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SEQUENTIAL ESTIMATION OF FUNCTIONALS OF THE
SURVIVAL CURVE UNDER RANDOM CENSORSHIP WITH
APPLICATIONS IN M-ESTIMATION

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MOHAMMAD HOSSEIN RAHBAR

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SEQUENTIAL ESTIMATION OF FUNCTIONALS OF THE
SURVIVAL CURVE UNDER RANDOM CENSORSHIP WITH
APPLICATIONS IN M—ESTIMATION

by

MOHAMMAD HOSSEIN RAHBAR

A DISSERTATION
Submitted to
MICHIGAN STATE UNIVERSITY

in partial fulfilment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY
Department of Statistics and Probability
1988

/" A7 I
/ ”as

6‘

.f

ABSTRACT

Sequential mtimation of functionals of the survival curve under
random censorship with applications in M-atimation.

by
Mohammad Homein Rahbar

Sequential point and interval estimation procedures for functionals of the
survival curve F (of the form NdF and leVz) are considered when the
underlying observations may be subject to random censorship.

In the point estimation problem, the loss is measured by the sum of the
squared error of the estimator and cost of observations made with per unit
cost c being constant.

The sequential estimator defined here is shown to be risk efficient and
normal as c tends to zero under certain regularity conditions on functions
w, F and the censoring distribution.

For the interval estimation, the sequential procedure is shown to be
consistent and the corresponding stapping rule is shown to be efficient as the
width of the interval decreases to zero.

In both estimation problems, the asymptotic distribution of the stOpping
rule is obtained.

Finally, as an application the consistency and efficiency of a sequential
fixed width interval estimation procedure using M—estimation is shown for the
location parameter in a location model when the error distribution is
symmetric and continuous and the censoring distribution is continuous but

unknown.

To my parents and my wife and daughter

iii

ACKNOWLEDGEMENTS

I wish to express my sincere thanks to Professor Joseph C. Gardiner for
his excellent guidance, continuous encouragement, careful reading of this thesis
and especially his patience during the preparation of this dissertation.

I would also like to thank Professors James Hannan, Hira L. Koul,
Habib Salehi and RV. Ramamoorthi for serving on my guidance committee.
My special thanks go to Professors James Hannan and Hira Koul for their
helpful discussions and careful reading of this dissertation and for giving useful
suggestions, which helped greatly in the final presentation of this thesis.

I would also like to thank Professor James Stapleton for his help in
doing some simulation studies.

I would like to thank my wife Afsaneh, my daughter Elaheh and my
parents for their moral support during the preparation of this dissertation.

Finally, my special thanks go to secretaries Cathy Sparks, Loretta

Ferguson and JoAnn Peterson for their help in typing the manuscript.

iv

TABLE OF CONTENTS

Chapter Page
0 Introduction and summary ....................................................................... 1
1 Preliminaries and some notations ............................................................ 6

2 Sequential point estimation of functionals of the survival

curve under random censorship.

2.1. Introduction ................................................................................... 11
2.2. Examples ....................................................................................... 12
2.3. Main results of Chapter 2 ........................................................... 13

3 Sequential fixed width interval estimation of functionals

of the survival curve under random censorship.

3.1. Model ........................................................................................... 20

3.2. Main results of Chapter 3 ........................................................... 21
4 Sequential fixed width confidence interval for a location

parameter under random censorship using M-estimation ..................... 24

Appendices

Appendix A: The almost sure representation and the

asymptotic distribution of the estimator of the

asymptotic variance, 02, of the estimator of functionals,

of the form 0(F)=]1/1dF and 0(F)=]qub, of survival curve F

under random censorship ........................................................................ 33

Appendix B: Rate of convergence of the estimator of the

2

asymptotic variance, 0 ......................................................................... 40

References ........................................................................................................... 45

Chapter 0
Introduction and summary:

The problem of estimation of various functionals of the survival curve
from censored data is of fundamental importance in epidemiological and
reliability studies, clinical trials and life testing. In most clinical trials ethical
reasons and the paucity of laboratory specimens compel consideration of
statistical procedures in which the sample size is not specified in advance of
experimentation.

In this thesis, sequential point and interval estimation procedures for
functionals of the survival curve F (of the form jtde and dew) are
considered when the underlying observations may be subject to random
censorship. When censoring is present, as is generally the case in several
survival studies, one observes a random sample {(Zi,&l): 1$i_<_n} where
2i: min(Xi,Yi) and 6i: [Xi_<_Y.] with life times’Xi's, having common survival
curve F and censoring times Yi's, independent of Xi's, having common survival
curve G and [A] denoting the indicator function of the set A.

Consider the natural estimator of 0, “011:! t/Jan, where PD is a suitable
estimator of F based on the above random sample.

In the point estimation problem we consider the loss structure
Ln(c) = a(dn- 602 + cn where a is a given positive constant. The
associated risk is Rn(c)=E(Ln(c)). The problem is to determine the sample
size which minimizes the risk Rn(c) for a given positive cost per observation
c.

In the interval estimation problem we construct a confidence interval for
0(F) of prescribed width 2d and coverage probability (1-20) where 0<Za<l.
In either case, the best fixed sample size procedure (BFSSP), say no,
possessing the desired property depends on unknown functions F, G and

therefore no fixed sample size procedure minimizes the risk universally for all
F, G. In order to resolve this difficulty we define a sequential sampling rule
(often called st0pping time) similar to the rule considered by Robbins (1959).
Essentially, this rule, say N, samples observations sequentially, updates a
suitable estimator of F, G in the BFSSP and stops sampling as soon as the
number of observations exceeds that of the estimated BFSSP. Thus one is led
to solve these problems using sequential procedures.

In the point estimation problem, the performance of the procedure
(N,RN) is usually measured by (i) Relative risk, (Rik/R0), and (ii) The regret,
r*(c) = R*- R0 where R* is the risk due to using stOpping time N and R0 is
the risk in the BFSSP. In (i) if lim (R*/R0) = 1, as c tends to zero, the
sequential procedure is said to be risk efficient. In (ii) if r*(c) = 0(c), the
sequential procedure is said to have bounded regret.

In the interval estimation problem, the performance of (N,IN) is
measured by the (iii) Coverage probability, P[0EIN] and (iv) Expected sample
size, E(N), where IN is the fixed width interval estimator using sample size N.
In (iii) if lim P[0€IN]=1—2cr, the procedure is said to be consistent. In (iv) if
EN/no—o 1, the procedure is said to be efficient where in (iii) and (iv) limits
are taken as (1 tends to zero.

We now give a brief review of the literature on sequential point and
interval estimation.

Sequential point and interval estimation problems have received
considerable attention ever since the fundamental paper of Robbins (1959) for

the estimation of the mean of a normal population. He considered

)(1,X2,...,Xn iid N(p,02) and the loss function Ln = al—X'n- pl + n, where in

is the sample mean based on n observations and a is some positive known

constant. He assumed that the cost of sampling is proportional to the sample
size and showed that when a is known the BFSSP is n0=(aa/J—2TI)2/3. For
this sample size the risk is R0=E(Ln )=3n0. All this presupposes that we

0

know a. When a is unknown he suggested to take the sample size

N = inf{n23: n2(aSn/J_2II)2/3} where S121 is the sample variance. Starr
(1966) proved the risk efficiency and Starr and Woodroofe (1969) showed that
the regret is bounded in the above case.

In the nonparametric context, Ghosh and Mukhopadhyay (1979) were
first to prove the risk efficiency for the mean problem under the condition
that the eighth moment is finite. Sen and Ghosh (1981) considered sequential
point estimation of estimable parameters based on U—statistics under the

2+8< 00, for some positive real number s, where g is the

condition that E|g|
kernel correSponding to the parameter. Estimation of the mean is a particular
case with g as the identity function. Chow and Yu (1981) have proved risk
efficiency for the mean problem provided an rth moment is finite for some
r>2. Sequential point estimation of location based on some R—, L-, and
M—estimators are discussed in Sen (1980) and Jureckova and Sen (1981).
Sen's book (1981) has an excellent survey of the above mentioned articles.

Gardiner and Susarla (1983) were the first to consider the sequential
point estimation of the mean problem, in a nonparametric context when
censorship is present. They did not find the asymptotic distribution of the
stopping time except for the case of an exponential survival time. (See
Gardiner, Susarla and van Ryzin (l985b)).

In this thesis we pr0pose a sequential point and an interval estimation
procedures for functionals of the form j¢dF and [Fdw when the underlying

observations may be subject to random censorship. We shall show that the

sequential point estimation procedure is risk efficient and asymptotically
normal as c tends to zero, under certain regularity conditions on functions

a, F and the censoring distribution. For the interval estimation problem, the
sequential procedure is shown to be consistent and efficient as the width of the
confidence interval decreases to zero. In both estimation problems, the
asymptotic distribution of the underlying stopping time is obtained. Thus our
results are generalizations of Gardiner and Susarla (1983) and Gardiner,
Susarla and van Ryzin (1985b).

