Weak Convergence of Progressively Censored
Likelihood Ratio Processes
by
Joseph C. Gardiner
*This research was partially supported by the National Science
Foundation under Contract MCS-78-01240.
JOSEPH CHITRANJAN GARDINER.
Weak Convergence of Progressively Censored
Likelihood Ratio Processes.
(Under the direction of PRANAB KUMAR SEN
and GORDON SHUNS.)
There are certain statistical experiments, notably clinical trials
and life tests, in which the observations are gathered sequentially in
time.
In these circumstances it is usually undesirable, from cost and
efficiency considerations, to prolong experimentation until the entire
sample of specimens has responded and therefore, in practice, sampling
is carried out under truncation or censored plans or tmder more involved
progressively censored schemes.
The asymptotic normality of a class of time-sequential statistics
arising in progressively censored schemes is obtained.
This also pro-
vides for the asymptotic normality of certain stopping variables.
Applica-
tions of these results to statistics arising in life testing are given.
Let Xn, l""'Xn, n be the order statistics corresponding to a random
sample Xl""'~ of independent variables with continuous probability
density function fe(x), XEIR, the real line and eE05::,IR. Let p(x k;e)
n,
denote the (j oint) probability density function of the vector X k:=
"1l ,
(Xn, l""'Xn, k)'
l~k:m,
and define for each k,
-k
p(~,k;eO+un 2)/P(~,k;80)
11._
fi,
k(u) :=
-k
where 80 , 8 0 +un 2E8. If {Tn; n~l} is a
sequence of stopping variables adapted to {o(X k): l~k~n}, then the
"1l,
weak convergence of the progressively censored likelihood ratio process
(u) in C(IR) is investigated. Further if {~(t): tE[O,lJ} are
,Tn
certain integer-valued nondecreasing right continuous functions the
u
+
II.n
asymptotic behavior of the functions (u,t)
+ ~,kn(t)(u)
is studied.
These results are applied to the derivation of the asymptotic distribu-
tion of the likelihood function when the observations are neither independent nor identically distributed.
classes of statistics {Tn, k
=
As an application, for certain
Tn,~,
k(X k):
l~k~n},
a lower bound is
furnished for the asymptotic variance of the variable Tn,T
n
AC KNOWL EDGEI"lENT S
This dissertation has been made possible through the assistance
and encouragement given to me by my advisors Professors Pranab K. Sen
and Gordon Simons.
Professor Sen suggested the central thesis of my
research and his counsel has been
instrum~ntal
Ll the rapid completion
of this work.
I
~lsh
to thank the Departrnent of Statistics of the University of
North Carolina at Chapel Hill for financial support throughout my
graduate program, and the faculty of the department of extending to me
their time and interest, and in particular the members of my examination
committee, Professors Wassily Hoeffding, Joseph eima and Richard
Shachtman.
My thanks to Ms. June :Maxwell for her enthusiasm and excellent
typing of the manuscript.
It is a pleasure to acknowledge the support of my colleagues and
friends in Chapel Hill.
I
benefited greatly from long late-night conver-
sations with NiT. Mohammed Habib and shall always remember Ms. Donna
Lucas for her cheerfulness and kindly advice on several matters!
Finally, I am tha.ilkful to my friend Janice, my parents and my
sister, Marie.
Their faith and confidence in me has been a constant
source of inspiration.
To them this dissertation is gratefully dedicated.
ii
TABLE OF CONTENTS
CHAPTER I:
INTRODUCTION
1.1 Progressively Censored Schemes
1.2 Outline of the Basic Mathematical Model
1.3 Outline of Proposed Research
CHAPTER II:
2.1
2.2
2.3
2.4
2.5
2.6
1
4
6
ASYMPTOTIC NORMALITY OF SOME TIME-SEQUENTIAL STATISTICS
Introduction
Formulation of the Problem
Assumptions
The Main Theorem
Limiting Distributions of Some Stopping Times
Further Results
13
15
17
20
34
42
CHAPTER I II: ASYMPTOTIC BEHAVIOR OF SOME STOCHASTIC PROCESSES
ARISING IN PROGRESSIVE CENSORING I
3.1
3.2
3.3
3.4
Introducti on
Notation, Basic Assumptions and Definition of Processes
Preliminary Lemmata
Weak Convergence of W
n
64
70
75
3.5
Limiting Distribution of
77
j,
Tn
11..
rl,
k (t)(U)
n
62
CHAPTER IV: ASYMPTOTIC BEHAVIOR OF SOME STOCHASTIC PROCESSES
ARISING IN PROGRESSIVE CENSORiNG II
4.1 Introduction
4.2 On the Assumptions Made in Chapter III
4.3 An Example
4.4 Asymptotic Distribution of the Likelihood Ratio Function when
the Observations are Given in the Series Scheme
CHAPTER V:
5.1
98
98
105
110
SOME APPLICATIONS AND SUGGESTIONS FOR FUTURE RESEARCH
Introducti on
115
iii
5.2
Lower Bound for the Variance of Tn
,Tn
116
5.3 Lower Bound for the Asymptotic Variance of Tn,T
n
5.4 Concluding Remarks
119
APPENDIX
129
BIBLIOGRAPHY
145
iv
127
CHAPTER I
I NTRODUCTI ON
1.1.
Progressively censored schemes.
There are certain statistical experlinents, notably those involving
clinical trials and life testing, in which the observations are gathered
sequentially in time and consequently collected in the order of increasing size.
Thus the first recorded observation is the smallest one,
followed by the next smallest, which is the second observation, and so on,
until the largest observation which emerges last.
From a practical stand-
point, experiments of this nature could be undesirable from considerations
of time and cost so as to preclude any general usefulness or wide applicability.
In these circumstances therefore, it is common practice to carry
out sampling under one of two schemes in which experimentation is curtailed
before the responses for the entire group of individuals have been obtained.
For example under the truncation procedure sampling is carried out for a
preassigned length of time and the observations recorded during this period
form the basis for statistical investigation.
Alternatively, in the
censoring plan experimentation is terminated once a prespecified proportion
of subjects from the target
sarr~le
have responded.
inference is based on the recorded observations.
Once again statistical
We note that in the trun-
cation scheme the actual number of recorded observations is random (while
the time of termination of experimentation is preassigned) whereas, in
censoring, the time of tennination is random and the number of recorded
responses preassigned in advance of the commencement of the experiment.
2
In the literature, truncation and censoring are sometimes referred to as
Type I and Type II censoring respectively.
The truncation and censoring plans themselves have several drawbacks
stemming from cost and efficiency considerations.
For instance, single
point truncation and censoring schemes are often inadequate for most practical problems where ethical reasons may demand high levels of efficiency
of statistical procedures based on them.
Second, the trw1cation scheme may
still necessitate prolonged experimentation to obviate the risk of erroneous decisions and thus has to be weighted against the increased cost and
sacrifice of experimental units which may not contribute significantly to
the sensitivity of the experiment.
Third, the time of tennination of ex-
perimentation in the censoring scheme being random may be at variance with
other restrictions on time and cost.
For these reasons progressiveZy censored schemes (peS) have been advocated in clinical trials and life testing.
Here the experiment is moni-
tored from the beginning and the results gradually accumulated, so that at
any stage of the experiment, if the current evidence warrants a clear
statistical decision, experimentation is terminated at that stage.
Accord-
ingly, pes can lead to a considerably shorter duration of experimentation
with the desirable concomitant reduction in cost and in lives of experimental units.
Progressively censored schemes, by their constitution, are sequential
in nature but differ from classical sequential designs in that, in pes
there is apriori a maximum number of observations that can be made in the
experiment, and more importantly, pes, by its very design, deals with
ordered observations and consequently nonindependent random variables,
3
whereas in most classical sequential schemes the observations are independent variables.
These differences introduce additional complications
and subtleties in the analysis of progressively censored schemes.
There is an
unfortunate~confusion
in terminology in the literature.
Armitage (1957) applies the term restricted sequential procedures to
sampling schemes that place a restriction on the maximum number of observations to be made in an experiment.
In life-testing and reliability the
term progressive censoring is sometimes applied to the data obtained when
the experimental units under consideration enter the investigation at
different points in time, consistent with the basic sampling design laid
out at the corrnnencement of the experiment.
For example, in the study of
a serious disease condition, such as carcinoma of the lung, it is seldom
possible to admit all the patients to the targeted study sample at the same
time because of the paucity of patients.
the patients as they enter for treatment.
It is then necessary to accept
Suppose now the study lasts only
for a limited period T and that there are on study n patients who arrived
independently at times z1"'" zn'
Let Ti be the maximum time for which the
i-th patient can be observed, i = l, ... ,n.
Hence Ti = T-z i and the only
information available is the set of values Tl~ ... ,Tn and the survival time
t i of the i-th patient provided t i s; T . Hence the tenn "progressively
i
censored" data is sometimes applied here. (Gross and Clark (1975)). We
advocate the use of the terms staggered entry or multiple point entry to
describe the situation here as opposed to single point entry plans in
which the individuals enter the experiment at the same point in time.
"Progressive censoring" has also been used to describe samples that
arise when at various stages of an experiment some, though not all, of the
surviving sample specimens are withdrawn from further observation.
The
4
sample units remaining after each stage of "censoring" are continued under
observation until ultimate failure or until a subsequent stage at which
another withdrawal of specimens is carried out.
Thus for sufficiently large
samples, "censoring" may be iiprogressive" through several stages.
not have occasion to deal with these situations any further.
ology agrees with Chatterjee and Sen (1973).
We shall
Our tenl1in-
We therefore envisage a gen-
eral environment in which the experimental units enter the investigation at
the same point in time and hence the recorded responses have a natural
ordering.
The term progressive censoring then describes the mechanism
through which the statistical evidence is continuously updated at each
stage of the investigation with a view to an early termination of experimentation whenever feasible.
1.2.
Outline of the basic mathematical model.
Let us consider the general situation in which n
units are under investigation.
2
1 experimental
IVe are interested in some particular char-
acteristic of these units and suppose that it is given in terms of measurements on a random variable X.
For example; if n
=
50 laboratory mice are
treated to a daily diet containing a suspected carcinogen, then X may
represent the length of time elapsed before the first appearance of a
Let Xl"" ,Xn denote the individual measurements on X
corresponding to a sample of n units. In the typical situation under study
malignant tumor.
these X's are not observable; the observable variables are the order statistics Xn, l""'Xn,n of the s~TIple. Now suppose that X has a probability
distribution on the real line with distribution fllilction Fe and probability
density function f e involving a parameter e, which we assume to be real.
e
5
For simplicity assume that Fe is continuous so that ties among the observations can be neglected in probability.
We may be interested in devising
an appropriate statistical procedure for a specified test of hypothesis
on e or in providing suitable estimators of e.
It is only reasonable that
our statistics be functions of the observab1es Xn, l' ... 'Xn, n. However, as
remarked earlier, it is often infeasible to prolong experimentation. until
the entire sample has responded and therefore our statistics must effectively involve only those observations gathered up to the point of termination of the experiment.
If sampling is performed under truncation for a
duration T, O<T<oo, and r*,
O~r*~n,
units respond, then we work with the
likelihood
r*
n-r* } ,
{n!/(n-r*)!}{ IT fe(X .)}{1 - Fe(T)
i=l
n,1
(1.2.1)
if Xn, r * ~ T < Xn, r *+1. In the censoring plan if the experiment is terminated after the r-th response, l~r~n, then inference is based on
Xn, l' ... 'Xn, r and the corresponding likelihood is
r
{nl/(n-r)l}{ IT fe(X .)}{1 - Fe(X )}n-r.
i=l
n~1
n,r
(1.2.2)
Note that in the first case T is predetermined and both the stopping
number r* and the observable X * (if r* > 0) are random variables. In
n,r
the latter case however, r is fixed in advance and only the stopping time
Xn, r is random. On the other hand, in progressively censored schemes
neither the stopping number nor stopping time are prespecified. Therefore
we are led naturally to consider a general class of variables {Tn;
such that for each n
~
1, Tn assumes values in the set {l, ... ,n}.
n~l}
In
practice Tn will be specified explicitly in terms of the observable variables and so typically Tn is adapted to the a-fields {a(Xn, l'···'Xn, k);
6
l~k~n}.
For example, Tn may be of the general form
Tn = min{k: Tn, k(X-u, l""'Xn, k)ECn, k}
where, for each k,
(1.2.3)
l~k~n,
Cn ,k is a certain admissible set and Tn, k a
statistic depending on the observables Xn, -l""'Xn, k only. We are there-
fore led directly to the study of Hstoppedl i statistics of the type
and of suitable classes of stopping variables {Tn;
n~l}.
T
n,\!
The statistics
described here are far too general to be anlenable to mathematical or
statistical analysis and moreover, in this generality, the problems are
neither well delineated.
In the following pages we shall provide a brief
outline of the results that have been obtained when an important simplification is introduced into the definition of these statistics.
Our research
focusses attention mainly on their asymptotic behavior.
1.3.
Outline of proposed research.
The class of statistics which are linear combinations of functions
of order statistics has been extensively investigated in recent years.
The
wide interest which these statistics have produced originates from practical considerations in which the search for suitable estimators of scale
and location parameters and for test statistics for these parameters has
led to statistics of the general form
T k
n,
=
n
-1 k
I c .h(X .) , lsksn
i=l n,l n,l
(1.3.1)
where cn, l""'cn,n are knO\~ constants called scores and h a known
fmction; Xn, l""'Xn,n are the ordered variables corresponding to a random
sample of n observations.
These statistics apart from their simple form
are often generally easy to calculate and exhibit desirable robustness
qualities.
7
The first researches in this area have been predominantly concerned
with obtaining sets of conditions that would ensure the asymptotic normality of the statistics Tn,n ,viz.
1
L{n~(Tn
,n -~)}
+
where ~ is the asymptotic mean and
2
N(O,o )
0
(1.3.2)
2 the asymptotic variance.
three different techniques of proof are available.
At least
Firstly that of
Chernoff, Gastwirth and Johns (1967) which employs a device of Renyi
(1953), secondly, the method of Stigler (1969) which is based on the projection technique introduced by Hajek (1968) and, thirdly, that of Shorack
(1969, 1972) based on the properties of a special empirical process.
(Pyke and Shorack (1968)).
We shall briefly comment on these methods later.
An excellent review is given in the paper of Stigler referred to here.
In
a recent article, Sen (1978) introduces a fourth technique of proof in that
his method exploits the "near" reverse-martingale characteristics of the
sequence {Tn, k; l~k$n} and directly leads to an invariance principle for the
class of statistics which are linear combinations of functions of order
statistics.
The discussion in Chapter II deals with the "stopped" statistics T
,
n,Tn
where {Tn,k} is given by (1.3.1) and {Tn} is the class of stopping variables
introduced earlier.
(1972).
Our technique is basically modelled after Shorack
In fact our treatment applies to a slightly wider class of statistics
as defined in (2.2.1).
The fundamental results of Chapter II are Theorem
2.4.1 and Corollary 2.4.1, where we establish the convergence to a limiting
normal distribution of the Tn,T under a set of regularity conditions on the
n
underlying distribution and scores, and under a restriction on the growth
of Tn with n.
other respects.
The stopping variables Tn used here are perfectly general in
In particular, no explicit definition of the Tn is needed
8
in order to establish the asymptotic normality of Tn
. There is also an
,Tn
interesting interplay between the restrictions imposed on Tn and on the
lIDderlying distribution.
In Corollary 2.4.1 we note that a slight weak-
ening of the conditions placed on one entails a compensatory strengthening
of the conditions that must be imposed on the other.
In Section 2.5 we
investigate the consequences of an explicit definition of the Tn'
We
furnish such a definition in terms of statistics of the type Tn, k and find
that, with only minor additional restrictions, our previous analysis can
be tailored to yield the asymptotic normality of the Tn'
This then furn-
ishes us with a more complete specification of the limiting distribution of
the Tn,T (Corollary 2.5.1).
n
In Stigler (1969) conditions are given which ensure the convergence
n Var(Tn,n )/0 2
-+
1 , as n
-+
(1. 3.3)
00
2 is the asymptotic variance of the T . Mason (1977) has also
n,n
established (1.3.3) under a different set of conditions. The question of
where
0
exhibiting an appropriate point estimator of
(1978).
0
2 has been taken up by Sen
He proposes the estimator
2
crn = n-
n-l n-l
2
L L c , c . ( (i Aj ) n - i j )(h (Xn,~~ +~ h (Xn,l~ ) )
i=l j=l n,l TI,J
-j )
-
x(h(Xn,J.+,)-h(X
.)) ,
1.
nJ
with n
>
1 and aAb
=
nlin(a,b).
Then under certain regularity conditions
consistent with those yielding the asymptotic nOTnlality (1.3.2), it can be
demonstrated that
1'2
cr
n
-+
2
cr almost surely (a.s.) as n
-+
00
•
(1.3.4)
The convergences (1.3.3) and (1.3.4) play an important role in Sen's
9
arguments leading to an invariance principle for the Tn, k of (1.3.1).
In Section 2.6 we proceed a step beyond (1.3.4) and derive the limiting
nonnal distribution of
az.n
For the stopped statistics T -r , a corresn"n
ponding estimator is defined for its asymptotic variance and a parallel
result is obtained.
In Chapter III we tum to the study of certain likelihood ratio pro-
cesses that arise naturally within the framework of our basic mathematical
model.
x
"'l1,k
To describe these statistics let us write, for each k,
l~k~n,
= (Xn, l""'Xn, k) for the vector of the first k order statistics of a
random sample of size n, having a density function f e (.) involving a
parameter e restricted to an open subset 8 of the real line lR'. Then for
eO' eo+un-~E8
we define the likelihood ratio statistics {An,k; l~k~n} by
(1.3.5)
With eO fixed in 8 and our definition of the sequence {Tn;
regard u
+
An
,Tn
(u) as a random process.
n~l}
we shall
The main content of Chapter III
is the derivation of an asymptotic expansion of this process. We shall show
that, under certain conditions, for each fixed
UE JR
"
the function
An,T (u) may be approximated by an exponential family.
Actually our dis-
n
cussion deals with a two parameter process (u,t)
An,k (t)(u) for which,
n
in this approximation, there appears in the exponent of the family, a re-
lated process t
+
Wn,T (t).
+
Section 3.3 and 3.4 are devoted to the analy-
n
sis of the latter process.
