :61” K.(Z£,t)d1‘1(t)
I Zz- Bdi
-1 00 Z A
(8.5) L22 = n Edi f0° f 5 t1 G 1(y)K1.( y,t )dG(y)du(t) .
Then

A _ .4 _2
(8.6) Var {Od(A - 8)} - flffl_ 0d E Var {LOz + L12 + L23}

(8.7) Var {Log + le + L22}
= Var (L03) + Var (Lg) + Var (L23)

+ 2 Cov(L LIZ) + 2Cov(L 2K) + 2Cov(L

Ofi’ IZ’L 2£’ LOZ)

Note the existence of the limit of (8.6) is guaranteed by (1.3.49).
By (1.2.11), we know

(8.8) 2 Var (101) = Edi 1: f { f52412 - G1(x)dF£(xx) - (1-F(?))(1-F(%))1dh(t)dfi(s)
Z

Z

45

By definition in (8.4), we have

(8.9) 1 Var (111) = E(Esz - £2111)

Edidj I“, - (F£(v) xv°°_6di K1.(v,1:)cm(t)r"°V_13d KJ(v, t)du(t t))dG(v)
Zd.d.([_: F£(V) f1,” 8_d1 K1-(v t)du(t)dG(V))(/_:F£(V)W°1,_j8d Kj(v t)du(t)
1

dG(v))

_ 2 _ 2
(8.10) E Var (ng) — 12: (512, E L23)

f_°°Mf'11(y)K (y)dG(y)du(t))d(GF£ )(V‘

1 e'1(y)K1(y)de(y)dfi(t)d(GFe)(v) °

1: 12311 6 11w (y)de(y)dd<t)d<eF )(v ) .

1121 = 2111115102 12 5102' 5112)

00')! ”-1
3 -2E d2 {wu(x-8d£)dF£(x) ZdJ

(8.11) 22 Cov(Loi,
2

-.., 1mg: 1,111.16 <y.t)d11(t)de(y))

_ -1 on"! - . on ‘ '1’
_ - 2n {mu(x+o+Bc)dF(x)%1Zd£di {:F£(y) fy_8diK1.(y,t)du(t)dG(y) .

(8.12) 22 Cov(L
2

= 2.111318 _:(ii(x-sd1.~)r:xxft1' e1K(y)K1.<y .t)de(y)d’{1(t))d(1-F,)<x)
1

L

02’ 22)

+ 2n “'11): Zd£d1(fu(X-Bd£, m)dF£(x))(-{:_:_xft1’6 1(y)K1.(y,t)dG(y)d1T(t)d(F£G)(X))
1

(8.13) 2% Cov(le, L22)

46

-2 co 00 I“ on A . _
= n 11d1-d- {oo— F£(v){ (v-8d.Ki(V1t)d11(t)£,.{V.,tJ G l(y)Kj(y,t)dG(y)dr1\1’(t)}
1
dG(v)

-n"2 z E didj(f_:- F£(u)fu°°_6d. K1.(u,t)d11(t)d6(u))(1:1:1Vftj - G'1(y)Kj(y.t)dG(y)
1

81(t)d(F£e)(v)) .

Now the calculation of (8.6) can be done by'using (8.8) - (8.13) and a computer.

 

9. Calculation of B
Let
_ A-1

(9.1) W1. -d1.6.G (Z1) .

1
Then by'using (1.3.3), (1.3.4) and the equality
(9.2) max (a,b) = (a + b + |a—b|)/2 ,
we have

(9.3) Q(A)

=22w1wjun((z1. + ZJ. - A(d1. + dj) + (21. - ZJ. - Md]. - dj)| )/2, on).

Hence(2(A) is continuous in A, if “n is continuous.

For the case {un} are Lebesgue measure over fixed compact set [-b0, b0]

(9.3) is simplified to be
(9.4) Q(A) b06111.)2 - E 1w1w1(z1 - dd1)

-)l

- .5 1 2W1lel1- - zJ. - 1(d1. - dJ

= B

O

(2W3)(1N1(Zi “ Ad1))

-5 Z Z wiwjlz‘i ‘ Zj - A(d.i - de

47

It is then clear that Q(A) is continuous and piece-wise linear in A with knots:

(9.5) {(2. - Z.)/(d.