In Chapter 1, we collect various preliminaries and necessary prerequisities
of asymptotic prOperties of the product—limit (P-L) estimator. Most of the
results are taken from Gill (1983), Foldes and Rejtii (1981), Cheng (1984), Lo
and Singh (1986), Gardiner, Susarla and van Ryzin (1985a) and Schick, Susarla
and Koul (1987). Hence some of the proofs have been omitted. We also
state the results regarding to the asymptotic normality and the almost sure
representation of the estimator of 02, the asymptotic variance of
n1/2(I¢an—]1/JdF), which are new and crucial for obtaining the asymptotic
distribution of stopping times in Chapters 2 31. 3 and we give an almost sure
representation of n1/2(f¢an-/¢dF), for (b in a class of functions \Ill (defined
later in Chapter 1).

Chapter 2 is divided into three sections. The first two deveIOp our
model and some examples are discussed. In Section 3 the risk efficiency of
the sequential point estimation procedure and the asymptotic normality of the
underlying stopping time are presented.

In Chapter 3 we discuss the prOperties of the sequential fixed width
confidence interval and the related theorems.

Chapter 4 deals with a sequential fixed width confidence interval

procedure for the location parameter of a location model under random

censorship when the distribution of the error is symmetric around zero but
unknown.

Finally, proofs of Theorems 1.3 and 1.4, the asymptotic normality, almost
sure representation and a rate of convergence of the estimator of 02, the
asymptotic variance of n1/2(]¢xan-jt/xlF), which are crucial for obtaining the
asymptotic distribution of the stOpping time in Chapters 2 &. 3, are placed in
Appendices.

Chapter 1

1.1 Preliminaries and some notations

Let {Xiz i21} be a sequence of nonnegative iid rv's (survival times) with
continuous survival function F on R+=[0,oo), F(0)=1. The corresponding
censoring rv's {Yi: i21} are also assumed iid, independent of {Xiz i21}, with
continuous survival function G taking values on R+ and G(0)=1. The
observable data are {(Zi,6i): i21} where 2i: min(Xi,Yi), 6i: [XiSYi] and [A]
denotes the indicator function of the set A. For an estimator of F we select
11’ which was first introduced by Kaplan
and Meier (1958), based on {(Zi,&l): 15i5n} defined by

the product—limit (P-L) estimator, F

 

111) F (t) =ll'll KwiH] for t<Z
‘ ° ° n .=. "Km— (n)
= 0 for t2Z(n)

n
h K t = 1 2 Z.>t, t>0, d Z < Z < ....< Z th d
W ere ( ) +i=l[ l ] _ an (1) (2) (n) are e or er

statistics of {Ziz lsign}. By the continuity of F and G ties among the
observations may be disregarded with probability one. Throughout we shall
need the following notations.

Let (Xj,Bj), 3'21 be copies of the R+x {0,1} with Borel a—fields and
ill * m
(X ,A ) = II(Xi,Bi). Let P = PF G denote the product measure induced by
i=1 ,

{(Zi,6i): i_>_l} on A* and E = EF,G denote the expectation under P.

Let P = {P: F, G 6F} where F = {the class of continuous survival
functions} and 0 = 0(F) be a real valued functional on F. The estimator
Rn: 0(Fn) of 0 has been studied by many authors including Breslow and
Crowley (1974), Wellner (1982), Gardiner and Susarla (1983), Gill (1983), and
Millar (1985). For {On} to be consistent for 0, certain conditions have to be

satisfied by the pair (F,G). For example, in Schick, Susarla and Koul (1987)
is stated that "when estimating 0(F), the p—th quantile of (1-F), we need at
least the condition G(0(F))>0, for the sample quantile to be consistent.
Generally, 0(Fn) is not consistent for 0(F) if 0(F) depends on parts of F that
lie beyond the upper support point 1G: inf{x: G(x)=0} of G." Similarly
define 7F and let 7=rH= min(rF,rG). Let (Z,6) be a mm of (Z1,61). Let

E(t) = P[Z5t,6=1], fi(t) = P[Z5t,6=0] and H(t) = P[Z>t]. Note that
E(s) = —(])stG, 17(3) = £ch and H(t) = 1 — E(t) — E(t).

Throughout, except in Chapter 4, all unspecified integrals are considered
on R+. Throughout this thesis L-r stands for (1 /L)r for any positive function
L; —2-1 denotes "convergence in distribution"; as. stands for almost sure with
respect to probabitity measure P; N(p,a2) will stand for the normal
distribution with mean u and variance 02 and the index in the summations
runs from 1 through 11 unless it is otherwise specified; \II denotes the class of
all real valued monotone functions on R+ and let

\II1 = {$91! : ti; is constant on [T,oc) for some T<r},

Ac) =tlew. Ans) thmg‘ndt,tzo.

,00

t _2 a
C(t) = [H dH, 0_<_t<r,
0

2_ 2 ‘2_ 2 2 —2”
a—IAdC,an—]AnanHn,

t . .
(1.1.2) Fij(t) = 1 A‘ (10’, tzo, for i,j = 1,2,3,4,
0
(1.1.3) £(Z,6;t) = C(ZAt) — {6 H’1(Z)}[zgt], t_>_o
and

J(Z,6) = - I§(Z,6;t)F(t)d¢(t)
(1.1.4) = 6A(Z)H'1(Z) — 131(2)-

The following lemma is taken from Theorem 1 of L0 and Singh (1986)
and Theorem 3.4 of Gardiner, Susarla and van Ryzin (1985a) and will be
stated without proof.

Lemma 1.1: If F and G are continuous‘ and T<r, then on [0,T] and for p>0,

(1.1.5) Fn(t)— F(t) = F(t){n‘12 ((Zi,6i;t) + rn(t)}

with

(1.1.6) suplrn(t)| = 0((11-1111 n)3/4) a.s.,

and

(1.1.7) sup urns)", = curl)

where H . "p denotes the LD norm and sup is taken over [0,T] and £(Z,6;t) is
as in (1.1.3).

Assumption 1.1: Let T be a positive constant such that T<r. Let it be a

monotone nondecreasing function defined on R+ such that (0(x) = h, for

sz, where b is a constant.

Theorem 1.1: Under Assumption 1.1 and assumptions of Lemma 1.1,

(1.1.3) h1/2(1¢dirn— was) - n—1/2EJ(Zi,6i) = 0(n—1/4{1n h}3/4), a.s..

Proof: Integration by parts and Assumption 1.1 and (1.1.5) allow us to write
n1/2(~0n_ 0) = n1/2(l¢an-l¢dF)

F -— F
= - Ill/21(‘—nF—) F (W
= 11"],2 E J(Zi,6i) — nl/zr;
where J is as in (1.1.4), rn(t) is as in (1.1.5) and r; = jranw. Note that
by (1.1.6) and Assumption 1.1,
:0:
r=rtFtdt_surt Fd
n 1,0014) 10<th|,()|}1¢

= O (n_lln n)3/4 as,

A

from which (1.1.8) is immediate. u

Corollary 1.1: Under assumptions of Theorem 1.1,
(1.1.9) n1/2(I¢d13n— was) .2.» N(0,02).
Proof: One can show that J (2,6) is a bounded random variable with mean
zero and variance 02. Therefore by the central limit theorem (CLT) and
(1.1.8), (1.1.9) is immediate. 0
Theorem 1.2: Under the assumptions of Theorem 1.1 and for s>0,

{us/21211,- 0|3: n21} is uniformly integrable (UI).
, Proof: Recall the representation

til/201; a) = n‘1/221(zi,6i) - n1/2r;.

Note that it suffices to show that {In-1/22J(Zi,6i)|sz n21} and
{Inl/zrDs: n21} are both UI. Since for t>T, A(t) = 0 and H(T)>0,
{J(Zi,6i): i21} is a sequence of bounded random variables. Thus an
application of the Marcinkiewicz-Zygmund inequality implies that for any p>0,

sup Eln‘l/ 2>3.I(zi,15i)|1’<te,
n21

_ 3k
which implies {In 1/223J(Zi,6i)|8: n21} is UI. Recall that rn=frn(t)F(t)d1/)(t).
By the Holder inequality, Fubini Theorem and (1.1.7) we have that for any
p>0,

(1.1.10) Elr:1lp= carp/2).

Note that (1.1.10) implies that {lnI/Zr;|s: n21} is UI, which completes the
proof of the theorem. 0

Remark 1.1: Schick, Susarla and Koul (1987) gave a sufficient condition for
an iid representation of n1/2(]¢an— [t/JdF), where well, the class of all real
valued monotone functions on R+. They have shown that

1/2EJ(Zi,6i) = op(1).