The asymptotic properties of likelihood ratio processes in a general
setting has been carefully investigated by LeCam (1960).
In an abstract
measurable space (D,G) let there be given a family of probability measures
10
{P e : eEe}, where e is a subset of the real line.
n-fold Cartesian product of
product measure on~.
Let (~,~) be the
(~,G)
and let Pn, e denote the corresponding
There are several statistical problems that lead
to a study of the functions e
Peas n + 0 0 . A familiar example would
n,
be problems related to asymptotic properties of maximum likelihood esti+
mators, where such questions can be posed in terms of the local behavior
of the process A (e,e O) = log dP e/dP e as n + 00. This rough descripn
n,
n, 0
tion applies to likelihood ratio processes arising from independent and
identically distributed observations.
Here the random functions in ques-
tion are defined by
n
Zn(t) = i~l{f(Xi;eO+t¢(n))/f(Xi;eO)}
(1.3.6)
where f(o; e) is the common density of the independent observations
X1"",Xn and ¢(n) a normalizing factor. An exceedingly elegant treatment
of the process t + Zn(t) has been given by Ibragimov and Khas'minskii
rne authors establish the weak convergence of (1.3.6) to a pro-
(1972).
cess t
±
00,
+
Zet) in the space CO(R) of continuous functions vanishing at
topologized by the supnorm metric.
rfhese results are then applied to
the derivation of asymptotic properties of maximum likelihood and Bayes!
estimators.
The generalization of (1.3.6) to independent, but not necessar-
ily identically distributed observations has been treated in a subsequent
article. (Ibragimov and Khas'minskii U975a)). The technique employed in
these papers does not rely on classical Cramer-type regularity conditions
involving the existence of two or three derivatives of the function
e
+
f(o;e) and additional uniform integrability restrictions.
For example,
the traditional assumptions on the existence and continuity (in e) of
a2f/ae 2 and log f(x,8) have been dispensed with in favor of conditions
11
involving only the first derivatives.
A somewhat different approach is taken by Roussas (1965) and Johnson
and Roussas (1969, 1970) for the more general situation in which the
observations {X.;
1
i~l}
form a stationary Markov process.
The independent,
but not identically distributed case is discussed in Philippou and Roussas
(1973).
The question of the weak convergence of the associated processes
is not considered by these authors.
The strategic methodology involved
here consists in exploiting the powerful tools of contiguity and in replacing the standard analytic assumptions by those concerning the differ1
entiability in quadratic mean of the entities ep(e,e*) = (f(X,e)/f(X,e*n7z •
(For nonidentically distributed variables one needs to consider
!<
cl>j(e,e*) = (fj (Xj;e)/fj (xj;e*)) 2.)
Another method of investigation of the weak convergence of likelihood
ratio random fields is demonstrated in Inagaki and Ogata (1975, 1977), for
both the cases III which tile observations are independent and identically
distributed or form a stationary Markov process.
Their technique utilzies
regularity conditions similar to Huber (1967) and Inagaki (1973) and also
include the multiparameter case.
For the random processes (u,t)
+ ~
n, k (t)(u) and t
+
n
Wn,T (t) which
n
we investigate in Chapter III, the observable variables are no longer
independent.
The statistics
k (t)
~
k (t)(u) may be expressed in the form
n, n
n
.
IT
1=1
{q
l(Xn 1·IX
ex
"11 1'l)/qe
- n
eo+un -x
2
"
·Ix"'l1 1. - I)}'
.
1
0'-
,
(1.3.7)
where qe(Xn , il~n , I" ~l) is the conditional density of Xn,l. given "'I1,1
X '-1'
The resemblence to (1.3.6) lies only on the surface, since in (1.3.7) we
have lost the enormous simplification that can be afforded through independent and identically distributed observations.
However, we find that the
12
basic approach of Ibragimov and Khas'minskii (1975) can be modified to
handle the analysis of (1.3.7).
For the treatment of the auxiliary process t
basically the argument of Sen (1976).
Wn T (t) we follow
, n
Our results are a generalization
-+
of his invariance principle in that it explicitly incorporates the stopping variables {Tn;
n~l}
into the general infrastructure and yields Sen's
results as a special case.
Furthermore, we have placed fewer analytic
restrictions on the underlying distribution.
In Chapter IV we make a few remarks on the regularity conditions of
Chapter III and derive the asymptotic expansion of
different set of conditions.
k (t) (u) under a
n, n
A considerable simplification is obtained
when the hazard rate is separable in its variables.
11.
We conclude the
chapter with an outline of results leading to the asymptotic distribution
of the likelihood function in the case when the underlying variables are
neither independent nor identically distributed.
In our final chapter we report on some results on ongoing research.
These pertain to applications of the decomposition of the likelihood ratio
processes of Chapter III.
We furnish an interesting lower
boun~
for the
variance of the stopped statistics Tn
large sample cases.
,both in the finite sample and
,Tn
This would form the groundwork for an eventual treat-
ment of efficiency of estimators formulated in our framework, perhaps along
the lines of Roussas (1972).
The discussion of the asymptotic properties
of maximum likelihood and Bayes' estimators can also be included under our
treatment, very much along the lines of Ibragimov and Khas'minskii (1973,
1975b). lhe entire subject of
asyrr~totic
hypothesis testing and estimation,
and of optimality under our framework will be reported elsewhere.
~
CHAPTER II
ASYMPTOTIC NORMALITY OF SOME TIME-SEQUENTIAL STATISTICS
2.1
Introduction
The class of statistics which are linear combinations of functions
of order statistics has received much attention over the past decade.
The literature of this period is largely devoted to examining sets of
conditions under which the asymptotic normality of such statistics can
be established.
A variety of technical strategies have been utilized in
obtaining this asymptotic normality but none of these lead to an invariance principle for the class of statistics that are linear combinations
of order statistics.
In one of the major researches in this area Chernoff,
Gastwirth and Johns (1967) admit that their techniques yielding the
asymptotic normality of linear combinations of order statistics lack the
sophistication of an invariance-princip1e type argument which would pave
the way to the "best possible" results in this area.
Recently an approach
along these lines has been initiated by Sen (1978).
The wide interest which linear functions of order statistics has received over the years originates from the fact that statistics of this
type are often appropriate, in a wide variety of contexts, for estimating
location and scale parameters, possess elegant robustness qualities and
can be calculated without much difficulty.
Such statistics have also
been used to devise suitable statistical tests in the area of life testing.
(Epstein and Sobel, 1954, 1955).
14
Chernoff, Gastwirth and Johns (1967) deal with linear combinations
of the form
Tn = n
-1 n
I c .h(Xn,l.)
i=l n,l
where the cn,l.'s are constants, Xn, 1"" ,Xn,n the ordered observations of
a sample of size n, and h a function. In essence their arguments leading
to the limiting distribution of Tn consists of a decomposition which
expresses Tn as a linear combination of independent exponentially distributed random variables, which has a limiting normal distribution,
together with a residual term which converges to zero in probability as
n tends to infinity.
Various regularity conditions are imposed on the
constants cn,l. and the underlying distribution F. Moore (1968) gives an
elementary proof of the asymptotic normality under a different set of
conditions.
In another investigation Stigler (1969) attacks the problem
using a device due to Hajek (1968) by representing Tn as a linear combination of independent random variables plus a remainder term which converges to zero in mean-square.
Again a different set of conditions is
considered for the c
(l969~
. 's and the 1~derlying distribution.
Shorack
n,l
1972) considered a somewhat more general class of functions of
order statistics for which asymptotic no:rmality
technique entirely different
et al. (1967).
fr~n
IS
established by a
those of Stigler (1969) and Chen10ff
Shorack obtains a decomposition in terms of a special
uniform empirical process and then proceeds to analyze the separate terms.
In this chapter we shall use the basic results of Shorack.
However,
we are concerned with a class of time-sequential statistics that arise in
progressively censored schemes into which a broad class of stopping
variables is introduced.
We begin the formulation of our problem by
~
15
considering two examples.
Our assumptions are listed in Section 2.3
and the proof of the main theorem is considered in Section 2.4.
In the
next section we apply our main results to a class of stopping variables.
Some examples from life testing situations are presented to illustrate
possible applications.
Finally in Section 2.6 we derive the limiting
normal distribution of a suitable estimator of the asymptotic vari&lce
of the class of statistics considered in Section 2.2.
2.2
Formulation of the problem.
Let us begin with two examples.
Example 2.2.1.
For n
2
1 items lliider a life test Epstein and Sobel
(1954, 1955) have devised statistical procedures based on the statistics
of the form
where Xn, l""'Xn,n are the order statistics of a random sample of size
n from the exponential distribution and k = ~(t) responses have been
recorded up to time t.
of the sample.
Then Tn, k is termed the "total exposure to life"
Variables of the fO:I111
where the cn are certain constants, arise naturally in this context. It
will be of interest to determine the limiting distributions, if any, of
Tn,k (t) and vn as n ~
n
Example 2.2.2.
00.
Let p(x"11 , k,8)
denote the joint probability density func-
tion of the vector of the first k order statistics "11,
X k
= (Xn, l""'Xn, k)'
16
1 ~ k ~ n, of a random sample of size n with distribution function Fe
and probability density function f 8 ; 8 is a real parameter. Sen (1976)
has developed a basic invariance principle involving the statistics
a~(lOg p(~,k,e)), when e is held fixed.
These have the form
In the next chapter we shall closely investigate these statistics and
discuss an invariance principle for them when a broad class of stopping
variables is introduced into their definition.
From the foregoing it seems natural to consider linear combinations
of functions of order statistics that have the form
-1 k
Tn, k = n i=l
I cn,l.h(XIl,l.)
~
(2.2.1)
d kh*(X k) ,
n,
n,
~
n, where cn, k' dn, k are constants and h, h* certain functions.
As previously stated, Xn, 1""'X.Il,n are the order statistics of a random
3.
k
+
sample of size n whose distribution function Gn lies in the class of all
distribution functions.
Suppose ~ n, l""'~ n,n denote the ordered observations
from a sample of n independent and identically distributed (iid) uniform
(0,1) variables. With gn = h(G-n1) and g*n = h*CG-n I ), the Tn, k of (2.2.1)
have the same distribution as does
Remark 2.2.1.
For each n
~
1,
we conceive of a stopping variable Tn taking values
in the integer set {l, ... ,n}, and defined in terms of the
sequence of random variables
{n
n,l.;
l~i~n},
~
.. For any
n,l
we define the variable
17
nn,T by nn,T = nn,l. if Tn = i, 1
n
n
s i s n.
Accordingly we shall be concerned with the statistics
(2.2.2)
For Tn
(1972).
=n
(2.2.2) closely resembles the functions considered by Shorack
By explicitly incorporating the stopping variable Tn into our
framework we allow for greater generality and wider applicability.
Remark 2.2.2.
{Bn, k;
l~k~n.
In practice Tn would be adapted to the a-fields
hk~n},
where Bn, k is the a-field generated by ~ n, l""'~ n, k:
This situation would arise, for exan~le,in the case when Tn
itself is specified in terms of statistics of the form Tn, k'
take up this point later in this chapter.
2.3.
We shall
Assumptions.
Let G denote the class of left continuous functions on (0,1) that
are of bounded variations in (e, I-e), for all
°
<
e
<
1/2.
For each
gEG there corresponds a Lebesgue-Stieltjes signed measure, also denoted
by g, whose total variation measure will be denoted by jg I.
Define J n on [0,1] by J n (t) = cn,l. foY (i-l)/n<tsi/n and l~isn, with
In(O) = cn,l' For fixed bl'b 2 andM>
define the "scores bounding
°
function" B by
-b
B(t) = Ht
For fixed 0 > 0, define
-b
l(l-t)
2, O<t<l .
18
and
-~+bl+o
D(t)
=
Mt
(l-t)
q(t)
=
[t(1-t)]~-o/2
-~+b2+o
, O<t<l
O<t<l .
Let q(t) = q(t) [t(1-t)]o/4 ,O<t<l.
Assume g EG for all n.
Let g de-
n
note a fixed function in G and J,g* fixed measurable hnlctions on (0,1);
let a be a fixed number,
0 < a <1.
The symbols ->a.s. ,->pr denote
respectively convergence almost sure and in probability. For a sequence
of random variables {Y } we express its convergence in distribution to a
n
random variable Y by L [Yn ] -+ L [YJ, as n -+ 00, or by Yn ->VY, as n -+ 00.
We now list our assumptions. In the interest of sirnplicity in script
the symbol Igi stands for the total variation measure as described at the
beginning of this section and for the absolute value function corresponding
to g.
[AI]
JlBq dlgl
<
o
[A2]
t
D, all
I~I ~
D, IJI
~
B, all IJnl
~
B on (0,1) and
00.
Except on a set of t's, not including the point a, of Igl-measure
zero, both J
of
~
Let Igi
as n
-+
1S
continuous at t and I n
o
J uniformly in some neighborhood
00.
J1Bq dig -gl
[A3]
-+
n
0, as n
-+
-+
00.
There is a neighborhood of a in which g'n is defined for all n ~ 1
and the family {g~; n~l}is equicontinuous at a with ~(o.) -+ g'(o.), as
[A4]
n
-+
00.
[AS]
There is a neighborhood of a in which
and the family {g*';
n
g*'
(a)
n
-+
n~l}
g*'(o.), as n
-+
00.
~I
is defined for all n
is equicontinuous at a with g*(o.)
n
-+
~
g*(o.) and
1
19
The prime (I) denotes the first derivative.
asstnnptions.
[AI] - [A3] are standard
We shall see later that [A4] may be weakened appropriately
in conjunction with the conditions we shall impose on the Tn'
For each xE(O,l) define
(2.3.1)
where d is a measurable function on (0,1).
function and
[A6]
Here
I
denotes the identity
J-dI integration with respect to Lebesgue measure.
Let d be differentiable at the point a and
-1
1
max {n 2Jd k - den k)l: In- k-al ~ d
l~k~n
n,
h
for some
2
E
a
n-'r CO
a
J J (SAt
a a
- st)J(s)J(t)dg(s)dg(t)
a
+
a as
Define
> 0.
o (a) =
-'r
2(1-a)d(a)g*l(a)J tJ(t)dg(t)
a
+
2
a(l-a){d(a)g*l(a)} . (2.3.2)
Note that under our assumptions both 0 2 (a) and ~n(a) are finite.
Up to this stage no conditions have been imposed on the sequence
• nd}.
The consequences of an eXvlicit definition of the Tn will be
n'
taken up in Section 2.5. For the purvoses of our main theorem, we shall
{T
consider one of the following condltions on the limiting behavior of the
Tn at the appropriate stage.
e
[Bl]
h
-1
n 2(n Tn-a)
=
opel)
as n
-'r
co,
[B2]
h
-1
n 2(n Tn-a)
=
opel)
as n
-'r
co,
[B3]
n -1 Tn - a
Opel)
as n
-'r
co.
=
or
20
We shall make an important remark on these conditions in relation to the
assumptions [AI] - [A6] in our main theorem in Remark (2.4.1).
2.4.
The Main Theorem.
Theorem 2.4.1.
Suppose that assumptions [AI] through [A6] hold and
]In(a) , o'Z(a) are defined as in (2.3.1) and (Z.3.2).
Then, if in addition
condition [B1] holds,
(2.4.1)
while, if [BI] is replaced by [B3] , then
(2.4.2)
Proof.
Following Shorack (1972), there is a probability space
on which there are n iid uniform (0,1) random variables
°<
(~,G,P)
~1""'~n
such
n, 1<"'< ~ n,n < 1. Let r n be the empirical distribution func~
1
tion of ~l""'~n and let Un = n~(rn-I) denote the corresponding uniform
that
~
°
empirical process on [0,1].
Then p(U.n ,U) ->a.s. as n -+ 00. Here p denotes
the uniform metric given by p(f1 ,f Z) = sup /fl(t) - f 2 (t) I, for functions
[0,1]
f l ,f 2 on [0,1], and U denotes a properly chosen Brownian Bridge in ~, that
is, a Gaussian process {U(t):
E{U(t)}
=
0, E{U(s)U(t)}
=
tE[O~lJ}
with continuous smnple paths and
sAt - st, s,tE[O,l].
Now V = -U is also a Brownian Bridge and the ill1iform quantile process
1: -1
Vn = n 2 (r - I) on [0,1] satisfies p(V ,V) ->
n
n
a.s. 0 as n
-1
f n (t)
=
inf{sE[O,l]:
fn(s)~t}.
-+
00.
Here we take
Let Q denote the class of all nonnegative
continuous functions q on [0,1] which are bounded below by functions q
nondecreasing (nonincreasing) on
([~,l])
and satisfy
If P (f1 ,f2) = sup Ifl(t) - f 2 Ct)l/qCt) for functions
q
O<t<l
and
q,
it
is
known
that
Pq(Un'U)
= 0pCl) and PqCUn,O) = 0pCl) , for
Z
Ilq -2dI <
°fl,f
[O,~J,
00.
21
each qEQ.
We shall have occasion to use these results in the proof of
Theorem 2. 4 .l.
We proceed to obtain a decomposition of ~(T(l) - ~(1)(n-1L ))
n,T n
n
n
along the lines of Shorack (1972), Now
and
x
~ (1) (x) =
fo gnJ n dI
l/ln(t)
f
n
Define
= -
O<x<l .
ex
JndI
t
a
b
with the customary convention:
f .
a
l/ln((i-l)/n)
= cn,l·/n,
for
l~i~n
=
-f·,
if b < a.
Then l/ln(i/n)
b
and therefore for each k,
l~k~n,
=
(2,4.3)
But rn (~ n,l.) = i/n , and the sum on the right hand side of (2.4.3) is
a.s. equal to
since, almost surely, no ~ n,l. takes on one of the countable set of values
at which gn is discontinuous. So from (2.4.3)
T(l)
n,L n
=
-l/l (O)Q" (~
)n ~ n,l
(2.4.4)
22
Similarly,
(2.4.5)
and
a.s.
=
(2.4.6)
by integration by parts.