1 J 1 - dj): d1 + dj } .

Therefore to find a minimizer of Q(A), we need only check the knots in (9.5)
instead of the exhaustive search over the who1e real line.
A computer program in BASIC is.used to calculate 6 under (9.4).
Inside the computer program, tie following notations are used:
(9.6) Q(A) = BO - S-A - .558, with
S = S(A) = §w1(z1 - adi) ,

A = W. ,
2 J

$5 = S = . . . . _ . - -
S(A) 52211111“;l ZJ A(d1. dj)|

For the general case, since u(t,<») is a monotone function in t,
we have monotone curve between any two adjacent knots in
(9.5) Hence it suffices to check the knots for finding the minimizers of
Q(A) again. But be aware that the caluculation of (9.3) requires, in general,
the Simpson's approximation for integration and yields different result from

that by (9.4). The optimal choice of u wfill not be discussed here any further.

48

 

5 REM ;HEART

10 REM **** INPUT DATA

11 LET NN = 6

14 DIN NP(69),GH(69),N(69)

15 DIN DL(69),D(69),Z(69)

15 DIN C(69)

17 DIN NND(24SO),Q(24SO)

20 FOR I = 1 TD NN

30 READ DL(I)

:5 DATA O,1,1,1,1,0

45 PRINT "DL(“§I§“)= “5DL(I)
SC NEXT I

50 FOR I = 1 TD NM

70 READ 2(1)

25 DATA —1,-2,10,—3,15,—28
85 PRINT "Z(";Is”)= "32(1)
RD NEXT I

190 LET CBAR = O

110 FOR I = 1 TD NN

120 READ C(I)

125 DATA 1,2,3,4,S,6

140 LET CBAR = CBAR + C(I)
145 PRINT "C(”;I;")= ";C(I)
150 NEXT I

160 LET CBAR = CBAR / MN

170 FOR I = 1 TD NN

18m LET D(I) = C(I) ~ CBAR
85 PRINT "D(";I;")= ";D(I)
190 NEXT I

230 REM *****CALCULATE N+
840 FOR J = 1 TD MN

250 LET NPP = 0

2?0 FOR N = 1 TD MN

280 IF 2(K) < = Z(J> THEN 300
290 LET NPP = NPP + 1

300 NEXT K

310 LET NP(J) = NPP

315 PRINT "NP(";J;”)= ";NP(J)
317 NEXT J

ZED REM ***** CALCULATE CHAT
322 FOR I = 1 TD MN

325 LET GGH = 1

327 FOR J = 1 TD NN

:28 IF Z(J) 3 2(1) THEN 340
329 IF DL(J) = 0 THEN 340
330 LET GGH = GBH * (1 + NP(J))

(2 + NP(J))

340 NEXT J

350 LET GH(I) = BGH

355 PRINT ”BH(”:I;”)= ":GH(I)
350 NEXT I

“*7 1
._l "

"I “7 P3
(‘1 5 L.

v7"!

-.. .,
I . 4
.__I I

'T’T-fi—f'

ZQQ
Q Q
1c»
4 1 5
420
430
440
450
460
470

80

4.35

490
500
510
520

arr-n:
\u-JJi-‘J

SEC

540
550
560
570

580
590
600
610
620

630
640
650

655
(E3CEJII)
670
6 B C)
690
7 C) O
710
720
730
735
7 4 C)
750

7531:)

49

REM ***%% CALCULATE A AND N
(I)

FOR I = 1 TD NN

LET N(I) = D(I) X DL(I) X EH
(I) _

PRINT ”N(”6I;”)x «”;N(I)
NEXT I
LET A = 0
FOR I 1 1 TD NN

LET A = A + N(I)

NEXT I

PRINT "A=”,A

REM **%** CALCULATE KNUTS

LET L = 0
FOR I = 1 TD (NN - 1)
FOR J = (I + 1) T0 NN

IF D(J) = D(I) THEN 49D

LET L = L + 1

LET HNO(L) = (2(1) - Z(J)) /
(D(I) - D(J))

PRINT "KND(";L;")= "5KND(L
)

NEXT J

NEXT I

LET NR = L

REM ***** NH IS THE # DE EN
0T5

PRINT "NK=",NH

REM ***** CALCULATE Q

FDR JJ = 1 TD NK

LET S = 0

FOR I = 1 TO NN

LET S = S + W(I) * (2(1) - K
N0(JJ) * D(I))

NEXT I

LET 88 = 0

FOR I a 1 T0 NN

FOR J = 1 TD NN

LET SS = 85 + H(I) * N(J) *
ABS (2(1) - Z(J) - KND(JJ) *
(D(I) - D(J)))