2 .
111’ (fwan- 11x11“) - n
Our Theorem 1.1 gives the corresponding almost sure representation for (1)0111,

a subclass of \II.

10

Theorem 1.3 (The almost sure representation and asymptotic normality of
2%): Let 1151111. For F, GEF,
n1/2(:7121- a2) = 11‘1/213vi + n1/2Rn

5
where
Vi = 2Wm + wi,2’
Q
Wi,l = II! €(Zi26i;8)F(8)d¢(s)}drlli
2 -2 -l 2
Wi,2 = A (Zi)H (Zi)6i- 2IH [Zi>-]dI‘21 + a ,
and
n1/2Rn,5—-i 0, 3.3..
Furthermore
- :1:
(1.1.11) n1/2(UI21- e2)_9—. N(0,7 ),
where

*_ 2 —1 -2 4
(1.1.12) 7 _ 611‘11 (11‘21 — 4111 1‘11 (11‘31 + [H (11‘41 — e .

Proof: See Appendix A.

Theorem 1.4: Let 1119111. For F, GEF, and for each (>0, and all r<oe,
(1.1.13) Plldfi- a2|2 c] = 0(h“).

Proof: See Appendix B.

Chapter 2
Sequential point estimation of functionals of the survival curve
under random censorship

2.1 Introduction:

In this chapter we consider the sequential point estimation of functionals
of the form jifidF and [Fdip where ¢€\II1 and FeF.

Given a sample of size n, {(Zi,6i): lgiSn}, we estimate 0 by, .911: j wan,
subject to the loss function
(2.1.1) Ln(c) = a(21n- 0)2+ on
where a is a positive constant and c is the cost per unit observation.
The object is to minimize the risk in estimation by choosing an appropriate
sample size. From Corollary 1.1, we have that
(2.1.2) 1.1/2a“- 10—12.. N(0,02), as n —» ac.
Recall that in Theorem 1.2, it is shown that (under certain conditions) the
sequence {us/zlbn- 0|8: n21} is UI for s>0. Therefore it follows from (2.1.2)
that
(2.1.3) E(bn— t1)2 = {102 + o(n_1), as n —-. .0.

Now if a is known, then the risk
(2.1.4) Rn(c) = E Ln(c) .—. h‘lae2 + on + 0(n-l)

1/2, with

is approximately minimized by the BFSSP, 110 g be, where b = (a/c)
corresponding minimum risk

(2.1.5) R0 = Rnog 2on0.

However, since a is unknown, the BFSSP cannot be used and therefore we
describe a sequential procedure for choosing a sample size whose risk will be
close to R0 for small c. Let

. ‘ —h
(2.1.6) Nc = 1nf{n2nlc: n2b(an+ n )}

11

12

where h is a positive constant to be selected later. Since NCZbNgh a.s., we
may assume file: int(bll(1+h)) where int(x) denotes the greatest integer g x.

Then our scheme utilizes the estimator “ON of 0 with associated risk
2 7 *- *- L — be 2 EN
( .1. ) R —-Rc—E Nc— a E( Nc- 0) + c c’

We now consider some examples before we present the main results of

this chapter.

2.2 Examples
1. Aformofwimorizedmean: Let T<r, and
(2.2.1) “)0 = (xAT)[x20]

and 0 = —[¢dF, FeF. Note that In: -j¢an= —231/)(Zi)tildin, where din is
the jump of the P-L estimator at Zi when a sample of size n is observed.
Since in the absence of censorship the jumps of the P—L estimator reduce to
(1 /n), In reduces to an estimator of 0 where the ordinary empirical process Fn
replaces Fn above. In this case In turns out to be the average of all the
observations which lie in [0,T) and all the observations greater or equal to T

replaced by T. When using the P—L estimator to estimate E(XAT), the

. T.
estimator p = I Fn(t)dt, T<r, is used. Susarla and van Ryzin (1979) have
0

.. M ..
generalized the estimator to get the mean by taking )1 = j n Fn(t)dt,
0

where Mnl as, as n --1 00, with certain restrictions on {Mn}‘ Gill (1983)
. M ..
indicates how Mn can be replaced by Z(n) in the estimator p = I n Fn(t)dt.
0

Remark 2.2.1 (taken from Remark 4.1 of Gardiner and Susarla (1983))

If the problem of interest is the estimation of the mean survival time observed
T

on the duration [0,T], where T<r, that is, E(XAT) = j F(u) du, then by
0

taking to as in (2.2.1) we get 0% = F21(T). One encounters this situation of

l3

estimation of E(XAT), T finite in some decrement models. See Gardiner

(1982) and Hoem (1976) and (1987). Analogous remarks hold for the problem

T
of sequential estimation of E(X|X<T) = r- (l-F(T))-l(]) (l—F(u))du, with, of

course, a different expression for the asymptotic variance, 02.

2. A form of winsorizcd sample moments:

Let k>0, ¢(x) = (xAT)k[x,>_T] and FeF. Consider in: 4an as an
estimator of 0 = -]¢dF. Note that if the mean of F is known, we can
estimate the variance of F by taking On: 411an where

ax) = ax) = (unites - 11%
and up denotes the mean of F.
3. A form of mean residual life:

The mean residual life function is defined by

p(t) = F‘1(t) {muses = - “1(t){°°(s-t)dr(s).

T
Our interest is in estimation of 0(F) = F-l(t)] F(s) ds, for a fixed t, t<T.
t

If we know F(t), for example at t = med(F), F(t) = 1/2, then our scheme
estimates 0 by On = [Fill/J, where 111(x) = F-l(t){(th)AT}[x20]} and therefore
the asymptotic variance of this estimator is 02 = F-2(t){I‘21(T) — 1‘21(t)}.
In the case that F(t) is unknown, one can use Fn(t) as an estimator of F(t)
with, of course, a different expression for the asymptotic variance.
4. Kaplan-Meier M—estimator

This example will be discussed in detail in Chapter 4.
2.3 Main results of this chapter:

In the sequel all limits are taken as c 1 0 or b I so. We shall drop the
subscript c in Nc’ nlc and on various entities when there is no possibility of

confusion.

 

14

Remark 2.3.1 Throughout all the proofs are given for 0(F)=]1/JdF, where 11) is
as in Assumption 1.1. Note that handling the case that 0(F)=]Fd¢ where (l;
is as in Assumption 1.1, is very similar. In the case that it is monotone
nonincreasing on KI and constant on [T,oo), we can reduce the problem to the
above case by writing 0(F)=-[—¢ dF. Hence all of the results hold for $6111
and 0 of the form It/XlF and [Fdw where FEF.

The following results hold under Assumption 1.1.
Theorem 2.3.1: With N defined in (2.1.6) and for each PEP,

(2.3.1) nalN -+ l a.s.,
and
(2.3.2) E|n51N — 1| —. 0.
Theorem 2.3.2 (Risk efficiency): With N and R”' defined in (2.1.6) and
(2.1.7),
* -1
(2.3.3) R R0 —-1 1.

Theorem 2.3.3 (Asymptotic distribution of N): Let h>1/2 and N as in
(2.1.6), then

(2.3.4) 1/ 2(01N - 1).—e N(0, 7 *4/(40 ))
and
(235) N1/2-n1/2D_.. No.7 “1604 ))

where 7 = 6] 13121111“ -4] it"lrlldr31 + f H’Zdr41 -
Proof of Theorem 2.3.1: By definition of Nc’ lim Nc=m, a.s., also if
0<cl<c2, we have
2 (a/el)1/2(£}NC + N311) > (a/c2)1/2(0N + Ngh).

1 c1

Hence by definition of N c we obtain Nc 2 Nc a.s.. Thus Nc is nondecreasing
1 2

as c10. From (1.1.13) and the Borel—Cantelli lemma, it follows that

c1

{0121: n21} is a strongly consistent estimator of 02. Since Nolan, a.s.,

15
(2.3.6) 01% -—i 02 a.s..
Recall that b = (a/c)1/2. By definition of N, we can write
‘ “ —h “ —h
baN$b(cN+N )$N<b(aN_l+(N-1) )+l.
So that on dividing all sides by 110 and using (2.3.6) we get
nalN —-+ 1, a.s..
The next lemma is very similar to Lemma 1 of Gardiner and Susarla
(1983) which gives a rate on the tail behavior of the stOpping rule N which is
crucial for the proof of (2.3.2).