~(T(l)
n,T
jJ
n
Therefore from (2.4.4), (2.4.5) and (2.4.6),
(l) (n-IT )) = _ (S(l)+y +y +y +y )
n
n
n
n, I n, 2n,3 4
n,
(2.4.7)
a.s., where
S(l)
n
~ n,T
=
J
~
n,l
with
A£
= An on
n
ex
A Undg n = 0J -n
A*Undgn
-n
[~ n, I' ~n,T
) and equal to zero outside this interval. We
n
have assumed for convenience ~ n,T (w) < ex. If ~ n,T (w) > ex the contribun
n
e
23
f
tion of
';n,T
nA*U
-n ndgn vanishes at the limit, as we shall see later.
a
Define
a
S
f JUdg
o
=
_ S(l)
+
(2.4.8)
d(a)g*'(a)U(a)
S(2) , say.
+
Then S has the distribution N(O,0 2 (a)) with o2(0.) given by (2.3.2).
show first S(l)
n
->
pr
S(l) as n ~
00.
Now
a
=
We
a
6 IJI
Iun-uldlgl
+
6 /A£Unldl~-gl
a
+
fo
lun"n
IIA*-Jldlgl .
By [AI],
a
IYn,sl ::; Pq(Un,u)
6 Bqd/g!
-pr
° as
n -to
00
•
Again, by [AI] and using Theorem 1 of Wellner (1977) there exists a
set Ac0, with peA)
1 such that the following inequality holds; for all
=
WEA, there exists an integer nW fOT which n ;: : nW iJnplies
(2.4.9)
for some constant M > 0.
O
Hence, by [A3]
a
/Y n ,6 1
::;
MoPq (Un ,O)6 Bqdl~-gl -pr 0 as n -to
00
•
To handle Yn ,7 we note that, from the definition of An(t), we have
Since fn(t)
->
a. s. t
as n -to
00,
also an (t)a-tost
... as n -to
00.
We show
24
In(an(t))
-~.s.
J(t) as n
+
00,
for each t except for a set of t's, not
By [A2], there is a set T
including the point a, of Igl-measure zero.
of Igl-measure zero such that for tiT, there exists some 00
=
00(t) > 0
for which J is continuous at t and I n + J uniformly on [t-oo,t+o O] as
n + 00. Now fix tiT. Then for some integer nO = no(t) , lan(t)-tl < 00
for all n
~
nO' except possibly on a P-null set.
Further, if € > 0
IS
arbitrary, for all n ~ some integer nl = n1(€,t), one has IJn (t i )-J(t 1 ) I <
€/2, for n ~ nOvnl and outside some P-nu11 set. Finally, by the continuity of J at t, with 00 sufficiently small, IJn(an(t))-J(t)/ < €/2 and
hence we get
€/2 + €/2
~
for n
~
= E ,
nOvn1 and outside some P-nu11 set.
A (t)
n
->
a.s.
J(t) , as n
for each t outside some Igl-null set.
·
[B3"'
cand1·tlon
j VIZ. n -1 'T
_.> a as n
n
pr
also E;n, 'Tn ->pr a as n
+
OD,
This establishes
+
(2.4.10)
00
In what follows we shall use the
+
OD.
I n view a
f te
h inequality
and for the variable e,'net) defined by
if E;
n,
l~t<E;
n, 'Tn
= 0 otherwise,
we can show sn(t)
pr I as n
->
+
OD,
for each tE(O,a).
trary and set Bnn = {w: IE; n,'T -al ~ n}. Then P(Bnn)
n
n
tE(O,a) and choose n < (a-t)/2. For wi Bn , E;n 'T
, n
Let E,n
+
0 as n
>
t.
+
>
OD.
a be
arbi-
Fix
Further, since
~
~
25
Sn,l -> a.s. 0 as n ~ 00, there exists an integer nO = n6(t) such that, for
Sn, I < t, except possibly on some P-null set. Therefore
P[/sn(t)-ll
5 €]
=
5
P(B~)
+
P{WEB~: Isn(t)-l! ~ €}
P(B~) , for n ~ nO .
So Sn (t)->pr I as n ~ 00. Thus, in view of (2.4.10) and the definition of
A~, we finally arrive at
A£(t) ->prJ(t), as
n
~
00
,
for tE(O,a) almost everywhere Igj.
Now from [AI] and (2.4.9) for WEA and n
2:
nw
and so we have
But p~(Un'O)
= Opel).
q
a
Ia
IA*-J!qdlgl
->pr 0 as n ~ 00 will follow
On
directly from Lemma 2.4.1 below. 1ilis then establishes s~l)-pr S(l) as
n
~
That
00.
Throughout the above discussion we assumed S.
< a. If otherwise~
s
n,T n
the contribution of J n,T n On
A*Undg n is zero in the limit since our argua
ment shows that for sufficiently large n and some constant M '
O
sn,T
1
1
II n~Undgnl ~ MOPq(Un,O) [I Bqdlgn-gl + I X(a s )Bqdlgl],
a
0
0 ' n,T n
where X(a,b)
Lemma 2.4.1.
1S
the indicator function of the interval (a,b).
Let Wbe a Borel-measurable set in (JR ,B) - the Borel real
line and v a measure on (R ,B) such that v (W) <
00.
Let h, hn , n
~
1
26
be measurable filllctions on
(i)
Wx~
such that
h, hn , n~l are vXP-integrable and for some vXP-integrable
function k, I~-hl ~ k, for all n sufficiently large;
(ii)
Except on a set of t's of v-measure zero
as n
-+
~(t,·)
-pr h(t,')
"
00.
Then E(JI~(t,w) - h(t,w)ldv(t))
-+
a,
as n
-+
00.
W
Proof.
For arbitrary
>
E
a define
for each n
~
I B = {(t,W)EWX~:
n
I~(t,w) - h(t,w)! ~ E}. Then B is a measurable subset of Wx~. Let
n
B~ denote the ~-section of Bn at tEW. Then by (ii) except on a set of
t's of v-measure zero, P(B t ) -+ a as n -+ 00. Now
n
E(JI~(t,w)
W
- h(t,w) Idv(t))
=
= J Ih -hld(vxP)
+
n
B
n
J Ih
Wx~
n
-hld(vxP)
J Ih -hld(vxP)
c
n
Bn
But
J Ih
B
n
n
-hld(vxP) ~
J kd(vxP)
B
n
and
A version of Fubini's theorem implies
(vxP)(Bn )
and since v(W)
<
yields (vxP)(Bn)
00,
-+
=
Jp(Bt)dv(t)
W n
and P(Bnt ) :$ 1, the dominated convergence theorem
a as n -+ 00. Thus the integrability of k entails
J kd(vxP)
B
n
-+
a as
n
-+
00
27
o
Since E > 0 is arbitrary, the lemma is proved.
Returning to our main argument, we shall show y n,l. ->pr 0 as n ~ 00,
for i = 1,2,3,4. That y n, 1 ->pr 0 and y n, 2 ->pr 0 as n ~ 00, under our
assumptions have been demonstrated in Shorack (1972).
To handle y n, 3
n =
define, for each n > 0, Bnn, 1 = {WEQ:'~ n,T -al < n} and Bn,2
n
{WEQ: In-1Tn-al < n}. Further, for each 0 < E < ~, let ME = sl~{B(t):
tE[E,l-E]}.
l~n(t)1
Then
With our
~
assun~tion
aE(O,l) we may choose E so that aE(E,l-E).
Men, for It-al < n,
ffi1d
n sufficiently small.
Now gn can
be expressed as a difference of two nondecreasing functions and we may
therefore assume, without serious loss of generality,
~
to be nondecreas-
n
Accordingly for wEBn
n, lnBn, 2 and n-1T n (w) < ~ n,T n (w)
~n T
ing.
Iyn, 3 1 ~ n~-l J
n
'n
~
l~n(t)ldgn ~ n~(Q'-'Il (~ n,T ) - '-'Il
Q (n-IT n))Men
n
and similarly for n-lTn(w) > ~_n,T (w).
n
But whatever W
n
-1
T
n)g'n (a*)
n
(2.4.12)
I
Since both ~ n,T , n -1Tn converge
where Ia*n - n -1 Tn I ~ I(1l,T - n -1 Tn :.
n
n
in probability to a, as n ~ 00, we have imnediately that a~ -pr a as n ~
00.
Summoning [A4], for arbitrary E > 0, there exists a o(c,a) > 0 such that
I~(t)
- ~(a)1 ~ c for It-a/ < o(c,a) and for all n ~ 1.
Hence
P{lg~(a~) - ~(a)' > c} ~ P{la~-al > o(c,a)} ~
as n
~
00, and we obtain
g'(a*)
- '-'Il
Q'(a)
n n
Also by assumption
~(a) ~
g'(a) as n
~
00.
= 0
P
(1)
Thus
°
28
g'n (a*)
n - g' (a)
Finally, ~(~
- n-IT )
n,T n
n
follows that
==
== 0
(2.4.13)
p (1) .
V (n-IT ) and p(V ,V) -> 0 as n
n
n
n
a.s.
+
00.
It
(2.4.14)
Hence from (2.4.12) and (2.4.13), (2.4.14) becomes
n~(g n (~ n,T ) - ~(n-IT n ))
n
==
V(a)g'(a) + opel)
(2.4.15)
n lnB n 2) + 1 as n + 00 and n is arbitrary we
and so in (2.4.11) since P(Bn,
n,
arrive at y n, 3 -> 0 as n + 00. Recall y n, 4' We can write
y
where Ib~-al
s
n,4
y
-n~(g: (~
) - g (n-IT ))J (b*)(n-IT -a)
"n n,T n
n
n n n
n
In-1Tn-al.
to (2.4.10) we get
this yields
==
In(b~)
Employing the same arguments
-pr J(a) as n+
n, 4 ->pr 0 as n
+
00.
00,
as
those leading
and together with (2.4.15)
Collecting our results we have established
(2.4.16)
under the hypotheses [AI] through [A4] and the condition [B3].
(2)'
(2)-1
the proof we need analyze the term n 2(Tn,T - ~ n
n
remaining assumptions. Now
h:
(n T))
n using the
-1
g:*(n -T1
n n,T n) - =n
n))d(n T)
n
h:
+ n2(g*(~
To complete
29
-1
~
~
In view of [A6] , n 2(d
- den T)) -> 0 as n + 00. For n2(~(~
)n, Tn
n
pr
n, Tn
g~(n-1Tn)) the same reasoning from (2.4.12) through (2.4.15) using [AS]
instead of [A4], leads to
~
n2(Q:*(~
'-'Il
1
n,T n ) - ~(n- Tn)) = -U(a)g*'(a) + opel) .
1
Also, by the now familiar argument, on
v*(C
> g*(a) d(n- T) > d(a)
"'n, Tn ) -pr
'
n -pr
as n + 00. Hence finally we obtain
_s(2) +
(1)
0
(2.4.17)
P
From (2.4.16) and (2.4.17) we get
~
n 2 (T
n,T n
-
~
n
(n
-1
(2.4.18)
T)) = -S + 0 (1)
n
p
Hence (2.4.1) of Theorem 2.4.1 is established.
Now let us consider
-1
~
n2(~n(n
1
1
n~(~ (n- T ) - ~ (a))
n
n
n
1
=
Tn) -
n~(
n -~
J0
~n(a)).
From our definitions
a
ng J dI - J g J dI)
nn
0 nn
The second term on the right hm1d side of (2.4.19) Clli1 be rewritten
(2.4.20)
From our earlier arguments
~
-1
~
. -1
n 2(g*(n
-T n ) - g*(a))
= n 2(n Tn -a)(g*'(a)
n
n
We now assume the condition [B2], viz. ~(n-lTn-a)
=
+ 0
p
(1))
Opel).
The expression
30
(2.4.20) then reduces to
Finally for the first term on the right hand side of (2.4.19) we can
write it as
!;:
n2
But
-1
J
g
n
d~n =
{g
n
(a)~n (a)
[n Tn,a)
~ dg we note that our previous discussion of y 3
n,
[n Tn,a) n n
J -1
To treat
is also tenable here.
!;:
We get n 2 J 1
~ dg -> 0 as n
) n n pr
[n Tn,a
+
00.
_
•
Also
~gn(n-lT )~ (n-lT ) = ~g (n-lT )(~ (n-lT ) - ~n(a))
nn
n
n
n n
n
(2 • t c. L
r
where Ic~ - al ~ In-1Tn-al.
pr g(a) as n
->
+
00.
Once again In(c~) -pr J(a)
So under [B2], (2.4.21) reduces to
-1
n 2(n Tn -a)g(a)J(a)
!;:
+ 0
p
(1) .
Collecting our results, under [AI] - [A6] and [E2] ,
!;:
-1
n2(~n(n
Tn) -
~n(a)) =
-1
n 2(n Tn-a){g(a)J(a) + (d(a)g*(a)) I} + opel).
!;:
(2.4.22)
Thus, from (2.4.18) and (2.4.22), we get finally
j
31
-S
+
h
-1
n 2(n Ln-a){g(a)J(a)
+
(d(a)g*(a)) I}
opel),
+
(2.4.23)
and this proves (2.4.2) of Theorem 2.4.1, and so the proof of the theorem
is terminated.
Remark 2.4.1.
0
We have established (2.4.18) under the hypotheses [AI]
through [A6] and the assumption [B3].
at a
Here the differentiability of d
is not needed - its continuity alone at a suffices.
The decomposition (2.4.16) was obtained under the assumptions [AI]
through [A4] and the condition [B3].
examination of y n,3 and y n,4
will reveal that we may establish the same decomposition under slightly
altered conditions.
An
We retain the hypotheses [AI] through [A3] but re-
place [A4] by the weaker assumption [A4*] , where
[A4*]
as n
Let the family {gn;
+
n~l}
be equicontinuous at a and
~(a) +
g(a) ,
00.
We shall now need to replace [B3] by the stronger condition [B2].
With
these modifications and the addit.ional assumptions [AS] and [A6] the
decompositions (2.4.22) and (2.4.23) will continue to hold.
We therefore
state
Corollary 2.4.1.
Suppose the assumptions of 1neorem 2.4.1 hold with [A4]
replaced by [A4*] , and in addition suppose condition [B2] holds.
Then
(2.4.18), (2.4.22) and (2.4.23) remain true and in particular (2.4.2)
obtains, while if [B2] is replaced by [BI] then (2.4.1) obtains.
Proof.
We only need examine y n, 3' Yn, 4 and the decomposition (2.4.22).
We have from our definition
32
Yn , 4
= -nk2 (n -1Tn-a)Jn (b*)(Q
n
where Ib~ - al
~
$
In-lTn-a/.
As
(~
-1
,
n,T n) - gn (n T))
n
before In(b~) -pr J(a) and in view of
-1
[A4*] and the fact that ~ n,T ->pr a, n Tn ->pr a, we obtain
n
(2.4.24)
Therefore under [B2]
Ta
Yn ,4 reduces to opel).
handle y n, 3 write
~n
k
- n2
n
-1
'
T
J
n
(2.4.25)
g d1/J
n n
Tn
(An appropriate change of sign will be required depending on the relative
magnitude of f;
n,T n
and n -IT .)
n
Now
(2.4.26)
where bn*, c*n ->pr a as n + 00. Once again we have J n (b*)
n , J n (c*)
n
as n + 00. In view of (2.4.24) and the fact that
prJ(a)
->
(2.4.27)
we find that under [A4*] and [B2] (2.4.26) reduces to -U(a)J(a)g(a)
+
0p(l).
33
For the second term in (2.4.25) write
k
+ n2(~ (~
~n(n
n n,T ) n
-1
Tn))Q'11 (a) .
(2.4.28)
+ 0 (1)
(2.4.29)
Of course
k
n2(~ (~
n n,T n ) -
~
n (n
-1
~
n))gn (a)
=
-U(a)J(a)g(a)
p
.
We shall show that the integral on the right hand side of (2.4.28) reduces to opel).
Then from (2.4.25) through (2.4.29) we will finally
arrive at y n, 3 = 0 p (1).
n l' Bn for each n ~ 1 and arbitrary n > 0 as in
Define the sets Bn,
n,2
the discussion following Lemma 2.4.1. By virtue of the equicontinuity of
the family {gn;
n~l}
at a, for arbitrary E
such that Ign(t) - ~(a)
I
<
>
0 there exists n = n(£) > 0
E for all It-al < n and n ~ 1.
With this n
n lnB n 2 we obtain
and wEBn,
n,
~n T
n
'
I~ J
-1
(gn(t)-gn(a))d~n
n Tn
I
~
Enk2l~n(~n T ) ' n
~n(n -1 Tn)
I=
n lnB n 2) ~ 1 as n + 00 and E is arbitrary, the integral in
Since P(Bn,
n,
(2.4.30) must reduce to opel). Combining our results we have thus proved
y
n, 3
= 0
p (1).
Finally, to complete the proof, it suffices to consider only
a
k
k (1) -1
(1)
n 2 (u
(n T) - U - (a)) = -n 2
n
n
n
n
J
-1
T
gd~
n
n
n
a
1
n'2
n
J
-1
T
n
34
k:
Now under [B2] the first term reduces to n 2 (n
-1
-a)J(a)g(a) + 0 p (1).
The argument leading to (2.4.30) will show that the second term is
opel).
T
n
Hence (2.4.22) remains true under [B2] and [A4*] and so does
(2.4.23).
0
This terminates the proof.
Example 2.4.1.
Let
be iid random variables from the simple
. -x/e
exponential distribution Fe(x) = l-e
, x > 0, e > O. Consider the
Xl'''''~
statistic
T
T
n,T n
=
n-
1
n
~ X . + n
i=l n,l
In the notation of this section
-1
(n-T)X
, n
n n,T n
~
1.
-1
= on
g* = F
d
e'. J n = 1 'n,k
l$k$n and d(x) = (I-x).
The hypotheses on the theorem now
rather pedestrian calculation yields ~n(x) = ex and 0 2 (a) =
~
'11
(2.4.31)
-1
= (l-n k)..
hold and a
e 2a, in this
case.
Therefore if the stopping variables Tn satisfy n-lT -> a as
n
we have from (2.4.18)
+
n
00,
Unk:2 (T n,T-en -1T n )}
+
pI'
2
N(O,e a)
(2.4.32)
n
Alternatively, the transformation Y1" = (n-i+l)(Xn,l.-Xn,l'1) l$i$n,
with Xn, a = a enables us to reexpress (2.4.31) in the form
Now the Y.'s
are independent random variables with the same distribution
1
Fe'
Thus the central limit theorem for random sums (Wittenburg (1964))
yields (2.4.32) under the same condition on T , viz. n-lT -> a as n
n
n pI'
2.5.
Limiting distributions of some stopping times.