NEXT J

NEXT I

LET Q(JJ) = — (S * A) - .5 *

38

PRINT "Q("3JJ;")= ";Q(JJ)
NEXT JJ

REM ***** MININIZE Q

LET QMI = 9999999999

FOR J = 1 TD NK

IF D(J) } = QMI THEN 730

LET QMI = Q(J)

LET JQ = J

NEXT J

PRINT "JG: “EJQ

LET B = HNO£JQ)

PRINT ”EHAT=",B

END

JRUN

DL(1)=
DL(2)=
DL(3)=
DL(4)=
DL(5)=
DL(é)=

p
v

(Whips—hp

2(1): -1
2'2): -2
2(3)‘ 1C)
2(4): -5
2(5): 15
2(6): -23
C ,

I')

p

H
IU‘m.pLJhJH

C(5)=

C(b)=

D(1)= 2.5
D(E)“ -1.5
D(TE): -.5
D(4)= .5
D(5)= 1.5
D(é)= 2 5

NP(1)=
NP(2)=
NP(3)=
NP(4)=
NP(5)=
NP(é)=
BH(1)= .éééééééb?
GH(2)= .656665667
GH(3)= .444444444
GH(4)= .853335355
GH(5)= .222222222
GH(é)= 1

N(1)= O

N(2)= —° ”=

N(3)= —1_125
N(4)= .6

N(5)= 6.75

N(6)= O

(‘1: 3. 97

Q-bF‘MtJ;

Ul

HND(1)=
KND(2)=
HND(3)=
KNO (4)
HNO(5)
KNU(6)
KND(7)=
KND(8)=
END (9) =
KND(10)=
HND(1 1)=
HND(12)=
KND(13)=
KND(14)=
KND(15)=
NK=
Q(1)=
Q(Z)=
9(3)=
Q(4)=
61(5)2
9(6):
D(7)=
0(8):
9(9):
B(10)=
Q(11)=
9(12)=
0(13)=
9(14)=
Q(l5)=
JD: 10
BHAT=

H il ll

5.66665567
-6.5
—13
2.5
-12.éébbéb7
18
—12.5
—43
15
~169.073437
-59.7167187
—168.774575
-94.725
-173.021052
775.7775
-168.624844
-33.6046875
~174.007968
—179.859688
-149.733281
—179.540525
1512.18553
—179.391094

-166.255312

-13

52

BIBLICISRAPHY

Bickel, P. J. (1982) On adaptive estimation. Ann. Statist.
Vol. 10, 647-671.

 

Buckley, J. and James, I. (1979) Linear regression with censored data.
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Fabian, V. and Hannan, J. (1980) Sufficient conditions for local asymptotic
normality. M403, Statistics and Probability, MSU.

Fabian, V. and Harman, J. (1982) (h estimation and adaptive estimation
for locally asymptotically normal families. Z. Wahrcheinlichkeits—
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Hajek, J. and Sidak, Z. (1967) 'Iheory of Rank Tests. Academic Press, N.Y.

 

Hewitt, E. and Stromberg, K. (1965) Real and Abstract Analysis. Springer-
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Hoeffding, W. (1948) A class of statistics with asymptotically normal
distribution. Ann. Math. Statist., Vol. 19, 293-325.

 

Ibragimov, I. A. and Has'minskii, R. Z. (1981). Asymptotic Theog of
Estimation. No. 16 in Appl. Maths. Springer Verlag, N.Y.

 

 

Kalbfleisch, J. D. and Prentice, R. (1980) The statistical analysis of
failure time data. John Wiley Intersfiénce, N.Y.

 

 

Kaplan, E. L. and Meier, P. (1958) Nonparametric estimation from
incomplete observations. J. Amer. Stat. Assoc., Vol. 53, 457-481.

 

K0111: H. L. and DeWet, T- (1983) Minimum distance estimation in a linear
regression model. Ann. Statist., Vol. 11, 921-932.

 

Koul, H. L., Susarla, V. and Van Ryzin, J. (1981) Regression analysis
with randonly right censored data. Ann. Statist. , Vol. 9, 1276-1288.

 

Millar, P. W. (1982) Optimal estimation of a general regression function.
Ann. Statist., Vol. 10, 717-740.

 

Miller, R. G. (1976) Least squares regression with censored data.

 

Miller, R. G. (1981) airvival analysis. Wiley, N.Y.

53

Parr, W. C. (1981) Minimum distance estimation: A bibliography. Carmun.
Statist. 'I’neor. Meth., A10 (12), 1205-1224.

 

Williamson, M. A. (1979) Weighted empirical type estimation of the
regression parameter. Thesis, MSU.

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