Lanma 2.3.1: For each O<£<1 and for any r<oo,

(2.3.7) P[N$n0(l—c)] = c(e("1)/2(1+h))

and

(2.3.8) 2 P[Nzn] = 0(c(”1)/2).
n2n0(l+c)

Proof: Recall that no: he and n1: b1/(l+h). Let 112 = 1120 = int(n0(1-c))

and 113 = 113 = int(n0(1+c)). By definition of N, N211], a.s.. For

c
sufficiently small c we have nl<n2, and on the set [N5n2], naban, for some
n e{n1,...,n2}. Therefore, for small c,

P[N5n2] g P[onsb"1n, for some n E{n1,...,n2}]

S P[cl21— a2Sb-2ng— 02, for some ne{n1,...,n2}]

IA

mei— 02${(l-e)2—1)a2, for some n€{n1,...,n.2}]
n

2 ..
2 PH 0121- 02|2¢(2-c)a2].
n=nl

IA

Thus (1.1.13) and the usual integral approximation for sums yield

P[N$n2] 5 const.[[12 x’rdx
n
1
= O(nI(r-l))

= 0(c(r—l)/2(1+h)).

16

To show (2.3.8), consider n2n3. Then on the set [N>n], we have that
«map k‘h), for all ke{n1,... ,n}. For c sufficiently small and n2n3,
P[N>n]< 17;], > b 1n- n‘h]

IA

P[an—a>b 13n -b1n0 -n]
P[an— a > (1/2) 60]
P[|s§- o2|> (1/4) 5202].

IA

IA

The last relation holds because
1231-02: (on-02)22n+2a(e-a)
2 (1/4) £2 02 + 026
2 (1/4) 62 02
Therefore using an integral approximation for sums and similar arguments as
in the previous case leads to (2.3.8). This completes the proof of Lemma

2.3.1. 13

Now we are ready to prove (2.3.2). Let 0<£<1, D = [n2<N_<_n3] and D
be the complement of the set D. Write

Nnol- 1 = Nn01[N<n2] + (Nnol - 1)[D] + Nn 01[N>n3] - [m
Note that N5n2, implies that Nnol S 1-6. Hence

Ean'l- 1| 5 (1-6) P[Ngn ] + c + n"1 EP [N>n] + P[D]
o 2 0 HM

= 0(c(r-l)/2(l+h)) + c + n610(c(r-l)/2) + 0(1)
= 0(1), since r>l and c is arbitrary.
This completes the proof of Theorem 2.3.1. a
Proof of Theorem 2. 3. 2 (Risk efficiency): Note that
R 1101 =(2en0)"1{a E(oN— s2 + c EN}
=an(2e)’1{0 lsz — (1)2 + c EN/no}.
Thus the theorem will be proved once we establish

(2.3.9) lim a(en0)'1E(bN— (1)2 =

17

For this, clearly it suffices to show that

(2.3.10) lim a(en0)’1 E{(bN- 22(9)} = 1
and
(2.3.11) lim a(en0)'1E{(bN — 102(5)} = 0.

First consider (2.3.11). Note that by the maximal inequality for reverse
martingales, (1.1.8) and (1.1.10),

sup Iln(0,, — 02H,= 0(1). M
n15n5n2

Therefore by the Holder inequality, Lemma 1.1, Lemma 2.3.1 and similar

arguments as in the proof of Theorem 1.2 we get, for s>2 and 0<h<s—2,
n

. 2 2 .. 2
Ein- o) [NSnzll = 2 E{(0,,- 0 [N=nl}

n=nl

IA

II
2 ..
2n {110,- 021, P1‘1/3[N=n1}
I]: 1

II

S { 22 Elan. 0'28}1/8{P1—1/S[NSD2]}
Il=Ill

s{sup nub,— 02II,}(§'3 n‘Sf/sipl’l/siNsna}
nISnSn,2 n=n1

z 0(,(s-1)/{2s<1+h)}) o(c(s—1)2/{2s(1+h)},

1l2), since 0<h<s—2.

= o(c
Similar arguments yield

‘ 2 _ 1/2

E{(0N- 0) [N>n3]} —- o(c ).

Thus by last two rates, (2.3.11) is immediate. Note that for (2.3.10), it
suffices to show that, for some c0,
(2.3.12) {a(m0)-l{(bN— 0)2[D]}: 0<c<co} is UL
and

(2.3.13) a(cn0)-l(l9N- ”MIDI—Q" Xi

18

where x? denotes the chi square distribution with one degree of freedom.
Recall that a(cn0)—1 = ba-l = non—2. For (2.3.12), it suffices to show that
for some s>1,

sup E{noc-2(3N - o)2[D]}8<a.

o<c<c0
Note that
E{n 17—260 — (1)2031)”: (11 0'2)3{E max(b - (1)23}

0 N 0 n

n <n$n

2 3
(2 3 14) < —2 s 28 * 28
, , _ const.(noa ) {(E max |Tn| ) + E max Irnl }

_ _ at _
where J n = n 123J(Zi,25i), and rn is as in Theorem 1.1. Since J n is a reverse

martingale, by the maximal inequality,

(2.3.15) E{ max ITHI28 = 0(n2-s).
n2<n$n3
Now using Lemma 1.1, and the usual integral approximation for sums yield
at: n :1:
(2.3.16) E max |rn|2ss 23 Elrn|23
n2<n5n3 n=n2
Q
S const.E Il—2S
n=n2
1-2
= 0(n2 8)
1-2
= 0(n0 8).
Now consider the first term in the R.H.S. of (2.3.14). By (2.3.15),
(2.3.17) (nor-2)8 E( max (‘1’ (23) = 0(1).
11
n2<n$n3

Similarly for the second term in the R.H.S. of (2.3.14), using (2.3.16), we

obtain

19

(2.3.18) (1100 2)SE max |r;|28}= 0(n1-8)
n2<n<n3

Since 3 can be taken greater than one, (2.3.17) and (2.3.18) imply
(2.3.12). As for (2.3.13), it follows from the fact that [n2<N$n3]-+ 1, a.s.,
Anscombe's and Slutsky's Theorems, Corollary 1.1 and Theorem 1.2. :1
Proof of the Theorem 2.3.3: By definition of Nc’

‘ —h “ -h
b(0'N+ N )$N<b(aN_l+(N—1) )+1.
By dividing all sides by no, adding (-1) to all sides and multiplying each side
by 11(1), 2, we get
n3/2(cN/ar —l)+a-1n(l)/2N.h< n1/2(N/n0— —1)
<nl/2(010N_1-1) +(N-l)"]:l

By (3.2.1) and since h>1/2, the limiting distribution of 110/ 2(N/n0- l) is the
1/2

n1/2 -1/2

[0 +n0
same as that of 110 (UN/0 -— 1). From (1. 1.11) we obtain
n1/2(aN - 01—117 No 7/0102»
which is equivalent to
n1/2taN/a - 1)—11—» No.7 /(4a4 ))
which implies (2.3.4). By taking the square root transformation, we obtain
N1/2- n01/ 2 —D-7 N(0,7*/(1604))

which completes the proof of Theorem 2.3.3. o

 

Chapter 3
Sequmtial fixed width confidence interval for functionals of the survival
curve under random censorship
3.1 Model
Suppose a random sample of size n, {(Zi,6i): 15i5n} has been observed.
We wish to construct a confidence interval In for 0 = 0(F) = Il/JdF and
0 = leifl, of prescribed width 2d such that, asymptotically as 11 tends to
infinity, the coverage probability is at least (l—2a). We assume F, G EF and
$8111. Note that by Remark 2.3.1, it suffices to consider 0 = fde and w as
in Assumption 1.1.
Notice that for each 11 an apprOpriate estimator of 0 is in: jwan,
where ED is the P—L estimator of F.
In the rest of this chapter all unspecified limits are considered as d tends
to zero. For a given positive real number (I and a€(0,l/2), in view of (1.1.9),
let us take 111 = (bu-d, Rn+d) with n = nd defined by
(3.1.1) nd = inf {k21z k 2 d-2 z: 02}
where 27 is the upper 1007 percentage point of the standard normal
distribution. Then we have

lim P [061%] = l-2a

and

. 2 —2 -2 _

hm {ndd za 0 }— 1.
Since F and G are unknown, the specification of the "Optimal" sample size in
(3.1.1) cannot be made. We are therefore led to construct a sequential
procedure in which the sample size is a positive integer valued random

variable N = N d’ and the desired confidence interval for 0 is

20

21

IN = (RN-d, IN+d). Motivated by (3.1.1), we define the stOpping time
N = Nd’ by

(3.1.2) Nd = inf {k2n M: k 2 b(&fi + {11)}

where b=d2 grand h isa positive constant. Since N d>b N—h, a..s ., we
may assume nl=nl d=b1/(1'1'h). Note that nd, the optimal sample size, is
asymptotically equivalent to 110 = ”M =b02, that is, ndnficll —7 1.

In the rest of this chapter we shall drop the subscript d in Nd’ 11d, n0d
and on various entities when there is no possibility of confusion. All
unspecified limits are considered as (1 tends to zero or b tends to infinity.