We remarked earlier that in most practical studies the stopping
variable Tn will be specified in some manner in terms of the variables
+
00.
35
For example, in the typical situation of n
~n, l""'~n, n'
~
1 items
under a life test, the observable random variables are the order
statistics Xn, l""'Xn,n of the sample. If sampling is performed under
the truncation scheme we may be interested in the case in which the T
n
are specified by Tn = min{l=:;;k;:;n: XI>
n, ( C}, for some constant c, O<c<oo.
In this section we shall examine the consequences of our main results
when Tn is given an explicit definition.
For this purpose let us con-
sider statistics of the type
~
T
n, k
=
n
-1 k
~
~
~
~
+ d
g*(C )
. L\ cn i gn (0c'n , i) n
k
1=1'
, n c'n , k '
(2.5.1)
c .,
l=:;;k=:;;n, where the constants n,l dn,l. and the functions gn' ~g*n will be
required to satisfy the same set of conditions listed in Section (2.3)
of this chapter,
except the condition [A4] which we now replace by [A4*]
of Remark 2.4.1.
Let us designate here these conditions by [AI]
Ft,J.
[A6]. Likewise define
t"-J"""2
~n(x) and 0 (a)
through
corresponding to (2.3.1) and (2.3.2).
Furthermore,
[A7] .
For each n
~
~
1, Tn,k is nondecreasing in k, l=:;;k=:;;n.
In most life testing problems the statistics of interest will possess
this property.
Let {rnn(a) ,
n~l}
be a sequence of constants and a be a
fixed number, O<a<l.
[AS] .
We now define
Tn
where c(a)
=
min{l=:;;k=:;;n:
and set Tn = n, otherwise.
.
Tn,k
>
lS
a finite constant
n (a)} , if Tn,n > mn (a)
ill
0
36
Theorem 2.5.1.
Suppose conditions [AI] through [A8] are satisfied.
Then, with rea) > 0,
~
-1
L[n 2 (n Tn-a)]
Proof.
+
We first show rea) ~ O.
2
~-2
(a),a (a)A
(2.5.2)
(a))
Consider two sequences {kl(n); n~l},
kl(n) , k2 (n) ~ n. If we
select {kl(n); n~l} so that n~(n-lkl(n)-a) = 0(1), then in view of
{k2 (n);
n~l}
~-l
N(-c(a)A
such that for each n
~
1, 1
~
Corollary (2.4.1), (2.4.18) yields
(2.5.3)
On the other hand choosing {k 2(n);
n + 00, will give us, from (2.4.23)
n~l}
-1
~
such that n 2 (n k 2(n)-a)
+
n~(Tn,k (n) - ~n(a)) = -S + 0 p (1)
2
0, as
(2.5.~)
From (2.5.3) and (2.5.4)
(2.5.5)
Now under our hypotheses (2.4.22) remains
tl~e
(see
Corolla~J
2.4.1).
(2.5.6)
-~
-~-€
at least for sufficiently large n.
Thus, from (2.5.5) and (2.5.6), with
Let kl(n) = [n(n 2y+a)] , with y > 0 and k2 (n) = [n(n 2 +a)] with € > O.
Then all conditions imposed on these sequences hold and also kl(n) ~ k 2(n),
the assumption [A?], it must follow that y~(a) ~ 0 and so ~(a) ~ O.
We shall assume in the sequel ~(a)
>
O.
Now, by definition, Tn ~ kif,
37
Tn, k
and only if,
k
P{n 2 (n
where
~
>
-1
mn (a).
T
n
~
-a)
-k
= [n(n 2 z+a)].
Let z be a fixed number.
~
z} = P{T k
>
n, n
-1
k
Then n 2 (n kn -a)
Then
m (a)} ,
(2.5.7)
n
as n
+ Z,
+
00.
Therefore
by our assumptions the lliialysis leading to (2.4.23) is valid.
So we
have
,. . .
= -S
=
-5
~
2
+ n (n
+
-1
zX(a)
,. . .
~-a)A(a) +
+
0p(l)
0p(l) .
Therefore, using [A8]
= P{5 - c(a)
+ 0
P
(1) ~ z~(a)} ,
(2.5.8)
and hence from (2.5.7) and (2.5.8) the conclusion of Theorem 2.5.1
o
follows.
Remark 2.5.1.
k
-1
n 2 (n Tn-a)
We have proved under the hypotheses of Theorem 2.5.1 that
-0
~-l
~
A (a) (S-c(a)).
~
From (2.4.8), (S,S) has a bivariate norm-
al distribution with zero mem1 vector and covariance matrix
Pa(a)
~2
a (a)
2
~2
with a (a), a (a) as in (2.3.2) and
a
a
Pa(a) =! ! (sAt-st)J(s)J(t)dg(s)dg(t)
a a
+
a(l-a)d(a)d(a)g*' (a)g*l (a)
+
(l-a){d(a)g*'(a)! tJ(t)dg(t)
a
oa
• ! tJ(t)dg(t)}
o
+
d(a)g*'(a)
38
Thus, if the Tn of Theorem 2.4.1 are given in terms of the statistics
Tn, k and the assumptions of both Theorem 2.4.1 and 2.5.1 hold, the
k
limiting distribution of n 2 (Tn,T
-~
n (a)) can be obtained through a
n
transformation on the bivariate (S,S). We are thus led to
Suppose assumptions [AI] - [A6] and [AI] - [A8] hold,
Corollary 2.5.1.
with Tn as defined in Theorem 2.5.1.
2
where crO(a)
=
2
~-l
cr (a) - 2PO(a)A(a)A (u)
= (d(a)g*(a)),
A(a)
Then, with X(a) > 0,
+
2 ~-2 ~2
A (a)A (a)cr (a) and
0
g(u)J(a) .
+
Remark 2.5.2.
We may replace [A4] by [A4*] , by Corollary 2.4.1.
Example 2.5.1.
Consider again n
~
1
individuals under a life test and
suppose Xn, 1""'Xn,n are the order statistics of the sample of size n
with common underlying distribution F, with F(O) = O. When sampling is
carried out under the truncation plan the variable Tn = min{l~k~n: Xn,K
,>c}
is of interest. Here O<c<oo. In the notation of Theorem 2.5.1, set
,..,J
f"'V
-1
=
F(c),
t"'owI
g*'(a).
O<a<l; g'l•
=
gn*
= F
;
n
~
t"'owI
'"
1; cn,1. ~ 0 and dn,1. = 1, l~i~n.
~
-1
~2
Also mn(a) = ~n(a) = F (a), n ~ 1, Therefore c(a) = 0 and cr (a) =
2
2
~
~
a(l-u) (d(a)g*l (a)) = a(l-a)(g*'(a)) . Also A(a) = (d(a)g*(a)), =
a
t"'owI
t"'owI
t"'owI
If F has a nonzero derivative at a, then g*'(a)
= I/F-l(a)
>
0
and so we have from (2.5.2)
k
-1
L[n 2 (n Tn - F(c))]
+
N(O,F(c) (l-F(c))) .
(2.5.9)
Alternatively, if we recognize that Tn = Fn(c) , where Fn denotes the
empirical distribution function of the sample, the result (2.5.9) can
be obtained from the well-known properties of the empirical process
k
{n 2 (Fn (t) - F(t));
t~O}.
39
Example 2.5.2.a. Consic.lcr once again n
:2:
1 items under a life test in
which the underlying distribution F has continuous positive density f;
F(O) = 0.
Let Tn(t) denote the number of responses recorded in the time
interval [O,t), O<t<oo.
k:
It follows from the preceding example that
-1
L[n 2 (n Tn (t) - F(t))]
N(O,F(t)(l-F(t))).
7
1 k
T k = n- I X .
n,
i=l n,l
and
+
Consider the statistics
1
n- (n-k)X k'
n,
l~k~n,
(2.5.10)
Tn (t)
T*
()
n,T n t
The quantity
=
nT~,T
n-
1
LX.
i=l n,l
+
n-l(n-Tn(t))t,
(2.5.11)
t>O.
(t) is the total exposure to life of the sample and
n
we are interested in obtaining the limiting distribution of
T~,T
(t)'
n
We begin by analyzing Tn,T (t) using Theorem 2.4.1. In the notation of
n
-1
that theorem, a = F(t), d(a) = (I-a); gn = g~ = F , In = 1, n :2: 1.
Therefore
a
-1
J F (x)dx
fln(a) =
o
+
(l-a)F
-1
(a)
t
=
J {l
o
F(x)}dx :: met) ,
.-
say .
In this case we use the decomposition (2.4.23).
S
=
a .
1
J U(x)dF- (x)
o
and
+
Since
'
(l-a)(F-L(a))'U(a) ,
~(n-lT_-a)
= UCa)
Ii
+
0pCl)
we find that
{-S
reduces to
+
k:
-1
n 2(n Tn -a)(g(a)J(a)
+
(d(a)g*(a)) I)}
(2.5.12)
40
a
{ -f U(x)dF- 1 (x)}
o
=
{-J
t
o
U(F(x))dx}
Therefore, from (2.4.23),
~2
L[n (T
n
2
(t)- met))] -)- N(O,Y (t)) ,
,T
(2.5.13)
n
Further, from (2.5.10) and (2.5.11),
(2.5.14)
1
We shall show n~(X n,T (t)-t) = 0 p (1). Indeed, if {T}
is the general
n
n
sequence of stopping variables introduced at the beginning of this
chapter, we shall show below, for the Brownian Bridge U discussed in the
proof of Theorem 2.4.1, that
(2.5.15)
-1
~
provided O<a<l and n 2 (n Tn-a) = 0p(1) , and the derivative of Fcontinuous in a neighborhood of the point a.
~
n 2(X
n,T
- F
-1
1
is
To prove (2.5.15), consider
- F- 1 Cn- 1T ))
(a))
n
n
- F- 1 (a)) .
.
-1
-1
Note Lhat Fn = F
(f
-1
n ).
,
So we have
(2.5.16)
41
where
Qn(t) = n~(p-l(r~l(t)) - P-l(t)) ,
Q(t)
Now, for O<a<b<l,
= -U(t)(p-l(t))' ,
sup IQ (t)-Q(t)!
[a,b] n
O<t<l .
~~
a.s.
0 as n
(2.5.16)
+
00,
and so in particular
{Q (n-IT ) - Q(n-IT )} -> 0, as n + 00. Finally, since the process Q has
n
n
n
pr
-1
continuous sample paths, Q(n T)
= Q(a) + 0 p (1), as n. + 00. Thus the right
n
hand side of (2.5.16) reduces to Q(a) + opel).
Of course, by the familiar
argument,
Collecting our results we obtain (2.5.15).
reduces to
k
n2(~'Tn(t)-t)
opel).
~
~
L[n 2 (T*
n,Tn
~
Example 2.5.2b.
So in view of (2.5.12) (2.5.15)
Hence from (2.5.13) and (2.5.14),
(t,)-m(t))]
Pinal1y we find the
variable of interest in this context.
v*n = inf{t>O: T*
2
N(O,y (t)) .
+
l~~iting
Define
(,) >
n~Tn L
where O<a<l and
variables
~*(a) =
JP
o
distribution of a stopping
~*(a)}
-1
(a){l - P(x)}dx.
vn = min{l~k~n: Tn, k >
~
-}
Introduce the auxiliary
~*(a)}
-1
.
Prom Theorem 2.5.1, we get n 2(n 'vn-a) -V A (a)S. Here A(a) =
((1-a)p-1(a)), + p-l(a) = (1-a)(p- 1 (a))', and S = JaUdF- l +
(1-a)(p- 1 (a))'U(a), so that n~(n-l~_-a) -0> A-l(a)!aUdP-l + U(a). Prom
n
0
(2.5.15), therefore, we also have n~(X
_p-l(a)) -0> (l-a)-l~UdF-l. A
n,vn
0
42
cursory examination of the functions Tn,T (t)' T~,T (t) reveals that
n
n
X
l
:
'
;
v*
< X
,n
~ 1.
Hence
we
finally
arrive
at
n,v n
n,v
n
n
L[n~(v~-F-l(a))]
->-
N(O ,0*2 (a))
where 0*2(a) = (1-a)-2Ja Ja(sAt - st)dF-l(s)dF-1(t).
o
arbitrary except for O<a<l.
that ~*(a) =
JF
o
Remark 2.5.3.
In practice it can be judiciously chosen so
-1
(a) {l-F(x)}dx may be given a suitable value.
Throughout this chapter we have taken O<a<l as is the case
usually encountered in practice.
-1
The constant a is
0
However, one may wonder what happens when
1 as n ->- 00. If {T k - ET k' O:,;k:,;n} forms a martingale sequence,
pr
n,
n,
as in the case of Examples 2.2.2 and 2.4.1, then by the Kolrnogorov inequal-
n
T
n
->
ity, for every
P{
€
> 0, 0 < 0 < 1,
IT
-ET -T +ET
I > E: (Var(Tn,n ))
k:l-o<n-lk:,;l n,k n,k n,n n,n
max
:,; E:-Zvar(Tn,n -Tn, [(1
n -u~)])/Var(Tn,n ),
1,;
2}
(2.5.17)
and the right hand side can be made arbitrarily small by letting 0 be so.
Thus, when n-IT -> 1 as n
n pr
->-
co?
(2.5.18)
and the desired conclusions can be made by appealing to the results of
1,;
Shorack (1972) on n 2 (Tn,n -
~n(l)).
This argument fails, of course, if
{Tn, k - ETn, k} is not a martingale. In that case, however, we could possibly
establish (2.5.18) by imposing stricter conditions on {Q*}
and {dn,l.}.
~
2.6.
Further results.
Consider again linear combinations of functions of order statistics of
the form
43
T* k
n,
=n
-1 k
I
hkm .
. hex· )
c
i=l n,l
n,l
As before Xn, l""'Xn n denote the order statistics corresponding to a
random sample of size n from a distribution F. The cn,l., l~i~n are
known constants and h is a known fW1Ction. We have remarked earlier that
if g = hoF- 1 and ~ n, 1""'~ n,n are the ordered observations corresponding
to n iid uniform (0,1) random variables, then for each k, l~k~n, T~,k has
the same distribution as does
n-1Ir~lCn,ig(~,i)' Accordingly, henceforth
we shall deal with the statistics
T k =n
n,
k
-1 \'
(2.6.1)
L c
.g(r .),
i= 1 n, 1 ""n, 1
Let G denote the class of all real valued functions of bounded variation on
(s,l-s), for all O<s<Yz, as defined at the beginning of Section 2.3.
For
each gEG we shall also retain the same interpretation on the symbols g
Define I n on [0,1] as before, and let {Tn ; n~l} be the general
sequence of stopping variables, defined in terms of the variables
and Igi.
~n,l""'~n,n which
was introduced at the beginning of this chapter.
Let us define the following bounding functions.
bl , b 2 , cl' c 2 and M >
° let
-b
B(t) = Mt
C(t) = Mt
and
For fixed 0 >
D(t) = Mt
-b
\l-t)
-c
For constants al' a Z'
2
O<t<l
l(l-t) "2
O<t<l
-c
-a
-a
l(l-t)
2
, O<t<l
.
° let
q(t) = {t(1_t)}Yz-o/2,
O<t<l.
Suppose gEG and J be a fixed measurable function on (0,1).
by
We designate
44
[E1] Let Igj $ D,
IJI
$
B, all IJn I
~
B on (0,1).
[E2] Except on a set of t's, not including the point a, of Igl-measure
zero, both J is continuous at t and I n
borhood of t as n +
+
J uniformly in some neigh-
00.
For some fixed number a, 0<0.<1, let us
[Bl]
!<:
desi~late
by
-1
n 2 (n Tn -a) pr 0, as n +
00
,
[B2]
[B3]
Define
)In (x)
and
=
x
rJ J n gdI
(2.6.2)
o
2
x x
cr (x) = J J (sAt-st)J(s)J(t)dg(s)dg(t),
o0
O$x~l.
(2.6.3)
As before, I denotes the identity function and JodI denotes integration
with respect to Lebesgue measure.
The following result is a special case
of Theorem 2.4.1 and Corollary 2,4.1.
Theorem 2.6.1.
Suppose conditions [El], [E2] hold with a l + b =
1
2
a + b = ~ - 0 and JIBq dlgj < 00. Then both )J-r~(o.) and cr (o.) are finite
°
and, if in addition, [B1J holds and g is continuous at a,
2
2
.<
L{~(T
- )J (a))} + N(O,cr 2(o.)), while if [Bl] is replaced by [B2J, or
n,T n
n
.
if [Bl] is replaced by [B3] and g admits a derivative at a, then
L{~(Tn T - )In(n-lT n))} + N(O,cr 2 (o.)).
, n
The main purpose of this section is to consider a suitable estimator of
the asymptotic variance cr 2 (o.) and to obtain its limiting distribution.
We propose the following estimator of cr 2(a).
45
Let r n be the empirical d.£. of E;, n, 1""
,E;,
n,n and for n > 1
define
n
=
-1
I
o
T
n
n
-1
I
T
° ncr
n
(SAt) -fn(S) fn(t))J (f (s))J (fn(t) )dg(s)dg(t)
n n
n
(2.6.4)
Before proceeding further, let us remark here that a result similar to
Theorem 2.6.1 is available for the statistics Tn ,n (see, for example,
Shorack (1972)). Indeed, if [EI], [E2] hold with a 1 + bI = a 2 + b 2 2
~ - 0 and II Bq dlgl < 00 , then both ~ (1),0 (1) are finite and
o
n
~
L{n 2 (Tn,n - .~~(1))}
n
2
N(O,o (1)) .
+
(The reference to the special point a in [E2] may be dispensed with here.)
Once again we wish to consider an estimator of the variance 02 (1) . We
propose to consider 0~ (n>l) , where
0~
=
(2.6.5)
llll(fn(SAt)-fn(S)fn(t))Jn(fn(S))Jn(rn(t))dg(S)dg(t).
In this section we shall obtain the asynvtotic normality of n~(0*2-02(a))
n
and n~(a~-02(1)) under appropria"ce regularity conditions.
will be
rr~in1y
confined to the latter.
additional difficulties we
Our discussion
Since the first case presents no
only provide a brief sketch of its proof.
s~2l1
We begin therefore with the additional restrictions we shall impose per" .
. norma I"Ity
tallllllg
to t h e asymptotIC
[E3]
OlC
n~(A2
0n-O 2 (1)) .