3.2 Main results of this chapter:
The following results hold under Assumption 1.1.
Theorem 3.2.1: For each positive real number h and F,G€F, (N,IN) is both

consistent and efficient. In fact we shall show that for each PEP,

(3.2.1) 11111 P[0e1N] = l—2a

and

(3.2.2) lim E |Nn31 - 1| =
Theorem 2.2: Let h>1/2 and F, GeF, then

(3.2.3) n1/2(n 01N - 1) —11-7 N(0,7 /a‘1 )

or equivalently
(3.2.4) N1/2— 19/211... N(0, 7*‘1/(4o ))
1|:
where 7 is as in (1.1.12).
Proof of Theorem 3.2.1: By definition of Nd’ lim Nd = 00, a.s.. Furthermore
if 0<dl<d2, we have N dlsz2 a.s., that is, N d is nondecreasing as (1

decreases.
Note that it follows from the representation (1.1.8) that {in} is strongly

consistent estimator of 0 and we have seen in Chapter 2 that {0121} is a

22

strongly consistent estimator of 02. Since N loo, a.s.,

2N -1 0 a.s.,

and
(3.2.5) 01% —-+ 02 a.s..
From (3.1.1) and (3.1.2), and for d sufficiently small,

2 2 2 2 2 2
(3.2.6) zaa Sdnd<d +zaa
and

-1 “ — ‘2 -h

(3.2.7) b no 01% S N/n0 5 b no1 { aN_1 + (N—l) }

whence (3.2.6), (3.2.7) together with (3.2.5) yield
dzno -+ z: 0'2
and
(3.2.8) nolN -7 1 a.s..
To show (3.2.2), we need a rate on the tail behavior of the stOpping time Nd‘
Since the method of obtaining this rate is analogous to Lemma 2.3.1, we state

a similar lemma without proof.

Lemma 3.2.1: For each 0<£<l, and any r<oo,

(3.2.9) P [NSno(1—c)] = 0(d2(“1)/(1+h))

and

(32-10) 2): P[N>n] = c(d2("1))
n_n3

where 112 = n2d = int(no(l-c)) and 113 = n3d = int(no(1+c)).

Now by (3.2.8) and Lemma 3.2.1 and arguments similar to those used in the
proof of the Theorem 2.3.1, (3.2.2) obtains. From representation (1.1.8), 2
Theorem 1.2, Anscombe's Theorem and (1.1.9), it follows that,

(3.2.11) N1/2(1N — a) —11-7 N(0,o2)

from which we get

23

P[0elN] = P[sN— d s o s bN+ d]
(3.2.12) = P[N1/2|IN— 0| 5 d N1/2].
Since d N1/2__. oz a.s., (3.2.12), (3.2.11) and Slutsky's theorem establish
(3.2.1). a
Proof of Theorem 3.2.2: From (3.1.1), (3.1.2) and similar arguments as the

one used in the proof of the Theorem 2.3.3, show that the asymptotic

a,

distribution of n(1)/2(Nnol - 1) is the same as that of "(Ill 20113 - 02)/02, from
which (3.2.3) is immediate. Now (3.2.4) follows from (2.3.3) by square root
transformation, that is,

nit/21111211151” -1) 2» mar/(47:41).

and the converse is similar. This completes the proof of Theorem 3.2.2. o

Chapter 4
Sequential fixed width confidence inteval for a location parameter
under random censorship using M—estimation

4.1 Model

Let c and Y be independent random variables and
(4.1.1) X = A + c, A69
where 9 is an Open subwt of the real line R. with compact closure.

The following notations will be usw only in this chapter. Let
F(t) = P[c>t], C(t) = P[Y>t], FA: F(--A), HA: FAG, 12A: ip(--A),

a t
H A(t) = — j GdF A’ {Xiz 121} be iid rv's with the same distribution function
_ 00

as X, {Yiz i21} be iid rv's, independent of {Xiz 121}, with the same
distribution function as Y, and (l-F), (l—G) are continuous distribution
functions and all of the unSpecified integrals are on the whole real line.

When dealing with survival time data, one can take Xi's to be log10 or
In of the survival times. The problem considered in this chapter is the
sequential interval estimation of A using M-estimation based on {(Zi,0i): i21}
where 2i: min(Xi,Yi) and 6i: [XiSYi].

Let Fn be the P—L estimator based on {(Zi,6i): ISign}. An M—estimator
of A is defined as the solution in t of
(4.1.2) An(t) = [fix-t)d11‘n(x) = 0
for some given function 111. In the absence of censorship, if
I/J(X,A) = —%log f(x-t)|t___ A’ where f is the density of the measure induced by
X on R with respect to the Lebesgue measure, then the solution to (4.1.2), is
the Maximum Likelihood Estimate (MLE). Huber (1964) proposed
M—estimation as a generalization of (MLE), with desirable robustness

pr0perties. Two important examples which are mostly used for the problem of

24

25

locating the center of a symmetric distribution, say A, are the Huber

M—estimate by taking Huber 1]) function defined by

(4.1.3) 7p(x) = {(—TVx)AT}
where T is a positive constant and Tukey's biweight
(4.1.4) 200 = x(1—x2)2 [IxISII

in which case the defining equation becomes
I¢(x—t)dFA(x) = 0.

In practice it is usually necessary to estimate the scale parameter of the
underlying distribution, but this will not be considered in this thesis.
4.2 Assumptions and some preliminary mults:
Assumption (Al): F is symmetric about zero and F,G are continuous.
Assumption (A2): Let M and T be constants such that IAISM, for all A69,
G(T+M)>0 and F(T)>0. Let 12 be a monotone nondecreasing, continuous,
skew symmetric function and has two continuous bounded derivatives 11),, (0"
on (-T,T) and 2 is constant on {x: x2T}U{x: xs-T}.
Assumption (A3): 7 = Ilfl’dF at 0.
Assumption (A4): t = A, is an isolated root of the equation
(4.2.1) AF(t) = [fix—t)dFA(x) = 0.
Remark 4.2.1: For nondecreasing 1/1, An may be written as

An = 1/2 (sup{t: An(t)<0} + inf{t: ln(t)>0}).

The next lemma is similar to Lemma 7.2.A of Serfling (1980) which has
been considered for the case of no censoring. Now we are following the same
lines of proof to get similar results in the presence of censorship.

Lemma 4.2.1: Under Assumptions (A2) and (A4), a sequence of solutions
{An} to the equation (4.1.2) exists and converges to A, a.s..

Proof: Let c be a given positive real number and An as in Remark 4.2.1.
Then AF(A—c)<0<AF(A+c). We will show that An(t)—-i/\F(t), a.s., for each

26

t for which ltISM. Assumption (A2) and integration by parts yields
|l¢(x-t)an(X) - l¢(X*)dFA(X)| = |l(Fn(X) - FA(X)) dWit-0|-
Note that, for any c< THA,
supIFn(t) - F(t)|—-1 0, a.s..
tsc

Thus for all t such that |t|$M, we conclude that An(t) —-+ AF(t), a.s..
Therefore
P[A—cSAn$A+c; i.o.] = P[ln(A-t)go and An(A+c)20; i.o.] = 1

where i.o. stands for infinitly often. Since 6 is arbitrary,

 

An—i A a.s.. 1:1
Lemma 4.2.2: Under Assumption (A2) and for {Tn} a sequence of random
variables such that ITDISM, a.s., and Tn—i A, a.s., where M is as in
Assumption (A2),
(4.2.2) j¢(x-Tn)an(x)—1]¢(x—A)dFA(x), a.s..
Proof: Using the triangle inequatity, Taylor's expansion and integration by
parts we have
IMx-Tn)dl1“n(x) - [Ax-A)dFA(X)I

sliax—Tnldtfintx) - FA(x))l+|l{¢(x—A) — ux-TnlldFAml

= II(11,,(X) — FA(x))d¢(x-Tn)l + Ilt'rn-Alr’tx-inldFAcll
where Tn is between T11 and A and tb’(x-Tn) = 3'? ¢(x-t)|t___f;n. Since 11),
w’are bounded, Tn —+ A, a.s., and
supMIFn(x) - FA(x)|-i 0 a.s.,

xST-l-
the lemma is immediate. 0

Remark 4.2.3 The results of our previous chapters which are given for life

times (nonnegative random variables) with or without the restriction of T< 1',

27

can be extended to the case of any random variable with similar restrictions
on the distributions (l—F), (l-G) and the function #1.
Theorem 4.2.1: Suppose Assumptions (A1) through (A4) hold and let All he
a solution sequence of (4.1.2), then
1 ‘ D 2
n /2(An- A) ——+ N(0,aA)
where
is
(4.2.4) 0: = 03(F,G) = 7‘2“: 1132 dHA
and
00
t
Proof: Since 111 is differentiable, so is the function
An(t) = j¢(x—t)an = 2312(Zi—t)0idin

where din is the jump of the P—L estimator at 21‘ Therefore we have
.. .. .. . , is .
Mic-An)an(X) - Mx—A)dF,(x) = [(An- Aw (x-An)an(x)
is . , is
where lAn- A|5|An— Al and 711 (x—An) = a? ¢(x—t)|t=z . Since
11