Except on a set of t's of Jgl-measure zero, JI exists and is continuous at t and IJ'I s C on (0,1) with
[E4]
Let us d
'
1
eSIgnate
Dy
n~2
II IJ
o
(f )-J(f ) Idlgl
n n
n
+
° s c1-b1 , c 2-b 2 s 1.
° in probability as n
+
00.
Write
J(l) = IJ,
J 2 = (l-I)J
(2.6.6)
46
and for each tE(O,l) define
(2.6.7)
Let
x x
2
= f f (sAt-st)LO(s)LO(t)dg(s)dg(t)
y (x)
o0
O:s;x:s;l .
(2.6.8)
For fixed 0 > 0, let
pet) = {t(l-t)}~-o ,
Theorem 2.6.2.
flBP dlgl <
o
00.
O<t<l .
Suppose conditions [EI] through [E4] hold with
Then 0 2 (1) and y2(1) are finite and
(Once again the reference to the point a in [E2] can be dispensed with
here. )
Theorem 2.6.3.
flBp dlgl <
o
00.
Suppose conditions [EI] through [E4] hold with
rnen 0 2 (a) and y2(a) are finite and, if in addition [B3]
holds ,
while if [B3] is replaced by [BI] mid g a&nits a derivative at a, then
~
222
(a))} + N(O,y (a)) .
L{n2(0~ -0
Remark 2.6.1.
In Theorem 2.6.3 condition [E4] may be replaced by the
following weaker condition.
[E4*] For some
€
>
a,
e
47
as n
-+ co •
Remark 2.6.2.
When LO of (2.6.7) is integrable with respect to the signed
measure g on (0,1), a more convenient expression for y2(1) can be obtained.
Define GO on (0,1) by
Then a pedestrian calculation will show
It may be interesting to note that ~n2 of (2.6.5) and 0*n 2
of (2.6.4) actually reduce to finite sums. For instance we can write
Remark 2.6.3.
We shall now commence with the proof of Theorem 2.6.2.
First observe that the condition flBp dig I <
Proof of Theorem 2. 6. 2.
stronger than flBq dlgl <
for
0
0
2
2(1) and °2
y (1)
00.
. 1·lCI. ty
SImp
For
respectively.
.
wrlte
0 2,
y2
t I l
+ f tB(t)dlg!{ f (l-s)B(s)d/gj}
1
°
°
°
y
2
easy to show
~
t
= 2(f fI(1-I)}o/2Bq dlgl)2
a
1
°
1
~ 2(flfI(1-I)}~Bdlgl)2
1
2
<
00
•
2(f {I(l-I)}~ILoldlgl) and from (2.6.6) and (2.6.7) it is
o
~
Therefore
·?ly
SlffiP
is
Now
~ f (l-t)B(t)djgl{f sB(s)dlgl}
Similarly
we s h a 11
00
-!-:
{I(l-I)} 4B,
on (0,1) .
48
< ()()
and so y2 <
,
00.
For each tE(O,l), let us define
t
L1 (t) =
6 J(l)dg
and
1
L2 (t) = - { J(2)dg
Then
0.
2 can be written
(2.6.9)
Likewise, define for each n
>
1
t
Ln l(t) = J fnJn(fn)dg
,
0
and
1
Ln ,2(t) = for each tE(O,l).
J (l-fn )Jn (fn )dg
t
Again
211
on :: 0J Ln,ldLn,2 - 0I L.n,2 dLn~l
: ;: - 2
1
J L dL
0 n?2 n;l
(2.6.10)
Hence, after some nllinipu1ations, we obtain
n
k2
-"2
2
(0 -0 )
n
-2(5n, 1
=
+
5n, 2
+
E )
n
(2.6.11)
where
(2.6.12)
1
5n ,2
= ~ L2d{n~(Ln,1-Ll)}'
and
1
En
1
=
(2.6.13)
1
~ (Ln;2-L2)d{n~(Ln,1-L1)}'
(2.6.14)
49
~
Let (Q,G,P) be the underlying probability space and let Un = n 2 (fn -I)
denote the uniform empirical process on [0,1].
Suppose U denotes a
Brownian Bridge process on [0,1] defined on the same probability space.
Let Q denote the class of all normegative continuous functions q on
[0,1] which are bounded below by functions qnondecreasing (nonincreasing)
on [O,~] ([~,l]) and satisfy
Jlq
°
-ZdI <
00,
If P (fl~fZ) =
q
sup Ifl(t) - fZ(t) I/q(t) , it is known that Pq(Un,U) = opel) and
O<t<l
Pq(Un,O)
=
0p(l) , for each qEQ.
Remark 2.6.4.
The probability space (Q,F,P) on which all our variables were
defined may not be rich enough to support a Brownian Bridge process U as
described here to be used in our proof.
However, by one of the usual tech-
niques of embedding (see, for example, Shorack E1972)) we may carry out our
argument in a rich enough probability space where the distributions of our
original variables are preserved and thus the final conclusion of Theorem
2.6.2 remains valid.
Now from our definitions, for each tE(O,l)
1
.J..
1
- 1{2/ (1- I) (J (fn) -J) dg
1:
t
=
J UnJn(fn)dg
o
+ n
~ t
2
J I(J(fn )
°
1
+ n~J
t
°
I(Jn(fn)
- J)dg
We begin with the analysis of Sn, 1 in (2.6.12) using the decomposition
(2.6.15). Now
50
111
dL { J U In(fn)dg - J UJdg} I
a 1 tnt
IJ
~
1
1
f IB
o
dlgl{
= ~n, 1
1
J IU
-UIB dlgl} + f B dlgl{ f IJ (fn)-JIIUnldlgl}
n a t n
t
+ ~n, 2'
1
(2.6.16)
say.
But
~
1
n,
1 ~ Pq(U ,U)! IB dlgl{
1
Bq dlgl}
J
nOt
(2.6.17)
For each tE(O,l)
t
J IB
a
1
J Bq
t
dig I
1
dlgl ~ (1-t)1/4 J (1_I)-1/4Bq dlgl
t
So we have
I I I
f
IB dlgl{
a
f
Bq dlgl}
t
1
~ ( f {I(1-I)}o/2Bp dlgl)( J Bp dlgl)
a
<
00
•
a
n, 1 ->pr a as n + 00. To handle ~n, 2' first note
I uniformly on [0,1], we obtain in view of [E2] that
From (2.6.17) therefore,
~
that since rn +
J n (fn ) ->a.s. J pointwise a.e. Igl, as n
Now invoke Theorem 2 of Wellner (1977).
+
Furthermore,
00.
There is a set Ac0, such that
peA) = 1 and for each wEA, there exists an integer nW for which n
~
n(;J
implies
(2.6.19)
~
~-o/4
where M* is a constant and q = {I(1-I)}2
.
For such
(;J
andn, therefore
51
and so by the dominated convergence theorem, for each tE(O,l) we obtain
(2.6.20)
Also
1
IB{
J IJnCfn)-Jllun!d!g!}
t
~ M*Pq(Un?O){I(l-I)}
0/2
1
Bp(
6Bp dlgl)
.J.
(2.6.21)
The right hand side of (2.6.21) is algi-integrable fllllction.
It
then
follows from Lemma 2.4.1 of Section 2.4 and (2.6.21) that £n, 2 ->pr 0 as
n + co. Returning to (2.6.16) we have shown that
1
1
J dL l { J U J (fn)dg}
o
t n n
1
->_
1
J dL1 { J UJdg}
pr 0
t
,
as n
+ co
•
(2.6.22)
The next term to be considered in Sn , 1 is
This integral is dominated in absolute value by
which in turn is bounded by
(2.6.23)
By virtue of [E4], the expression (2.6.23) vanishes in probability as n
Finally we consider
111
R 1 = J dLl{~f (l-I)CJ(rn)-J)dg}.
n,
0
t
Now
+ co.
52
IJ(fn)-JI/(fn -I) s C(f n
)VC '
Once again invoke Theorem 2 of Wellner (1977).
For some A*c0, with
P(A*) = 1, there exists, for each w*EA*, an integer n W* such that n
implies
~
n W*
(2.6.24)
for some constant M* and
1
R1 =
Then
J dL1{ f
a
q as
in (2.6.19).
Define
1
(l-I)UJ'dg}
t
I~ l- R1 1 s
,
f
+
f
1
a
1
IB dlgl{
1
a
J (I-I) IJ(fn)-J)/(fn-I)-J' I
t
1
IB dlgj {
:: ~ ,1
J (I-I) Iv
n
t
+ ~ , 2'
-vl!J' Idlgl
(2.6.25)
say.
Now
1
1
k 1 s (f {I(1-I)}o/2Bp dlgl){f {I(1-I)}3/4 1 (J(fn)~J)/(fn-I)-J' I Iv !d!g!}
n,
a
t
n
1
'l/4
s M**PqCun,O)b {IC1-I)}J Cq dlgj
Here M** is a constant.
[E3] and (2.6.24).
from [E3].
(2.6.26)
The passage to the second step above utilizes
The finiteness of the integral in (2.6.26) also follows
In view of the continuity of J' we have also proved, by use of
Lemma 2.4.1, that for each tE(O,l),
1
J {I(1-I)}3/4{(J(fn)-J)/(rn -I)-J!}Vn dg ->pr a
t
as n
-+
00.
It then follows that kn, 1 ->pr 0, as n
1
kn 2 s Pq(Vn,V)(J {I(1-I)}O/2Bp
' 0
-+
00.
Similarly,
1
dlg/)(J {I(1-I)}3/4Cq
a
dlgl)
e
53
and so kn, Z ->pr 0, as n + 00. Recalling (Z.6.Z5) we have shown that
Rn, 1 ->pr Rl as n + 00. Combining our results we finally arrive at
S
Il"
'1 ->
pr
Sl' as n -~ 00, where
1
III
(2.6.Z7)
51 = f dL l {f UJdg} - J dL 1 {f (1- I)UJ' dg}
tOt
The analysis of 5n, 2 is entirely analogous and in the interest of brevity
°
we shall dispense with the formal ma.nipuladons here,
as n
+
00,
Thus 5n, Z ->pr 5Z'
where
1
1
5Z = J LZUJdg
°
Furthermore, that
same lines.
J ILZUJ'dg
4-
(Z.6.28)
°
0, as n + 00 can be readily demonstrated along the
pr
Hence from (Z.6.l1) we finally obtain
€
n
->
"z
n (0 -0 2) ->
pr -Z(5 1+S Z) , as n
n
In Lemma Z.6.1 below we shall show S
+
00.
(Z.6.Z9)
-Z(5 l +5 2) where
=
1
5
J ULodg
a
==
,
(Z.6.30)
with LO defined by (Z.6.7). Therefore, with yZ == y(l) given in (2.6.8),
5 has the distribution N(O,y2) and so our theorem is proved.
0
Outline of Proof of Theorem 2. 6. 3.
We proceed very much along the same
lines as in the proof of 111eorem 2.6.2.
n
L~,Z(t)
-1
1"
n(l-fn)Jn(fn)dg ,
== - {
and
n
Define
-1
T
= - J nJ(Z)dg
2
L (t)
t
\.
Then it is quite easy to see that
-1
n 1"
*Z = _Z J nu: dL
on
0 n,2 n,l
and
2 (-1
°
n
,
t)
n
n
-1
T
- ,. L nL*dL
2 1
_'If'
u
54
Therefore corresponding to (2.6.11) through (2.6.14) we have
n 2(on*
!,;
where
2
n
-0
-1
2
-1
(n Tn)) = -2(S*n,l +S*n,2 +s*)
n
T
!,;
S* = J n n 2(L* -L*)dL
n,l
0
n,2 2 1
n
-1
T
1
* 2 -- 0J n L*d{
Sn,
2 n~(Ln, 1- L~ 1)I} ,
and
Also the decomposition corresponding to (2.6.15) is
-1
-1
n TIn T
~(L* -L*) = J n U J (fn)dg - n~
J n(l-1) (In(fn)-J(fn))dg
n,2 2
t
n n
t
1
1
n~
n
-1
T
J n(l-1) (J(fn)-J)dg
o
Examining (2.6.16) it follows immediately that
-1
-1
-1
nT
nT
ill:
~ n dL 1{ {n UnJn(fn)dg UJdg} -pr 0 as n
{n
llilder no restrictions on the growth of T.
n
vided [B3] holds,
n
-1
-T
Jn
o
n
dL1{
--I
-T
Jn
t
a
+
00
,
We shall now show that, pro-
a
UJdg} -> f dL {
pr 0 1
J UJdg},
as n
->-
00
•
t
Observe that we can write
-1
-1
n TnT
J n dL1{ f n UJdg} =
o
t
where X(a,b) denotes the indicator function of the interval (a,b).
in (0,0:) and fix s in (t,a).
Then X
(t,n
-1
(s) --:>r1 as n
Tn)
P
+
00
Fix t
and an
55
a
a
application of Lemma 2.4.1 yields J X
-1 UJdg -pr J UJdg, as n
1
t (t,n Tn)
t
If s > a, then J X
-1 UJdg ->r 0 as n + 00. Hence, finally
a (a,n Tn)
P
1
1
JX
-1
o (O,n T)
n
as n
+
dL l {
00.
a
UJdg} ~>r f dL1{ J UJdg}
(t,n T)
POt
n
JX
t
a
+
-1
00.
Similarly examining (2.6.25) and following a very sjmilar argument we
arrive at
n-1T
In-IT
Thus
+ co.
Sr =
f
pr 0
t
S~,l
a
o
a
f n(l-I) (J(rn)-J)dg} -> J dL1 { f (l-I)UJ'dg}
°f n dLl{~
as n
a
dL 1{
f
t
-pr Sy where
a
a
a
UJdg} - fdL 1 { !(l-I)UJ'dg }
tOt
=
Note that if [E4] is
assun~d
then following the argument up to (2.6.23)
we find that the quant i ty
-1
n fT n d.L (_!Z
m
1
o
n
-1
f
t
T
n(l-I)CJ~(rn)-J(r ))dg}
1i
n
is dominated by the expression in (2.6.23) and so vanishes in the 1hnit
(in probability).
If, however, [E4] is replaced by [E4*] and used in con-
junction with [B3], we will arrive at the same final conclusion.
We shall not connnent on the terms S*n, 2 and s*n since our remarks so far
clearly suffice. So we have S*n, 2 ->pr S*2 and s*n ->pr 0 as n + co with
S2
=
r
LzUJ
b) dg .
We have there fore shown
n!zCcr*2n cr 2Cn -l Tn ))
->
pr
-2(S*+5*)
1 2 ,as n
+
00
•
56
A minor modification in the argument of Lerrnna 2.6.1 will show that this
right hand side limit can be re-expressed as J~Ladg, with the obvious
a
definition of L6 corresponding to (2.6.7).
This establishes the first
part of Theorem 2.6.3.
For the second part observe that we can write
~2
2 -1
2
~ n
n (O' (n Tn) - a (0:)) ::: 2n 2
-1
JT n
0:
~2
But In
-1
n
JTn
0:
L1J(2)dgl ~
~2
Mcf
-1
lg(n Tn) - g(o:)
I,
L1J(2)dg
for some constant
This follows from an argunlent similar to that leading to (2.4.11).
MO·
It
follows from our assumptions that if g admits a derivative at 0: and {Bl]
holds,
o
The proof is now complete.
Lemma 2.6.1.
With 51' 52 and 5 given by (2.6.27), (2.6.28) and (2.6.30)
respectively,
Proof.
From (2.6.27)
1
51 ::: -
.r J(l)dg{ f
o
t
'
1
UJ (2)dg}
Now integrate by parts; we obtain
(2.6.31)
We shall show
(2.6.32)
57
where the limit is to be taken in each of the two cases t
-+ 0+
and t
->-
1-
0
In what follows this is to be interpreted whenever the limit is not exp1icitly stated.
Now for each tE(O,l),
1
IL I (t)J UJdgl ~
(U,O)(
p
q
t
f
1
t
°
IB dlgl)(
J Bq
(2.6.33)
dlgl) .
t
From the analysis of (2.6.17), we find that, for each tdO,l)
(J
1
t
o
IB dlgl)(
J Bq
dlgl) ~
t
1
Since J Bp dlgl <
o
00
we obtain
t
lim
f
t-+O+
Q
and
{I(1-I)}O/2Bp dlgl
=
0 ,
1
lim J Bp dlgl = 0 •
t-+1- t
So, from (2.6.32)
I
lim LI(t)
J UJdg = 0
(2.6.34)
.
t
Again, for each tE(O,l),
1
IL 1 (t)! (l-I)UJ i dgj s
t
p
q
f
(D;O)(
1
t
IB dlgl)(
0
J (l-I)Cq
dlgl)
t
Once again we have the inequality>
1
t
( J IB
°
dlgl)( J (l-I)Cq dlgl) ~
t
( J {I(1-I)}O/2 Sp
t
o
dlgl)
I
x (
J {I(I-I)}3/4Cq
t
for every tE(O,I).
Following the usual argument again we get
dlgl) ,
58
f
lim Ll(t)
1
(l-I)UJ'dg =
a.
(2.6.35)
t
Combining (2.6.31), (2.6.32) and (2.6.35) we arrive at
1
l Ll UJ(2)dg ,
Sl = -
(2.6.36)
and so using (2.6.28)
1
t
2J UJ(2)dg{ J J(l)dg}
a
0
S=
We have thus shown S
Example 2.6.1.
with E!Xl /r <
Let
00
=
f
1
o
o
ULodg.
Xl""'~
be iid random variables from a distribution P
for some r > 4.
A2
an
We consider the sample variance
n
- 2
= n -1 . L\ l(X.
-X)
In
,
1=
where Xn
n -lL~_lXo,
11
Then in the notation of Theorem 2.6.2 we have c n , 1 =
1, for all i, l~i~n, and g = p-l. Since EIX11r < we have, using the fact
=
0
00
that P-loP(t) ~ t for all tE(-OO,OO) ,
(l-t)/P-l(t) Ir
~
~
l/p-J.(s) jrds
t
as t
+
1-.
Similarly tIF- 1 (t)!r
+
0 as
t +
Joo !slrdP(s)
F-1(t)
+
0
0+, and so we arrive at the
~
D on (0,1) with a1 = a 2 = l/r.
Also J = I n = 1. So [E4) holds trivially. Moreover, bl = b 2 = O.
Thus if 6 is given by llr = 1/4 - 6 we have 6 > 0, provided r > 4. By an
inequality Ig/
integration by parts we find that JlBp dig I <
theses of Theorem 2.6.2.
o
00.
This verifies the hypo-
For the situation under consideration LO of (2.6.7) reduces as follows.
59
1
t
LO(t) = 2f (l-s)dF-l(s) -2f sdF-l(s)
0
t
=-2F
-1
(t) +
1 1
2f F- (s)ds
o
~
for tE(O,l).
,
1 _]
For simplicity let us take EX l = 0 so that f F '(s)ds = O. In view of
"
~
0
2
Remark 2.6.2 we may tw,e GO(t) = _{P-L(t)}L ffi1d then we obtain y (1) =
- 04 ~ where
~4
~4
= b~X4 and 02 = EX2
~
'3)
02 l'1) ot- (2 00..
'l'hus our ~. heorem
yields
a result which is obtainable under the assumption r = 4 from the theory
of U-statistics.
In this context Theorem 2.6.2 "just fails" to yield the
slightly stronger result.
Remark 2.6.5.
The previous example presents a very interesting observation
pertaining to the different sets of conditions that suffice to yield the
almost sure (a.s.) convergence of the statistics Tn,n ,its asymptotic
normality and the as~nptotic normality of the estimator 0n2 of its asyn~totic
variance.
We have noted in the discussion following (206.4) that if condi-
tions [EI], [E2] hold with
(2.6.37)
and
I
I
o
Bq
dlgl
<
00
,
then both ~(l) of (2.6.2) and 02 (1) of (2.6.3) are finite and
~
L{n2(Tn,n~~n(1))} +
~n(l)
2
N(O,0 (1)).
2
Now (2.6.37) ensures the finiteness of
and (2.6.38) that of 0 (1) under our
i~damental
assumption [El].
In practice (2.6.38) is verifiable directly by an integration by parts and
60
(2.6.37).
For example, (see Shorack (1972)) if we consider the sample
mean
n
\'
T
= n -1 i=l
L X· = X
n,n
1
n
(2.6.39)
corresponding to n iid r.v.'s XI"",Xn from a distribution F with
Elxl r < for some r > 2, then in our notation J = I n = 1 (so that
bl = b2 = 0) and Igi = IF-II ~ D with a l = a Z = 1/r. In view of (2.6.37),
00
o is
given by 11r = ~ - 0 and so 0 > 0 provided r > 2.
Then (2.6.38)
follows by an integration by parts.
A2
-l\,n
- 2
In Example 2.6.1 we considered the sample variance an = n Li=l(X i -Xn)
and strengthened (2.6.38) by assuming
1
J Bp dlgl
<
o
(Z.6.40)
00
~
which is a consequence of
a1 + b1
= a Z + b Z = 1/4 -
and the assumption that E/xl r <
00
(Z.6.41)
0
for some r > 4.
Wellner (1977) has given
sufficient conditions Uflder which the a.s. convergence of the statistics
Tn, n of (Z.6.1) can be obtained.
Indeed if [El] holds with
(Z.6.4Z)
and
1
J Bv dlgj
o
<
00
,
where vet) = {t(1_t)}1-o / 2, O<t<l, then ~n(l)
(Tn,n-~n(l)) -~.s.
0 as n
7
00.
(Z.6.43)
IS
finite and
Of course (2.6.43) is weaker than (2.6.38)
or (2.6.40). For the particular case of Tn,n given by (2.6.39) with
E/xl r < for some r > 1, we have a1 = a2 = 11r, b1 = bZ = 0 so that 0
is given by Ill' = 1 - o.
00
61
For the "stopped" statistics T
n,T
~
L{n 2 (T
-1
2
the convergence
n
T ))} + N(O,o (a)) can be obtained basically under [El]
n, Tn n
n
and [E2] together with (2.6.37), (2.6.38) and the condition [B3]:
-~
(n
n- 1T - a -> 0, as n
n
pr
+
00,
provided. g admits a derivative at a,
is strengthened by assuming [B2]: n
assume g to be continuous at a.
-1
~n(n
2en
:k
~'l
If [B3]
Tn -a) :: 0p (1), then we need only
In order to replace the "random centering"
c·
!'--1
Tn) by ~n(a) we need to assume further tB1]: n 2 (n
Tn-a)
:=
opel).
' t "he COllSld era'lon
t'
(*2 -0 2 (-1
paL. t ern appears ill
0 f- n ~ \.0n
n Tn))
:k 2 -1
2
-]
and n 2 (0 (n Tn) (a)). FOT the a.s. convergence of (Tn,Tn-~n(n 'Tn))
A very
" 1ar
Slllll
°
we will only need (2.6.42), (2.6.43) together with [EI] and the condition
n -1 Tn ->a.s. a, as n
+
00.
CHAPTER III
ASYMPTOTIC BEHAVIOR OF SOME STOCHASTIC
PROCESSES ARISING IN PROGRESSIVE CENSORING I
3.1.
Introduction.
In this chapter we shall cons:cder the problem of deriving the asymp-
totic distribution of certain stochastic processes that arise naturally in
progressively censored
sch~nes.
To help motivate our development, consider
a typical life or dosage-response study in which n
under observation.
2
I
individuals are
A characteristic feature of such experiments is that the
individual observations are time ordered and it is common practice to cease
experimentation before all the specimens under study have responded.
We
may, for example, conduct the investigation for a preassigned length of time
(truncation) or alternatively, curtcdl the experiment once a pre-specified
proportion of units fyom the sample have responded (censoring).
We observe
that in the truncation scheme the number of responses actually recorded
during the period of monitoring of the experiment is a random variable, whereas in the censoring plml the tline of termination of experimentation is
random.
Under progressive censoring however, one monitors the experiment
from the onset and continuously updates the data at each stage of the experiment with a view towards an early termulation whenever feasible.
For the purposes of constructing a mathematical model, let the random
variables Xl"" ,Xn denote the individual lifetimes of the experimental
subjects. Typically these are nonnegative random variables, and in the
simplest case, are assumed to have a common underlying distribution.
Now
63
the observable variables are the order statistics Xn, l'Xn, 2""'Xn,n
corresponding to Xl"" ,~. If sampling is carried out under truncation,
the experiment is continued for a predetermined length of time 1', 0<1'<00,
and if r*,
O~r*~n,
responses have been collected in this period, it is
reasonable to base statistical investigations on the observed variables
Xn, l""'Xn,r *, if r* ~ 1; for r* = 0, one uses the likelihood function of
none of the observations falling in the interval [O,T]. On the other hand,
if the experiment is to be terminated once a pre-specified proportion r,
l~r~n,
of individuals have responded, the time of tennination is Xn , y' and
one uses, as before, Xn, l""'Xn,r as a basis for statistical inference.
In progressive censoring, however, we allow the possibility of terminating
the experiment prior to Xn,r through monitoring from the onset. Therefore,
if for some k, l~k~n, the data Xn, l""'Xn,(1 advocates a clear statistical
decision, the experiment is terminated following the observation Xn ,k' Thus,
both the stopping number k and the stopping time ~ ,k are random variables.
Accordingly, we conceive of a broad class of stopping variables
{Tn;
n~l}
~
defined on the observables Xn,l, ... ,Xn,n' for each n
1.
Suppose
p(x"11, k;8) denotes the joint probability density function of the vector
X k = (Xn, 1" .. ,Xn, k) of the first k order statistics, l~~n. Here 8 is
a parameter; 8Ee ~ lR i , the real line. We define the progressively censored
"11,
likelihood ratio statistics (PCLRS) by
An , k(u)
l~k~n,
of e.)
e.
and
UE ill
'.
= p(~a, k; 80
+
-~
un- 2)/p(X
-' 8 )
"11,k' a
'
(u may be restricted in range depending on the geometry
80 is the "true" parameter and is in essence any fixed element of
We set An,T = An,k. if Tn = k, l~k~n. In this chapter we shall obtain
n
tit
64
the limiting distribution of II.n T (u) when u is held fixed. The weak con, n
vergence of the random process {lI.n T (u); UEIR'} will also be investigated.
, n
There is some generality afforded by focusing attention on a two-parameter
n ,kn (t); UEIR', tc[O,I]} ~heYe {kn(t); tE[O,l]} is a sequence of
certain integer-valued nondecreasing right continuous functions on [0,1].
process
{lI.
j
We shall find that
11._
n,T n
behaves essentially like
11.
1
n'~n
(1) in large samples.
The exact definitions of these stochastic processes are given in
Section 3.2 together with the necessary notation and basic assumptions on
the underlying distribution and on the sequence {Tn; nd} under which our
results are obtained.
Section 3.4 deals with the weak convergence of an
auxiliary process {Wn,T (t): tc[O,l]} (see (3.2.10) definition).
Some pre-
n
liminary results required for its proof have been placed in the Appendix.
We have also included in the Appendix the fundamental dependent central
limit theorems which form the probabilistic machinery needed for our investigations.
In Section 3.5 we turn to the study of the local asymptotic
behavior of the progressively censored likelihood ratio process
k (t)(u): UEJR', tE[O,l]}, (see (3.2.12) through (3.2.15) for its definin, n
tion). A series of preliminary lemm.ata al~e :first proved before we obtain
{II.
the asymptotic expansion, in a probaDiUstic sense, of the process
K~,k (t) (u),
n
3.2.
Notation, basic assumptions and definition of processes.
Let {Xi;
i~l}
be a sequence of independent and identically distributed
(iid) random variables (r.v.) whose distribution ve on the Borel line
ern. ' ,B) depends on a parameter e, eE8 s::. IR'. We suppose the family of measures {ve : eE8} is dominated by Lebesgue measure ~ on eIR' ,B) and write
fe(o) = dVe/d~ for a version of the Radon-Nikodym derivative involved and
65
Fe (0) for the corresponding distribution function.
Let (IR j ,B), j
;e:
1
be copies of the Borel line and set ()(',A I) : IIj:l (ffi.j ,Bj ) with Pe denoting the product measure of the v e induced on A I . Ee will denote the
expectation taken with respect to the measure Pe .
For simplicity in script denote the k-th order statistic Xn, k of
the random sample
XI""'~
lsksn; ~(O) = Zo =
by Zk' lsksn.
Also write
~(k) =
(Zl"",Zk)'
The a-field generated by ~(k) is denoted Bn ,k'
Osksn, with Bn,a denoting the trivial a-field. Obviously Bn,k-l C Bn,k'
lsksn.
a.
By virtue of the assumed continuity of the underlying distribution
ties among the observations Xi may be neglected in probability, and so for
each 1;ksn the joint probability density function (pdf) of ~(k) can be
written
(3.2.1)
defined on the domain An,k = {~(k): -oo<zl<",<zk<oo}.
The conditional pdf
of Zk given Bn, k- 1~ is
qe(Zk!Bn,k-l) = (n-k+l)f e (zk){l - Fe(Zk)}n-k/{l - Fe (Zk_l)}n-k+l
(3.2.2)
defined for zk > zk-l'
We shall also need the hazard rate
(3.2.3)
For all eEe and XEIR' for which fe(x) >
a ffi1d
Fe(x) < I we have re(x) > O.
It will prove convenient to set
(3.2.4)
In the context of life testing Ge is called the survivaZ function. For
any measurable non-negative function he(o) let he(x) = a~(IOg he(x)). We
now state the basic assumptions which will pervade our entire development
66
:in this chapter.
We shall exam:ine them more closely later and give suffi-
cient conditions for their validity.
Assumptions.
[AI]
The parameter set 8
the real 1 ine IR I
[A2]
an open interval, bounded or unbounded, of
IS
•
The set {ZEIR I: f e (z»O} is independent of eEe.
by X.
[A3] (a)
We denote this set
For each XEX, fe(x) is differentiable with respect to e, on 8.
Clf (x)
(b)
e
For each eE8, fe(x) and -38---are continuous with respect to
x on X.
[A4]
For each eE8
k, ls::ks::n, and
Cl
ae f
x,.
.
I8(z)d~(z) =
-00
[AS]
x df e
J
-ae-(z)d~(z) , for each XEX .
-00
There exists some 8 > 0 such 'i:hat for each 8E8,
28
Ee!r'e(X) 12+8 < co and E'8 1If·-e (X) 1 + <co.
[A6]
d
0
For each 8E8, dX Y8(x) is continuous with respect to x on X.
Let {Tn;
n~l}
be a sequence of stopping variables such that for each n
~
1,
Tn is adapted to (Bn,k)~l; that is, for each n ~ 1, Tn takes values in the
set {I, ... ,n} and the set [T_=klEB_
k for every k.' ls::ks::n.
- a
11,
67
For each eEe, there is a number a,
[A7]
n
~ co
O<a~l
-1
for which n Tn
~
a as
in Pe-probability .
We shall discuss these conditions further in Section 1 of Chapter IV.
Remark 3.2.1.
From (3.2.4) observe that, for all eEe and XEX,
x
Ge(x) = 1 -
J fe(z)d~(z)
.
-co
Hence from the continuity of fe(o), we have for each XEX,
aGe (x)
ax
-----;::-_::: - f (x)
for each eEe.
e
On the other hand using [A3](a), [A4](b) gives
aGe (x)
ae
:::
x afe(z)
-J ae d~(z), and so
2
a Ge(x)
axae
-co
for each XEX and eEe.
Hence
::: re(x) (Ge(x)
= -re(x)re(x)
fe(x))
, from (3.2.3).
We also note here that a~ Ge(X) is continuous in x, for each eEe.
To formulate the stochastic processes of interest in this chapter,
let
=
(k)
~,k - ~n,k(~
and
~n
_.
,e) - Pe(~
, 0 ::: O. For each eEe define
(k)
,n),
(3.2.5)
68
2
J n, k(8):: E8{t~n, k} , for hksn
and J n, 0(8)::
Note that J n, k(8) is nondecreasing in k for each 8Ee.
of standard notation we set, for each 8Ee,
°.
With a slight abuse
Now if
(3.2.7)
and ~* :: 0, write 0*2 (8) ~ E {~*2 IB
n,O
n,k
e n~k n,k-l } and define
V k
n,
= Vn, k(8)
~
k
2
I 0* "
i=1 n,l
(3.2.8)
Introduce a sequence of random integer-valued nondecreasing right continuous
functions
{~(t):
tE[O,l]}, by
~ (t)
The
~ (t)
::
~ (t)
:: max{k: V keG) s tJ
n,
n, Tn
may depend on the choice of 8Ee.
(8)}.
(3.2.9)
We are now in a position to
define several processes which we wish to investigate in this chapter.
Let
1'1
1",
,_ (t) :: ~
n, In
2
n, k (o_)/I
L n, _[ (8), Ostsl .
n
(3.2.10)
11
For each n ~ 1 the sample paths
of the random process Wn,T ::: Wen,T
n
n
{Wn T (t): tE[O,l]} belong to the space D[O,l] of all real valued right
, n
continuous functions with left hand limits on [0,1], equipped with the
usual Skorohod topology.
In our definition "8" is held fixed in e.
If
W:: {W(t): tE[O, I]} denotes the standard Brownian ivlotion process in D[0 91],
we shall prove, (Theorem 3.4.1) when
e obtains,
Wn
-> Win D[O,l]
,Tn W
(3.2.11)
69
Provided n -ITn ->~ a.E: (0, lJ as n
-+
The symbols -> ,-> denote convergence
00.
~
~
in probability and in mean; -> denotes weak convergence.
w
gences here are under the probability measure
P8
with
8
All the conver-
held fixed in 8.
With a view to defining certain likelihood ratio statistics let 80
be a fixed but otherwise arbitrary element of 8 and consider the sequence
n:::::l} where
{8'
n'
(3.2.12)
For 8 0
-!.:
+ un 2E:8
and for each k l
An, k(u)
If 8 =
=
l~k~n,
define
P8 (ZCk)
~
, n)/p 8 (z(k)
~
, n)
n
(3.2.13)
0
we consider the two-parameter random process
:IT{'
{A
8
tdO,l]}
(u): udR',
(3.2.14)_
n,~O Ct)
where kn Ct) is defined in (3.2.9), but now we give 8 the fixed value 8 0 ,
Whenever k Ct) is used in the context of the process (3.2.14) it would mean
8
0
n
8
kn a(t). We can then dispense with some awkward notation. The definition of
An,k Ct) Cu) is not quite complete if
e
ef
We are assuming here
lR I .
n
=
(a,b).
TI1en we define a modified process
DC
e
(u):
UE]\
tdO,lJ}
I,
n,k OCt)
n
as follows:
A
(u)- A
8
n,k OCt)
n
8
n,kn
oCt)
(u)
!.:
n 2(b-8 )) aJld
0
A 8
(u)
a
n,kn (t)
=
a,
(3.2.15)
•
70
e
Here again we abbreviate k 0 (t) by
n
In Theorem 3.5.1 we establish that, under the probability measure
and continuous in all other intervals.
~ (t).
'
a
the finite dlinensional distributions of
{~
k (t)(u): UEIR', tE[O,l]}
n
converge to those of the process {A(u,t): UElR', tE[O,l]} where
Pe
'
(3.2,.16)
and
(3.2.17)
Some alternative expressions for J a (e) are worked out in the Appendix.
3.3.
Preliminary lemmata
In this section we state a few basic results required for the proof
of (3.2.11).
Some of them are based on auxiliary arguments which we have
relegated to sections 1 and 2 of the Appendix.
Throughout this section
is to be considered a fixed but otherwise arbitrary element of
Lermna 3. 3. 1.
For ever./v n ~ I,- {~-c'n , -,
B _}:n a
k n , lC .1(=
lS
e
e.
a zero mean martingale
under the probability measure Pe'
Proof.
From (3.2.1) and (3.2.2) it is immediate that
and so, by our definitions (3.2.5) and (3.2.7), also
(3.3.1)
Now for each k,
l~k~n,
71
(3.3.2)
and
Assumption [AS] assures the Pe-integrability of
~~,k'
To establish the
lemma we need only show Ee(~~ ,klBn ,k-l) = 0, for each k, l:::;k:m.
Now
00
Ee(~~ klBn k-l)
=
"
f qe(zlBn k-l)qe(zIBn k_l)d~(z)
zk-l'
,
00
J 3~{qe(zIBn,k-l)}d~(z)
=
zk-l
00
=
a~ J qe(zlBn k-l) d~(z)
zk-1
a
= ae
{l}
'
°,
0=
where the interchange of integration and differentiation is validated by
[A4](a).