An(An) = AF(A) = 0, integration by parts and some algebra yield
n1/21An- A) = in’1in1/2ltrn- FA)d¢A}[2rn#01
= gn'1n1/2Lnlgn#0]

where 7n = ;¢’(x—Zn)drn(x) and Ln = [(1311- F AWA—

Recall the representation (1.1.8) and note that under Assumptions (A2)
and (A3) and similar arguments as is used in the proof of Lemma 4.2.2 yield
13111—7 7, a.s., and [%n#0]—-i I, a.s.. Thus by Slutsky's Theorem (4.2.3) is
achieved. 0
Remark 4.2.2: Under the Assumption (A1), A is the median of (l-F).
Gardiner, Susarla and van Ryzin (1985a) estimated A by the sample median,

28
m = 13‘;1(1/2) = inf{t: sumo/2} which is shown to have the following
properties:
(i) For each positive constant 8,
(4.2.5) um — All, = 0(n‘1/2)
(ii) If F A has a density f A and is positive at A, then
(4 2 121/20}: - A) .13.. N(0 CA(A)/(4 172m)»

where C A(t) = It H A2dH A Also they have suggested the use of
cum/(413m)) as an estimator of CA(A)/(4fA(A)), where

C n(t) = I (Hn +1/n)-2dHn , and In is a suitable estimator of the density f A'

.mit
In this chapter our results do not require the existence of a density.
However we use m = F;1(1/2), as a preliminary estimate of A in the

estimation of the asymptotic variance 0:. Let

(4.2.7) 02(111): ( 17p (x—m)an (71))‘2 |A2dc

where

An(t) = {magmas-m).

Remark 4.2.3 In the absence of the censoring, Carroll (1978) has an almost
sure expansion for M—estimates. In his paper he states that for (4.2.7) to be
a consistent estimate of ”1231 we need F to be symmetric and lb skew
symmetric. Reid (1981) calculated the influence curve for Kaplan—Meier
M—estimate and has found the above asymptotic normality by the influence
function approach.
4.3 Sequential fixed width confidence interval for the location parameter A

In the rest of this chapter all the limits are considered as (1 tends to
zero. In view of (4. 2.,3) for a given positive real number (1 and ae(0,l/2)

we take I: (An -d, A n+d) with n = 11d defined by

29

(4..31) nd = inf{k21: k 2 b a: }
where b = d 2.2: Then we have
lim P[Aeln] = 1—20
and
lim {nb-lozz} =
Since F, G and A are unknown, the specification of the "Optimal" sample size
in (4.3.1) cannot be made. We are therefore led naturally to construct a
sequential procedure in which the sample size is a stopping time N = N (1
(4.3.2) Nd = inf{k2nl: k 2 b(.}2(m)+lr'11)}
where 111 is as in (3.1.2) and 02(111) is given by (4.2.7).
Before we present the prOperties of the above sequential procedure and
the st0pping time, we provide some preliminary results.
The following lemma is stated in Sriram (1987).
Lemma 4.3.1 (Lemma 1 of Sriram (1987)): Let Un, Vn be any sequence of
random variables and a, b at 0 and s>0 be real numbers. If
P[IUn-a|2£] = O(n-s) = P[an-bIZc], for every c>0,

then
P[|Un/Vn— a/blZc] = 0(n‘1), for every (>0.
Now we shall show that for each positive c and some r>l,
(4.3.3) P[|a§(m) - 7A| 2 c] = 0(n’1).
Note that by Lemma 4.3.1 and the Assumption (A3), it suffices to show that
(434) Pllw’ (x—m)dF,, (x) - 7| 2 e1= 0(n )
and
2 —2 —
(4.3.5) P[| IA nan- [A AHA dHA| 2 c ]= 0(n 1')

First consider (4. 3. 4). By Taylor's expansion
Ill) (x-m)an (X)= I 1/1 (x-A)<1Fn (X) + (In-MN (x-An )an (X)

30

where An is between A and m. Now by the above expansion, Assumption
(A2) and integration by parts we have

s| /¢’(x—m)dfrn(x) — 7|215 const.{s|m-A|21+ Isle“. FA|21d¢’(x—A)}.
Now by (4.2.5), the representation (1.1.8), Lemma 1.1 and Marcinkiewicz—
Zygmund inequality we have

EII¢'(x—r§1)dfin(x) - 7121 = 0(n‘1)

which, by Markov's inequality, implies (4.3.4). A similar calculation to that
done in the proof of Theorem 1.4, which is shown in Appendix A, and
consideration of the extension explained in Remark 4.2.3, leads to (4.3.5).
This completes the proof of (4.3.3).

Recall that nd, the "optimal" sample size, is asymptotically equivalent to
no = no d = bag. In the following, whenever there is no possibility of
confusion, the subscript d of N (1’ nd and nod will be dropped. Now we state
the main results of this chapter.

Theorem 4.3.1 Under Assumptions (A1) through (A4) and for h>0, (N,IN) is
both consistent and efficient, that is, for each A69 and PEP,

(4.3.6) lim P[AEIN] = 1—2a
and
(4.3.7) lim E|N n51 - 1| = 0

Proof: Note that by definition of Nd’ Nd -—-1 co, a.s.. Furthermore it can be
shown that N d is nondecreasing as (1 decreases (see Chapter 2 or 3 for similar
arguments). Therefore by Lemma 4.2.1,
AN -+ A, a.s.,

and (4.3.3) together with Borel—Cantelli lemma yield

.312, (m) ——7 oz, a.s.. .
From the definition of Nd and 11d and arguments similar to those in Chapter
3 we obtain, for d sufficiently small,

31

(4.3.8) 1161 N -» 1 a.s.,

and

(4.3.9) d2 no -» 221 7:

Note that, as in Chapter 3, for showing (4.3.6) we need to show that
(4.3.10) N1/2(AN- A) ll. Mom/2;).

Recall that

1 2 ‘ - 1 2
N / (AN- A) = 7N 1N / LN [71.1110]
and
_ -1
where J A and 1* NA are the same as J and r; ,given by (1.1.4) and Theorem
1.1, respectively when F is replaced by F A' Now write
1 2 “ -l — * —l 1 2*
N/(AN-A) = {(7N -71)N1/21NA+7N/NA
- 7‘1N‘1/2TN -(1‘1N‘1- 7'1) N‘1/2 TNAlliNtol

where

TN, A = N'1EJ A(zi,5i).
Therefore by (1.1.8), Anscombe and Slutsky's Theorems, (4.3.10) is immediate
and so is (4.3.6). To show (4.3.7) we need similar rates as the one given in
Lemma 3.2.1. Note that all we need to get such rates is (4.3.3). Hence,
following the same lines of proof of Lemma 3.2.1 and Theorem 3.2.1, we
obtain (4.3.7) which completes the proof of Theorem 4.3.1. 1:
Remark 4.3.1 The asymptotic normality of the st0pping time Nd’ can be
obtained by arguments similar to those given earlier in Chapters 2 & 3 but
obtaining the exact form of the asymptotic variance is animmersely tedious

calculation. Thus one can establish

n1/2(Ndn1 no - 1) —1-1—» No.11)

32

and
1 D
Nd/Z- n5” -—-1 N(0, /4)
but the exact computation of fl is difficult.

APPENDICES
Appendix A

The almost sure representation and the asymptotic distribution

oftheestimatoroftheasymptoticvarianoeoftheestimator

of functionals of the form 0(F)=l¢dF and KF)=IFd¢, of survival

curve F under random censorship

We shall present here the proof of Theorem 1.3. To the best of our
knowledge this is a new result. In the sequel $9111 and all the limits are
considered as 11 tends to infinity. Note that by Remark 2.3.1, it suffices to
consider (1; as in Assumption 1.1. Recall that

t . .
131(1) = 1 Mad, t 2 o, 1,) =1,2,3,4,
o

t z
K(t) = 1 + 2 [Zi>t], C(t) = 111‘de, KT,
0

A(t) = I Fdzp, An(t) TI gnaw, ost<oo

too 13,00

and
$2 — [A2n2K-2dfi a2 — [AZH'2dfi
n — n n’ — '
Proof of Theorem 1.3:
. . . __ 2 2 —2 _ 2 -2 .
To s1mp11fy notatlon let ‘pn - n An K and (p — A H . We wr1te
1 ‘ 2 2 2
n ”(0,2, — a) = Ill/2(lspndHn -17:dH )

= nl/ZUUpn- ¢)d(§n- :1) + mp,- (0)7111} + I “10:71- 131)}

(A.1) = nl/2( Dl+ D2+ D3 ), say.
Note that
1 2 1 1 2
901,- so = (2,] - so /2)2+ 2 vl/2(¢,1,/2- cp / )

33

34

and
”2 (pl/2 = nK-1(An— A) + A(nK—l— H_l)

son -
H‘1(An— A) + A(nK-l- 11'1) + (An— A)(nK‘1— H‘1)

= Bn + Rn1
where 811: H_1(An- A) + A(nK-l- H-1) and R111: (An— A)(nK-1- H—l).