0
Corollary 3.3.1.
For a sequence of stopping variables v n adapted to
{Bn . k : l:::;k:::;n}, Ee(~n v ) = 0 •
,
, n
V
Proof.
',~
r-
.Gell:.n., vn
)
_. E (
e
I
n
E,* .)
3-=1 n,l
vn
= Ee (i=l
I
=
Remark 3.3.1.
_
Ee(E,* ·IB
°.
_.
If eO,elEG and An,k -
n,l n,l·-1))
o
P~(~
(k)
,n)/Peo(~
(k)
,n), l:::;k:::;n, then
for each n ~ 1 {An ,k,Bn ,k}~=l is a martingale Wlder the probability measure Pe ' even when [A4] (a) may not hold.
a
72
Lennna 3.3.2.
For any sequence of stopping variables vn adapted to
-1
{B k; l~k~n}, for which n v -> a*E(O,l] as n + 00,
n,
n pr
(3.3.3)
where the convergences hold under the probability measure Pe.
v
Now Vn,v (e) = Li=10
\' n *2, i
(e)) an.d 0 *2,i (8) -- Ee (r*2
"'n,i IBn,i-l )
n
11
Proof.
r-
n
Therefore from (3.3.2)
(3.3.4)
The first term of (3.3.4) we re-express as
and the third conditional expectation in (3.3.4) can be rewritten as
Ee((n-i+l)(Ye(Z.)Ge(Z.)-re(z.
l)G e (z.1- 1)) IBn,l. 1)
1
1
1-
n~i~l,
For each
let Yl""'Yn-i+1 be
i~d
random variables with distribu-
tion function
if
z
~
Zi-l and zero, otherwise.
The corresponding density function is
Now the (conditional) distribution of Zi~ given Bn,l·-1 is the same as that
73
of min {yl' ... ,Yn _i +1}.
Appealing to Result 1 of the Appendix we have,
for each n2::i2::1,
convergent equivalent almost surely (a.s.) to
Similarly E ((n-i+1)(T (Z.)-T
(z.l- 1)) IBn,l. 1) is convergent equivalent to
l
e
e
e
and Ee ((n-'i +l)(r e (Zi)(;e (Zi) -re (Zi -l)(;e (Zi -1)) IBn,i -1) is convergent equivalent a.s. to
a·
-1
.
{r e (x)ax(re(x)Ge(x))}x=Z.
l-l
a•
-1'
= {r e (x)(re(x) ax Ge(X)
'2
= -r 8 (Zi_l)
Finally Ee
-1·
+{r e
•
a•
Ge(X) ax re(x))}X=Zi_l
d •
(x)Ge(X)dX re(x)}x=Zo
({r e2(z o) - re2(z.
l
+
l-
l-l
l)}jSH,l"
. 1) must be convergent equivalent a.so to
zero.
Hence we find that for each n2::i2::1, 0*2.
n,l (8) is convergent equivalent
·2
-1
to r e (Zi_1) a.s. Thus n Vn v (8) is a..s. convergent equivalent to
, n
-1 \,vn 2
n Li=lr 8 (Zi-l).
Summoning Result 2 of the Appendix we get
o
Corollary 3.3.2.
n-1J
(8)
n,vn
+
Under the hypotheses of Lemma 3.3.2, also
J *(8) , as n
ex.
+
00.
74
Lemma 3.3.3.
If the sequence of functions {kn(t): tE[O,l]} is defined by
(3.2.9), then for each eEG and tE[O,l]
(3.3.5)
under the probability measure Pe ; at is given by
Proof.
By definition ~(t) ~ j if, and only if
Fix eEG and tE[O,l].
V . s tJ
. The case t == 1, a :::: 1 is trivial. In what follows we
n,]
n, Tn
assume this is not the case so that O<a~<l.
For arbitrary E >
L
°,
and
Therefore
,
(J
-tJ )] + P [n
a +E
a
e
t
- (J
a
t
-E
-tJa )
~
- (Ja
t
-E
-tJa )]
-.1
(V
n [nea -d]+l
'
t
- tnJT )
' n
(3,3.6)
+ - tJ > a and J
- tJ", < 0. It then follows from Lemma
at E
a
at-E
~
3.3.2 and its Corollary that each term on the right hand side of (3.3.6)
Note that J
converges to zero as n
+
00.
Hence (3.3.5).
o
75
3.4.
Weak convergence of Wn,T .
n
We now have the technical machinery necessary to establish the weak
convergence of Wn,T to standard Brownian Motion W in D[D,l].
n
Theorem 3.4.1.
Suppose conditions [Al]-[A7] hold and let Wn,T =
n
(t): t€[D,l]} be the stochastic process defined in (3.2.1D).
{W
Then
n,T
n
for each fixed 8Ee, W
-> Win D[D,l] under the probability measure P .
8
n,T n W
Proof.
[D,l].
Throughout the proof 8 is a fixed element of 0 and
t
is fixed in
By definition
k (t)
W
n
T
, n
(t)= ~n
n
1
L
,on (t)/~ ' Tn (8) - i=l
k
1
~* ./~
n,l n, Tn
(8)
To establish the theorem it suffices to verify conditions (a), (b) and (c)
1
of Result 5 of the Appendix, for the normed array {~*
n,l. /pn,Tn (6): l::;iskn (t)}.
Now condition (c) follows trivially in view of Lennna 3.3.1. To verify
condition (b) recall Lennna 3.3.2 and its corollary.
k (t)
J
n,T n
(8)
n
L
i=l
2
·18 . 1)
u,l n,l-
E8(~
=
J
-1
n,Tn (8)Vn, kn (t)
By [A?], n -lJ
(8) -+ J (8) and using Lernrna 3.3.3 also n -IV 1 (J_) -L>
n,Tn
a
n,Kn~L
1
tJa, (8) . So condition (b) obtains. We are left with verifying condidon
(a).
To this end we utilize [AS].
Observe that for every
The "8" appearing here comes from [AS].
each i, lsi::;n
E
> 0, lsisn
In view of the above we have for
76
k (t)
n
2
1
I
s*
.I(ls*
.
I
> €~
)) ~
e i=l n,l
n,l
n,Ln
E (
€
-0
(n
-1
I n L)
-0/2 -0
, n
n
-1
Since n J n,L (e) + J ex (e), the Lindeberg condition (a) of Result 5 will be
n
verified once we show
<
00
(3.4.1)
0
Recall (3.3.2).
and so we have
~* .12+0
I "'n,l
<_
o
20+1{l r e
(Zi )1 2+0 +
I(n -1.+])(Go
.
e (Z)
i
2 0
- G· e (Z i -1 ))1 + } '
(3.4.2)
Now surrrrnoning Result 1 of the Appendix and following the argLUIlents given in
Lemma 3.3.2, we see that for each
n2i~1,
is convergent equivalent to f(3+0)1}~e(Zi_l)12+0 a.s,
trarj
n > 0 and
rlnerefore, for arbi-
n sufficiently large,
k (t)
n
n- 1Ee ( I l(n-i+1)CG e CZ,) - GeCz. ~) 12+0)
. 1
1
1-L
1=
k (t)
n
r·
2 0
.
•
= n -1Ee ( L\' EeC I Cn-1+1)(G
(Z.) - Ge(Z'_l)) 1 + I~ "-1))
e
i=l
1
1
. n,l
~ n
-1
k Ct)
n
\'
,.
12+0
Ee ( i~l f(3+0) reCZi_l)1
+ nn)
~ fC3+0) Ee lr e LX) 12+0
+
n.
C3.4.3)
77
It then follows from (3.4.2) and (3.4.3) that (3.4.1) holds and so the
proof of the theorem is complete.
Remark 3.4.1.
o
We have proved that for the normed array
1
{~* ./J~
n,l n,Tn (e): l~i~n}, the conditional Lindeberg holds, that is, for
every E > a
(3.4.4)
in P -probability, for each GEG.
e
3.5.
Limiting distribution of A k (J )(u).
n, l1-'-'-'c_'---_
In this section we shall derive the limiting distribution of the
statistics An,ku(t) (u), for each pair of (u,t), with UEIR' and tE[O,l]
such that e +
un-~€0.
For the definition of An,kn(t)(U) see (3.2.12)
through (3.2.15).
We emphasize here that the kn (t) referred to in this
section are given by (3.2.9) with G fixed at the value 80 , With this
G
remark the cumbersome notation kna(t) will be interpreted for ~ (t). Thus
~ (t) = rnax{k: V "k(e ) ~ tJ
n,- O
n,Tn (8 0) L The basic assumptions [AI] through
[A?] of Section 3.2 will continue to hold. In addition we shall impose
the restriction,
[A8]
For each UE JR ' ,
...
78
Our first main concern of the present section is to establish that, under
[AI] - [A8], the finite-dimensional distributions of the process
(u,t)
+
An,k (t)(u) converge to those of (u,t)
A(u,t), where A(u,t)
+
n
was defined in (3.2.16).
.
o
All convergences are valid under the probability
Note that in view of [AI], for any
measure Pe
sufficiently large
11
eO+OO
IR i
UE
-!,;
2E
8 for
and thus the quantities under discussion are meaning-
fUlly defined.
Let us introduce the notation
n~, i (u)
for all l:::;isn and UE lR' with eO
+
un
-!,;
2
We first derive several auxil-
E8.
liary results for the array {n*
.: l:::;i:m}
TI,l
In what follows u and tare
held fixed and probabilities are evaluated illlder Pe
. In addition to the
a
basic assumptions [AI] - [A7], the condition [A8] will be assumed to be
valid.
Lemma 3.5.1.
FOT any UEIR
i,
tE[O,l]
'. (t)
tn
'
\
2
limsup E ~ L n*~.Cu)~ s ~u tJ~(80)
n-+OO 8 0 l i = 1 n, 1 )
'-"
Proof.
Since [kn(t)
kn (t)
2
i]EBn , i=l' for each lsisn, we have
kn (t)
,.,2
\
E { I
*2 . ~ = E {-- L E ( " . lB. ) ~
8 0 i=l nn,l)
80
i=l 80 ~,1 n,1-1 J
\
.
(3.5.1)
Now
Eea(n*2;
n,..L IBn,l'-1)
=
!,;
2
- q~o(ZIBn,i~l)] dv(z)
(3.5.2)
79
qe~ (zlBn i-I) - qe~ (zlBn i-I) = ~
n'
0'
Jen
eo
qe-~2 aea {qe(z IBn i_l)}d~(e) .
'
Hence, with the assistance of the HC51der inequality, we have for each
(3.5.3)
Hence, from (3.5.2) and (3.5.3) we have
-~
e
I E (n*2· 18 ·-1) ~ ~
Je n Vn, kn (t)(e)d~(e) ,
i=1 e0 n,l n,l
~
o
kn(t)
(3.5.4)
and thus using (3.5.1) and Fubini's theorem, also
k (t)
n
E
eO
Let eE8.
(
L
i=1
11*2;)
Il,1.
(3.5.5)
80
(3.5.6)
where
a~o
qe (ZIBn ,i-1) =
o
(a~
qe(zIBn ,i-1))e=e
o
.
The following
sru~le
inequalities will be used repeatedly in the sequel.
For a, bE JR' and
y > 0,
labl
~
2
ya /2 +
b2/2y
,
and
(3.5.7)
Therefore
+
(qe-~o(z IB·h,l
n • -1)
a {qe lZ
~. I
})2 ] .
'\'ea 0
0 B.n,1. -1)
The second tenn on the right hand side of (3.5.8) is
and the first tenn can be rewritten
s~ly
(3.5.8)
81
Therefore from (3.5.6) and (3.5.8) we get, for any e such that
le-eol s; luln-\
(3.5.9)
Write yn(u,e O) for the expectation appearing in [A8], and set Y = 2 In
the above inequality. We obtain
a s;
-1
n J n, kn (.L._) (e) s;
-1
n (u,8 0) + 3n J n, kn (t) (eo) ,
By
for le-eol s; luln-~.
Since yn(u,e O) + 0, by assumption and n-1Jn,k (t)(e o)
n
converges by the corollary to Lemma 3.3.2 ffi1d Lemma 3.3.3, we see that for
each uER
I
-1
sup
k {n J. 1, ( ) (e)}
le-eol~iuln-2
n'~l t
is bounded in n.
-k
Invok]11g the inequality (3.5.7) with Y = Y 2(u,8 0 ) we
n
obtain
+
y~(u,eo){
!2
+
Hence we obtain, for any UE ill. I
s~
-k
le-eols;luln 2
-1
n-1Jn k (t)(e)}
, n
Yn(u,eo){n In,~(t)(eo)}'
82
and
Therefore we have established
o
for each UE:JR " tE [0,1] .
Remark 3.5.1.
Lemma 3.5.2.
A stronger result in this connection will be obtained later.
For each UEIR' and tE[O,l]
I
e0{ i=l
lim E
n fOO
Proof.
k (t)
n
d
Cr£·,1 (u)
Here ~*n,l" = ~e
(log
a 0
2
_~
.. !-zun 2t;,,! .J } :;: 0 .
n;~
qe 0 (Z,"- IBn,l_
"-1)). Consider the identity
83
Hence
-
- 'led
o
-> 0 ,
0
d
~e
o
k
2
qe
(ziB ·-1)]
0 0
n,l
2
d~(e)}
qke2 (z lB. -1) ] 2d]1 (z) }
0
n,l
o
for each fixed UEIR' and tE[O,l].
Remark 3.5.2.
The preceding argument contall1s the proof of the statement
k (t)
n
L E {(n* 1·
i=l e0 n,
2
1
-
~-~~*
n,
1
0
)
IBn,l·-l}
+
0
(3.5.11)
in Pe-probability, for each UEIR' tE[O,l].
Lemma 3.5.3.
For each uEIR'and tE[O,l]
max
l~i~k (t)
n
Proof.
Let
€
> 0,
In*
.(u) 1+ 0 in Pe -probability.
n,l
0
n > 0 be arbitrall' We have (see Dvoretzky
(1972))
84
kn(t)
P
[ max
In* "I > c:] ::; n + Pe [ I Pe ( I~ i
80 l::;i::;k (t) n,l
a i=l a '
n
I
>
c: IBn , i -1)
;>
nJ .
(3.5.12)
Now
k (t)
n
I=l
i
k (t)
Peo(!n*,l"
I
n
n
1
I
i=l
> siBn,l-i
0')::;
.- ~lnl-~~
o!
0 Cln*
n 1
11,1
Pe
>
c:/zlsn,l. 1)
k (t)
n
Yz
\' P (IE/-.j >~n~_
i~l 8 0
n,l
lui
+
18
)
n,i-l'
By Chebychev's inequality and (3.5.11), for UElR
k (t)
n
\'
i~l
eO
in Pe -probability.
o
k (t)
n
IP
i=l
(In*. -
P
e0 (, ~*n,l. I
n
1
1
Yzun-~~*
n
tE [0 ,1]
I,
. I > c:/2IB
.
n,l-
1
(3.5.13)
1)
To handle the second term in (3.5.13) we write
~
>
kn(t)
~un2i8
°-1) =
I E (1(IE*.o
I
I~I
n,l
i=l e0
~,l
::; c:
k (t)
n
- 2 2 -1 \'
P
("* 2 I ( I "oJ:
L.c e ':>
..
i=l
0 n,l
u n
':>
n, i
I
Yz
c:n )1 B
)
> -u,I n , i -1 .
(3.5.14)
We have seen earlier, in the proof of Theorem 3.4.1 that the conditional
Slllce n-1J
(8 0) + J (eO) as n + 00, it follows
n,T
a
n
that the right hand side of (3.5.13) converges to zeroin P -probability.
Lindeberg (3.4.4) holds.
Since
n > a is arbitraly
Lennna 3. 5. 4.
eo
tile lemma now follows from (3.5.12).
For each liE R' and tE [0 , 1]
kn(t)
\' no"
*2. _
L
i=l n,l
~ 2tJ~
1
(8)
~..0
+
a,
in Pe -probbl·
a i lty .
a
0
85
Proof.
1
-1<::*2
I]
wn
s
n,l
2
0
k Ct)
+
Cl/YE)E e {
o
2
I
Cn*
i=1 n,l
n
0
+ ~u
2 -1 2
n ~* :)} ,
n,l
(3.5.15)
where we have used Chebychev's inequality in the first step and the basic
inequalities (3.5.7) in the second.
Here Y > 0 is arbitrary at present.
Now from Lennna 3. 5. 2 we have at once
k (t)
n
E
n
=E { I
eo
(n*
n,l
0
i=l
\un-~~n* lo)2}
-
,
Also
n
-1
+
k (t)
n
I E (s*
o i=l e0 n
Ee (
- n-1Jn,kn(t) ceo)
as n
+
00
•
Finally from Lemma
0, as n
+
2
olB
1
n
right hand side converges to zero in the limit as n
I
i=l
Cn~
2
'
+
00.
Then the entire
It follows that
in Pe -probability
o
i
for each UEJR' and tE [0,1].
0-1))
1
3.5.1~
-k:
n
00.
tJace O)
Now we select y = En 2 in the inequality (3.5.15).
k (t)
+
Further
(3.5.16)
86
In view of the Remark following Result 5 of the Appendix we have
kn(t)
L n-l~*2.
i=l
°
tJ (eO) , in Pe -probability
a
+
n,l
for each UEffi' , tdO,l].
(3.5.17)
The lennna then follows from (3.5.16) and
(3.5.17).
0
Remark 3.5.3.
The arguments presented in the preceding lemma also lead
to the result
~(t)
° n,
L Ee (n*2 1· IB ·-1)
i=l
n,l
+
~2tJ~(eo) ,
in Pe -probability, for each UE ffi' and tdO, 1].
°
recognize that, for any
°
Pe
[
k (t)
n
L
i=l
£
To show this we need only
> 0,
2
(n* . IB.
° n,l
(E e
(3.5.18)
~
I
) 2 -II:: (c* 2 B
))
~ n LeO sn,i n,i-l
. -1)
n,l
I
>
£
]
so that
~(t)
k (t)
_
\' E
(*2. IB
L
en
i=l
° n,l
. 1)
n,l-
J
2n -1 nl'L
-!4U
i=l
° n,l IBn
Ee (c*2
s·
)
. 1
1-
converges to zero in Pe -probability, for each UEIR', tE[O,l].
o
course
n
k (t)
n
1
\' E
L
i=l
°(c*2·
n,l
e
S
.,
1
n,l. -1)
=
n
°
°
-IV
(e )
tJ (e )
- k (t) 0 ->
11'·'n
L1 a
under the probability measure Pe -again, for each tE[O,l].
claim (3.5.18) is established.