Therefore
2 -l
(pn- (p = (Bn+ RnJ) + 2AH (Bn+ RIM)“

We simplify to get
.2 —1
(A2) 9011‘ (p = 2AH {(An- A) — AH (Hn— H)} + Rn,2

where
2 —l 2 -3 2 -1 -1 1
R112 (Bn+ Rm) + 2AH Rn,1+ 2A H (Hn— H) - 2A H (H nK )
Therefore it follows from (A.l) and (A.2) that
_ —1 2
D2 — 2/(An— A)dl‘ll - 21H (Hn— H)dI‘21+ [Rmde
Note that we can rewrite U1 and U2 as
a) A
U1: 1{[ (Fn— F)d1/)}dI‘11
and
N
= n'121A2H‘3{[zi>.] — H} dH
n-12[H_l{[Zi>o] — H} dI‘21.

Recall the representation (1.1.5). We rewrite U1 as
U1 = ”3 + R11,4

where
U,= “'12n(”4(zi,6,;s)F(s)d¢(s)}dr11

35

and

co
R11,4" “I r111111221131

and rll is as in (1.1.5). Hence

Ill/2G: - a2) = 2nl/2U3 - 2n1/2U2
(A.3) + n-1/22{A2(Zi)H-2(Zi)6i- 772} + n1/2Rn 5
where Rn,5= D1 + R?’3 + 2 Rn,4° Ajmr some alg/ebra on (A.3) we get

1 2 ‘2 2 _ -1 2 l 2

n (an—o)—n 2Vi+n Rn,5
where

,1 '2 Wi,2’

w,l =111°°4(z,,6,;s)F(s)d“snarl1

and

2 -—2 -1 2
Under assumptions of Theorem 1.3, {Viz i_>_l} is a sequence of bounded iid rv's
with mean

EVl = EW

=EW =0.

1,1 1,2
To obtain the almost sure representation of 02 we need to show that
nl/ 2R115 -+ 0 a.s.,
*
in addition if we show that the variance of V1 is 7 , then (1.1.11) will be
achieved by the central limit theorem. First we compute the variance of V1.
Let (Z,6), V, W1 and W2 be copies of (Z1,61), V1, W1,l and W1,2
respectively, then

(A.4) Var v = E(2w1+ w )2 = 4wa + 153w2 + 4EW1W

2 2 2’
In the following we use repeatedly the Fubini Theorem, integrations by

parts and the identities

36

:3 t m is
(IgdH)2 = mung) gamma),

Es(Z) = Isd(-H)
and
fl
E{s(Z)5} = IsdH-
Now we are ready to compute Ewg. Since W2 has mean zero,
E{A2(Zi)H_2(Zi)6i- 21H'1[zi>-]dr21} = — 02.
Therefore
Ewg = E{A4(Z)H‘4(Z)6} + 4E(1H“1[Z>-]dr21)2
— 4E{A2(Z)H‘2(Z)61H‘1[Z>-]dr21} - 04.
Note that

E(fH-1[Z>-]dl‘21)2 = 2E{(f((f)8H_1[Z>o]dI‘21)H—l(s)[Z>s]dI‘21(s)}
= 21((/)°11‘1111~21)¢1r21

= 21H—1([de‘2l)dI‘21

and

_ m
E{A2(Z)H 2(2)5111 1[Z>-]dI‘21} = [H 1([ dI‘21)dI‘21.
We simplify to obtain

2 —2 —1 ‘2 4
(A.5) 13w2 = [H dI‘41 + 41H (1 dI‘21)dI‘21 - a .

To simplify notation, throughout £(Z,&,t) will be abbreviated to ((t).
Since E((s)§(v) = C(sAv) for s,v<1',

w? = Hummus) WNW?

no t on
= 2E{I{({ 51342,!) (111 {Pdcdrnondrncn

on t on
= 21{{ Fang) {‘Il C(sAv)F(v)dw(v)}dr,,(u)}d¢(s)}druc).

37

Now consider the most inner integral on two sets [85v], [s>v] and note that

on the set [35v], uStSsSv. Hence

swf= 2/{{°°Acrd7p}rn(t)drn(t)

no t 8
+ 21% F(SHé {I CFdi/Jldl‘11(U)}d¢(S)}dF11(0-
u
Integration by parts on the inner integral of the second term on the R.H.S. of
the last equation yields

2 _ 2° 2
Now integration by parts on the middle term of the R.H.S. of the last
equation gives
2 _ 2
(A.6) EW1 — [(2I‘11 + 1‘22)dI‘21.
Now we consider EW1W2.
Q
EW1W2= E{A2(Z)H 2(2):)”; C(ZAs)F(s)dw(s)}dI‘11}

— E{A2(Z)H-3(Z)5f{£mFd¢}dF11}
— 2E{][mC(ZAs)F(s)d1/J(s)d1‘1llH-l[Z>'ldrzl}
+ 2E{6H’1(Z){j{émF(8)d1/J(8)}d1‘ll}IH-1[Z>'1dr21}

(A°7) = E{Q1- Q2“ 2Q3"’ 2Q417 333'-
To compute EQI, consider Ql on the two sets [Z53] and [Z>s]. Therefore, by

Fubini Theorem
EQl = I{[mE(C(Z)A2(Z)H’2(Z)61258])F(s)d¢(s)}dl“11
+ [{[wmA2(Z)H-2(Z)5[Z>8]}C(s)F(s)d(b(s)}dI‘11
= (1/2)1(1°°r22Fd7/7)drn+ lumccxfdrm)F(s)d4<s)}dr11.

Similar arguments yield

38

m, = 1{1”E(A2(Z)Ir2(2)tzsslnitswsndr,1
= [{[m((I)SH”ldP21)F(8)di/J(8)}d1‘11-
Similar arguments as is used in the computation of so, yield
m, = E(ll"C(Z)lzsle(s)d7/4s)dr,,)(lH‘1lz>-ldr2,)
+ E (l[mC(s)[Z>le(8)d¢(8)dI‘11)(IH'1[Z>~ld1‘21)

= I[m{(l)sH-l(u){?l Cd(-H)}dI‘21(u)}F(s)d1()(s)dI‘11

co co _
+ I! C(s){ (1) H l(u)H(sVu)dI‘21(u)}F(s)d¢(s) or“.
We consider the last term in the last equation on two sets [sSu] and [s>u]

and simplify to obtain

EQ3= I{Im{ésH-l(u){C(u)H(u)—C(s)H(s)+1118HdC}dI‘21(u)}F(s)d¢(s) }dI‘11
+ [[mC(s)H(s){(])8H-ldl‘2l}F(s)d(b(s)dI‘ll

+ ltl°°0(s){£°°dr2,lF<s>d¢<s>ldrN
Finally, similar but simpler arguments yield
so4= E(I{[mflZSSIH—1(Z)F(S)d¢(8)}dI‘11)(IH_1[Z>°ldF21)
= 1115311“(nillllsn‘1dfildr2linsidwtsndrl1.
By substituting EQl through EQ4 in (A.7) we obtain

nwlwz = —/[°°((1)80d1‘21}r~(s)d¢(s)drll
(A.8) — [[mC(s){£de‘21}F(s)dt/1(s)dI‘11

co 8 _
— 1) {(1)11 1dr21}r(s)d¢(s)drn.
Now substitution of(A.5), (A.6) and (A.8) in (AA) and some algebra yield
_ 2 -l -2 4
EV _ tijrndr21 — 41H I‘lldl‘31 + [H dI‘41 — o .

39

We need the following lemma to obtain the almost sure representation of
n1/2(o12l - 02).
Lemma A.1: Under Assumption 1.1, for continuous F, G, T<r and a<1/2,

(i) nallHn - H|| —» o a.s.,
S Q
(ii) n"‘||Hn - H|| -—7 o a.s.,
(iii) n"||nK'1 - “lug—7 o a.s.,
(iv) nalan- F": ---7 o a.s.,
(v) nallAn— All}; —» o as.
where H ~ ll denotes the sup norm and II - ll: means the sup is taken over

the closed interval [0,T].

Proof: (i) and (ii) follow from Glivenko—Cantelli Theorem. Let (>0. Since
H is monotone nonincreasing and H(T)>0, (iii) also follow from
Glivenko—Cantelli Theorem. (iv) follows from Theorem 2 of Shorack and
Wellner (1986) page 308. To show (v) note that

°° ‘ ‘ T
nallAn- An}; = "n“! (Fn- mot/7“}; 5 oonst.||n“(Fn- F)||0.
Hence (v) is implied by (iv). This completes the proof of the lemma. n
Recall that Rn,5 = Dl+ Rn,3+ 2Rn,4° We shall show that

(a) nl/ZDl —-+ 0, a.s.,
(b) nl/ 2R113 —-o 0, a.s.,
(c) n1/2Rn,4 --7 0, a.s..

Now recall the representation (A2) of (on— (p). To show (a) holds, we
shall show that each term has such a pr0perty. Consider the first term, we

want to show that

N %
n1/ 212AH—2(An— A)d(Hn - H) -7 o, a.s..
This follows from Assumption 1.1, Lemma A.1 (ii) and (v) and similar

arguments as are used in the proof of Lemma 2 of L0 and Singh (1986). All

40

other terms can be handled similarly. By similar arguments, it can be shown

that (b) holds. To show (c) holds note that 11114"! { [ran¢} dI‘1 11, where
rn is as in (1.1.5). Therefore it easily follows, from Lemma 1.1, that
nl/anA—o 0, a.s..
Thus we have the almost sure representation of :7: and by the CLT (1.1.11)
follows. :1
Appendix B
Rate of convergence of the mtimator of the
asymptotic variance, 02: [A2H—2dfi

In this section we shall present the proof of Theorem 1.4. We shall
show that for each e>0, and I'<ao, (1.1.13) holds. Note that it suffices to
prove the theorem for r>1.

The following arguments are very similar to that of Gardiner and Susarla
(1983) Appendix A, except we are giving a shorter proof using the
representation (1.1.5) and Lemma 1.1.

Proof of Theorem 1.4: Let us write
7}:- a2: jnK‘2(A§— A2)dfin+ [A2(n2K‘2- H-2)dfin + [AZH’2d(Hn- H)
= Tn,l + Tn,2 + Tn,3’ say.
First we examine Tn,2' Since K = l + an,
H2- n-2K2 = H2- (Hn-t- n-1)2
= -n‘2- 2n’1H - (Hn- H)2— 2n‘1(Hn- H) — 2H(Hn— H)
so that on substitution we have
5
2 l|T

(13.1) |Tn,2|<

n ,2j|
To handle the terms in (8.1) we shall use the following result for Binomial

moments.

41

Lemma B.l: Let U be a Binomial random variable with parameters (n,p).
Then for any It 2 l

E(l + U)_k _<_ k! (np)- 1‘.
Proof: See Koul, Susarla and van Ryzin (1981), Moment Lemma, Page 1283.
0

Recall that

2 2 -2 —2 —2 -1 2 —1 2
Tn,2=[A (n K H ){-n -2n H—(Hn-H) -2n (Hn-H)-2H(Hn—H)}dH.
We want to compute the rth moment of TH 2, for r>1. Consider the TH 21
term. Note that
_ -1 2 -2 -2
Tn,2l — EdiA (Zi)H (Zi)K (Zi)'
Therefore

E| g n‘12: E{5iA21(zi)H‘21(zi)K‘21(zi)}.

Tn,2lIr
In the following El stands for a conditional expectation given (Z,6), and
all ci's are constants may depend only on r. Note that given (Zj,6j),
(K(Zj)-1) is the sum of (n—l) Bernoulli random variables with probability of
success of H(Zj). Thus it follows, from Lemma B.1, that
EIT I2 s cln‘1{E{6A2(Z)H’2’(Z)El(K‘2’(Z))ll

5 c2E{6A21(Z)H-2r(z){nH(Z)}—2r}

n,21

is
= c2n'211A2’H‘4’dH.
Under Assumption 1.1 and for F, GEF, it follows that the last integral is

finite. Hence

(3.2) ElTnmlr = 0(n'21).
Exactly in similar manner we can bound E|Tn 22| and show that
(B.3) E|Tn’22|r = 0(n'1).
Recall that
—1 2 —2 2 -2 2

42

Thus

s. .<. c3E{(n2K'2A2H‘2(Hn- H)2)’(Z)-6}
= c3E(n216A2’H‘21)E1{(Hn— H)21K’2’}.
By the Holder inequality, for p>1 and q = (1 - 1/p)—l,

/ /
1 1

Hence by Lemma 3.1 and an application of the Marcinkiewicz—Zygmund

r
Tn,23 '

- lp _ l q
E1{(H,,- Hl2’K 2’} s E (K 2% (IHN- leq’).

inequality yields

N
ElTn 23(1’ 5 c4n2’jii‘2‘A2’(nH)'2’(n'q’H)1/qu
which implies
(13.4) ElTn,23|r = 0(n").
Similarly for
1111,24 = —22(K‘2A2H‘2(Hn- H))(Zi)-6i,
and
—l — 2 -l
T1135 = —(2n )E(n2K 2A H (Hn- H))(Zi)-6i,
it can be shown that
2
(3.5) ElTn,24|r = 0(n‘31/ )
and
(B6) Errngs)?‘ = 0(n“).
From (B.2) through (B.6) we conclude that
(13.7) Err”)2r = 0(n").

8
Now we consider the term TIll = jnzK—2(AI21- A2)dHn. Note that by
Cauchy—Schwarz inequality we have
2 2 _ 2
An— A — (An- A) + 2A(An- A)

= (firing- F)F1/2d¢v)2 + when. F)d1/))

5 (lwr’zd‘n- F)2Fd7p)([°°rd¢) + 2A([°°(frn- F)d¢).

43

Therefore

2 —2 °° —2 ‘ 2 2 2 —2 ‘2 “ 2
|Tn,1|$|]n AK ([ F (Fn- F) th/J)dHn|+2|]n AK ([ (Fn— F)d1/))dHnI.
Integration by parts on last two integrals yields

'2 -22 -2‘ 2 '2 -22 ‘
lTn,1lSc5”({)n AK dHn)F (Fn— F) Fd¢|+2c6| [(6 11 AK dHn)(Fn— F)do|

= c5'Tn,11l + 2‘16'Tn,12'1 223"
Since Tn,11 and Tn,12 are very similar, we just show that
E(TWP’: 0(n").
Note that
(rm) 5 E(“){|F'1(Fn— F)|{)'n2AK'2d§n}
where E00 is the integral on [0,oo) with respect to PM).
Then by the Holder inequality, for r>1,

E(“){ | F‘1(Fn— F) | é'nzax‘zdfin}

IA

(E("lur‘1c2n- F): 5'n2AK'2dfinlrf/21E2h1)l1‘1/ 2

IA

. . fl
c7E(n){ |F"1(Fn— F) (1] n2’A1K‘21dHn}.
0
Hence
2 -1 “ 2 ' 4 2 -4 2
ElTnjzl 5 c7E$“){|F (Fn- F)| 15o 1A 1K rdHn}
where E1“) = E G E(n). Let p_l+ q_1 = 1 and p>l. By the Holder
inequality
. . fl
0
n) -l‘_ 2rpl/p n) '4r 2r -4r‘3 ql/q
5 {ES IF (Fn FM 1 {El {5n A K dHnl} .
We shall show that the second term in the R.H.S. of the last inequality is

finite and the first term is of order 0(n-r), r>1. Recall the representation

44

. is
(1.1.5) and Lemma 1.1. First we show that E91)” n4rA2rK—4rdHn}q<oo.
0
. at . is
E (n) { I n4rA2rK--4rdHn}q S c813(In) U n4qrA2qu—4qrdHn}
0 0
. is
5 c9E{] n4qrA2qu—4qrdHn}A(0)
0

= c9n4rqE{n-l26iA2rq(Zi)K—4rq(zi)}

= c9n41‘13{6A21‘1(Z)31K“11‘1(2)}
N
Now by the Marcinkiewicz—Zygmund inequality and (1.1.7), it follows that
3£n)|F‘1(Fn— F)|2rp = 0(n‘P’).

Since p>1,
2
ElTn’bl = 0(n_r).
Similarly
2
EITnjl' = 0(n-r).
Therefore
(3.3) Elrnfll) = om").

Finally, note that Tn,3 = [A2H-2d(fi - H) is an average of n iid mean
zero bounded random variables. Thus an application of the Marcinkiewicz—
Zygmund inequality yields

(3.9) ElTngl = 0(n‘1), r>1.

Therefore (B.7), (B.8), (3.9) and Markov's inequality imply (1.1.13). 0

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