Of
Thus our
87
Remark 3.5.4.
From Lemma (3.5.4) and Fatou's lerrnna we have
~(t)
( I n*2.) ~ ~2tJ (eo) ,
1iminf Ee
n~
a
i=l n,l
for each udR' and tdO,l].
a
Therefore it follows from Lemma 3.5.1
that
(3.5.19)
for each UER' and tE:[O,l].
Lennna 3.5.5.
For each UER' and tE:[O,l]
in Pe -probability.
° We first observe that for each b;b;n,
Proof.
={
qe (2.
n
lB.
1)
n,lqe (2 ·1 B . _J
1
n,l-l.
1
°
°
I} - 2n*n,l.
so that Ee (n*2· IB . 1) = -2E o (n* . i8 . 1)'
n,l n,lVo n,l' n,l-
In view of (3.5.18)
it follows irrnnediate1y that
kn(t)
1
I
Ee (n* ·18 .) ~ - -8 u2tJ~(SO)
i=l 0 n,l n,l
~
, in Pe -probability
0
for each uER' and tE [0,1].
Let
€ >
0 be arbitrary.
kn(t)
PSo [/ i£1 n~,i -
Then
k (t)
lin
zun-Y;2
if1
-/
+
i
2
u tJa(SO)! >€]
(3.5.20)
88
kn(t)
[I L (n*" o i=l n,l
~ Pe
+ Pe
k (t)
n
L
o[I i=l
-k
~ 2~n*,1'
Ee (n* "IB "-I)
0 n,l n,l
- Ee0 (n*n,l·IBn,l"1)1
1
> E/2]
2
(3,5,21)
-8 u tJ (eo)1 > E/2] .
a
+
By virtue of (3.5.20) the second probability on the right hand side of
(3.5.21) vanishes in the limit as n
-;>-
co.
To handle the first tenn we
observe that if
r*
= n* ~n, i n , i
~
2
-k
2"*
(n* in,
I B i-I ) '
"'n, i - EeOn,
].-<1"<n
then, recalling Lemma 3,3.1, we may conclude that {r*
B 'l~i~n} is
~n,i' n,i'
a centered sequence and so if
k
r
~n
-
,
r*
k -. L\ ~n
i '
1=1
\k r*2 . Bn k'"l~k~n}
'
2
then {I;;n,k
Li=P n, 1"'
l~k~n
'
1S
a zero-mean martingale.
There-
fore
The first step uses Chebychev's inequality; the second, the martingale
property referred to above.
kn(t)
E
eO
(
L
i=l
1;;*2 .) =
n,l
Now
89
o
And so from Lemma 3.5.2 the result follows.
We shall now formally establish the main theorem of this section.
Theorem 3.5.1.
Suppose the progressively censored likelihood ratio
statistics An,k (t) (u) are defined by (3.2.13) for each UEJR' , tdO,l]
n
such that eO + un
-k
2
E8, and
Sl~pose
conditions [AI] - [AS] hold.
Then
A k (t)(u) can be expressed in the form
n, n
where, for each pair of (u,t), 0
n
=
0 (eO,u,t)
n
Moreover, under the probability measure Pe
o
~
a in
Pe -probability.
0
the finite-dimensional dis-
tributions of the process {Kn,kn(t) (u): UEIR', tE[O,l]}, defined in
(3.2.15), converge to those of {A(u,t): UElR', tE[O,l]} where
Proof.
Since
k
An, k = i=l
IT {qe 0 (2·IB
. l)/qe 0 (z·IB
. I}}
1 n,l1 n,lfor each k,
l~k~n,
and by the definition of n*TI)l. we have
1<:n (t)
log A 1 (t)(u)
n,Kn
By Taylor's theorem, log(l+x)
o<
A < 1.
and so
x -
'7
x~
~-
"I
+
Take Ixi < s, with 0 < s < 1.
I (1+Ax)-3/ ~(1-s)-3.
to ensure (1-s)-3
where IA
~
= 2 i=l
I 10g(1+n!'il,1.) .
<
3/2.
xJ
~(l+Ax)
-3
(3.5.23)
for Ixl < 1, where
Then Il+Ax/ ~ l-Alxl ~ l-s
We may choose s sufficiently small in order
Therefore, we obtain, for l~i~n,
. I < 1 and max In* ./ <
n,1
1<'<
-1-n n,1
E,
with s sufficiently small.
Now
90
kn(t)
2 I 10g(1+n*·)
i=l
n,l
kn(t)
= 2{ I
i=l
k (t)
n
- { i=l
I
n~ i
'
2
n* .
n,l
(3.5.24)
In view of Lemmas 3. 5.4 and 3.5.5, the first two terms on the right hand
side of (3.5.24) vanish in Pe -probability.
o
kn(t)
kn(t)
I 1=1
.L
Also
An
'
iln~ , il31 ~
{
max
l~i~k (t)
n
and so by Lemmas 3.5.3 and 3.5.4
In*. I}{ I n*2.} ,
n,l
i=l n,l
the third term in (3.5.24) also vanishes
in Pe -probability. Therefore, if 8 = 8 (eO,u,t) denotes the sum of
o
n
n
the first three terms on the right hand side of (3.5.24) we have proved
that for each pair of (u,t) , on
-+
e -probability.
0 in P
(3.5.23) and (3.5.24) we obtain
Hence from
o
and thus the first parL of the theorem is proved.
To establish the second part of the theorem it suffices to show that
under the probab il i ty measure Pe
o
r,s
L[
a· .log 71. k (t )(u.)]
i,j""l 1J
n, n i
J
for arbitrary aU"" ,arsElR' ,
r,s
~
1.
r,s
I
Now
U
j
->- L[
I
a .. 1og A(u.,t.)]
1,3=1 1J
,." ,USElR i
J
1
and tl"'" trdO, 1], and
91
r S
t
a .. log A ,k (t.)(u.)
i , j =1 1J
n n 1 J
1 1
r,s
8
= n-'2J '2
(8 )( L u.a ..W 0 (t.)) . . lJ 1J n,T n 1
. n,T n 0 1,J=
a
rfs
2
L a .. u.t.)
0 l',J'=llJ J 1
J-.-;J (8 )(
r,s
L
+
0 (8 ,u.,t.) .
0 J 1
. . 1 n
1,J=
It follows from Theorem 3.4.1 that the finite dimensional distributions
8
of W a
converge to those of W.
n,T n
L[
Hence under P8 '
a
eo
rfs
u.a .. W
(t.)]
i,j=l J 1J n,T n 1
L
-r
L[
r,s
L u.a .. W(t.)].
i,j=l J 1J
1
!z
-h: h:
Finally, since n 2J 2 (8 ) -r J (eO) and 0 (8 0 ,u.,t.)
n,T n 0
a
n
J 1
ity for each u.,t., it follows that
J
-r
a in Pe -probabila
1
r,s
L[
I a .. log An, kn (t i )(u.)]
i , j =1 1J
J
-r
1
r,s
r,s
2
L[J~(eo). I. 1 u.a .. W(t.) - (J-.-J (8 0)) I a .. u.t.]
u.
1,J=
J 1J
1
a
i,j=l 1J J 1
=
L[
r,s
I
a .. log A(u.,t.)]
i,j:=l
1J
J
1
under P
8
0
o
This completes the proof of the theorem.
In
Theor~n
3.5.1 we have obtained the
asyrr~totic
progressively censored likelihood ratio process (u,t)
distribution of the
-r ~
k (t)(u), for
n, n
each pair of (u,t), uEffi' and tE[O,l], when the Wlderlying probability
measure is P8 . Our derivation follows closely the method of Ibragimov
a
and Khas'minskii (1975a) and is based on the analysis of the entities
9Z
n*n,1. (u)
,
l~i~n
.
Let us define
(3.5.24)
for each i,
l~i~n
and UER r with 8
0
n*n,1.(u)
+
un -~ EG.
Then we have
n,1.(u)) - 1 .
= exp(~A*
(3,5.25)
The following is an immediate consequence of Theorem 3.5.1.
Corollary 3.5.1.
sequence
If the hypotheses of Theorem 3.5.1 hold and the
{A*n,1.(u):
l~i~n}
is defined by (3.5.24), then for each pair
of (u, t)
and
kn(t)
L[
L n*n,l:(u) IPE''0 ]
';--1
+
.
1
2 1
2
N(- -8 0 '-4 0 ) ,
... -..L
In the case of independent, but not necessarily identically distributed observations {X.;
1
i~l},
the A*n,1. of (3.5.24) are replaced by
r f" (X" . e0+un -~'1) J
l
I?
¢n,i Cu ) = lOgl f (X ;8 )
i i 0
where f i Co,8) is the pdf of Xi'
, l~i~n
In LeCam (1966) it is shown that, under
certain conditions, the asymptotic normality of I~=l¢n, i implies and is
93
implied by the asymptotic normality of L~=l[exp(S¢n,i) - 1], for some
SE(O,l].
Indeed, it can be proved that, under certain conditllions
if and only if for some SE(D,l] (and hence for all)
L[
I {exp(S¢n,lo)-l}/Pe0]
i=l
+
N(S(S2- 1) 02,S2 02)
k (t)
This corresponding equivalence between Li~l
k (t)
A~,i and Li~l [exp(SA~,i)-l]
does not necessarily hold in our specific situation where the variables
{Zi:
l~i~n}
are dependent.
CorollarY 3.5.2.
However, we can easily show
Under the hypotheses of Theorem 3.5.1, for each pair of
(u,t), and SE(O,l]
(3.5.26)
Proof.
Consider a Taylor expm1sion of the function x
+
x 2S about x = 1.
We have
x 2S _l = 2B(x-l) + S(2B-l) (X-l)2
1
f (1-~)[(1+~(x-l))2(S-1)-1]d~
o
+ 2S(2S-l) (X-l)2
Take x =
n,l0) in the above expansion and write s*n,l0 =
[exp(SA*n, 1 )-1], l~i~n. Then (3.5.26) yields
exp(~A*
0
s*n,l
0
-
2
2Sn*n,l -S(2S-1)n*n,l
.
0
= 2S(2S-1)n*2. I
n,l 0
1
(1-<';)
[(l+l;nn* lo)2(S-1)-1]dl;
'
94
Now
I (l+~n*n,l.)2(S-1) -11 ~
2, for each ~E[O,l] and all i.
It follows
from Lemmas 3.5.3, 3.5.4 and an argument similar to that in Lemma 2.4.1,
that
k (t)
n
k (t)
n
L l;;~
i=l
'
i
2B
k (t)
L n* .
. 1 n,I
g(2S-l)
1=
n~
2
i'-'l n,I
L n* .
+
0
in Pe -probability. 'fherefore Lemma 3.5.4 together with Corollary 3.5.1
o
leads to the desired result.
0
The next lemma will be instrumenta.l in obtaining the weak convergence
of the stochastic process u -+!J.n;r (u).
We shall continue to assume that
n
assumptions [AI] through [A8] hold true.
Lennna 3.5.6.
i=1,2,
(3.5.27)
Proof.
Fix t, upuZ as in the statement of the lemma.
Here En, k is a measurable set in IRk on which
Now
On (t) = k, An, k
k
is the
domain of definition of Pe(f(k)n) and ~k denotes Lebesgue measure in
m
.II'.
k.
Also, en,l.:::; eO
+
u1·n -k2, i :::; 1,2.
The rest of the proof is direct.
95
Write
e
k
Pe
n,I
(~
(k)
,n) - P~
1
n,2
(~
("k)
n,2
1
= ~ f p~~(~
en,l
,n)
(k)
,n)
ape
-ae d~(e)
so that by the Cauchy-Schwarz inequality
k
{p;
n,l
(~
(k)
,n)
- Pe
(k)
k
(~-
,n)}
2
n,2
=~
e
n;.2 2
2(u 2-u l ) e J,~n,k(e)Ped~(e) .
-k
n,1.
Thus, following the usual steps, we have
o
and so (3.5.27) is proved.
Remark 3.5.5,
by
Tn'
The inequality (3.5.27) remains valid if
Thus for ul ,u 2EIR' with ul
$
~(t)
is replaced
u 2 and eO + uin-~E8, i = 1,2"
(3.5.28)
Let us assume for a moment that 8 = ]R'.
traj ectories of the random process
L
-)-
!I.
n·T
, n
By our assumptions, the
(u) lie in COR) , the space
96
of all continuous real valued functions on ffi.
We shall endow COR)
i.
with the topology of uniform convergence on compacta.
(1970)), COR) is a complete separable metric space.
on C(lR) appear in Section 5 of the Appendix.)
>
(Some connnents
In order to demonstrate
the relative compactness or tightness of the process u
suffices to show, for each L
Then (see Whitt
+
An,T (u), it
n
0 and u1 ,u ZE[-L,1],
(3.5.29)
where K is a constant.
To this end we note that condition [A8] can be
written in the following form.
[AS]
For each L > 0,
With this assumption ~ in (3.5.28) can be replaced by
K* =
sup
{n- 1J
(e)}. The argument following (3.5.9) in Len~a
n
le-e I~Ln-~
n,T n
3.5.1 leaHs to the conclusion that K*n is bounded in n and so (3.5.29)
j
Therefore, the process u -" A~ 'T (u) is tight in C(ill.) , and
,.
'n
since the function x + x 2 , XEC (ill.) ~.s obviously continuous in C(lR) ,
follows.
it also follows that the process u
n,T n (u) is tight.
If e fill.', the process u + A
(u) has to be defined more precisely.
n, Tn
Here we suppose e = (a,b) and consider u + Kn,T (u) where we set
~
A
n
(u)
A
n,T
n
= An,T
(u)
if
n
=0
and take 1
n,T n
(u) linear and continuous in all other intervals.
Clearly
97
the functions u
+
An,T (u) lie in C(lR) and (3.5.29) continues to hold
n
for the modified process.
Theorem 3.5.2.
process u
+
Hence, recalling Theorem 3.5.1, we have
Suppose conditions [AI] - [A8] are satisfied.
An,T (u) in C(lR) converges weakly to a process u
n
where
and
s
is a standard normal variable.
Then the
+
A(u);
CHAPTER IV
ASYMPTOTIC BEHAVIOR OF SOME STOCHASTIC
PROCESSES ARISING IN PROGRESSIVE CENSORING
II
Introduction
4.1.
In this chapter we shall investigate further the random processes
introduced in the previous chapter.
We begin with a brief discussion
of the assumptions under which Theorem 3.4.1 and Theorem 3.5.1 were
proved.
It appears that if somewhat stronger analytic conditions hold,
a simplification can be obtained in the proof of Theorem 3.5.1.
In
Section 4.3 a general situation is considered where the validity of our
results is directly demonstrable.
Our entire analysis in Chapter I II has dealt with the ordered
variables for a sample of Illdependent and identically distributed (iid)
observations.
In Section 4.4 we briefly outline two theorems parallel-
ing Theorems 3.4.1 and 3.5.1 where
in the series scheme and are
~he
neithe~
underlying variables are giverl
independent nor identically distri-
buted.
Throughout this chapter, unless othel\Nise stated, we shall adhere
to the same basic notation, ternlinolo&'l and definitions of Chapter III,
4.2.
On the assumptions made in Chapter III.
The principal assumptions under Which the weak convergence of the
process Wn,T = {Wn,T (t): tE[O,I]} of (3.2.10) was demonstrated are
n
n
[AI] through [A7] of Section 3.2. For the discussion of the asymptotic
99
behavior of the progressively censored likelihood ratio process
An,kn(t)(u) an additional assumption [AS] was imposed in Section 3.5.
It will be noticed that these conditions involve at most the first
derivatives of the logarithm of the underlying density f e and hazard
rate r
e, the derivatives being, of course, with respect to the parameter
e. Assumption [A4] (b) is utilized in the discussion in Remark 3.2.1,
leading to the relation
On the other hand
for each i,
l~i~n
If Ee(~*n,l·IBn,l. 1) = 0
and all eE0, then Lemma 3.3.1 holds without this assump-
[A4] (a) is really not necessary.
[AS] is instrumental in deriving a Lindeberg Condition for the
tion.
array
{~*
1:
n,l n,T n ; l~i~~(t)}.
In Section 3.5 we employ the additional restriction [AS].
./J2
Note that
this condition involves only the first derivative of r e though now in
a slightly more complicated form.
Suppose the functions
e ~ q~(ZIBn,i-l)' l~isn are differentiable
(wi th respect to e) for all z.
Then live may write, for each i, hhn and
z
and therefore one has, by use of the Cauchy-Schwarz inequality
100
If
y
(u) = E {
su~
-k nn
eO le-eol~luln 2
2 n
d2 . ~
L z.J 12 qe(zIB,TI,l'-1) I
Cle~
i=1
00
Z
d]l(z)} ,
1-1
(4.2.1)
then [A8] follows from the condition
lim
n-+oo
y
n
(u) ::::
a,
for each 1lEJR I
•
The verification of (4.2.2) itself can be quite tedious even in relatively
simple situations.
However, an alternative proof of Theorem 3.5.1 can
be given if we impose stricter conditions on the functions f e and reo
Notice that since Te == fe-G e we may formulate some of our conditions in
terms of Ge instead of reo
We now propose the following alternative conditions for [A8].
Suppose that for each XEX, fe(x) is twice differentiable with respect to e on 8, and for each eEe, (aZfe(x))/(ae Z) is continurn~ in x on
[A9]
X.
Further, suppose
(*)
for each XEX, 6Ee, i
Suppose for some
k(X,2)
==
2
0
> 0, the
a<
2
==
< SO' and
lim Ee (k(X,s)) -
2"+0
f~1ctions
su~ I f~ (x) -£~ (x) I ; h(x,2)
le-eol<2
0
are defined for all
(i)
1,2.
=
a;
0
either
(ii)
hex,s) is monotone nondecreasing in x for all s
<
So
and
101
lim E (h(X,E)) = 0 , or
£-+0 e0
the function H(x,£) = sup h(y,E) is defined for all 0 < £ < £0
y::;x
and lim Ee (H(X,E)) = O.
£-+0 0
(iii)
Suppose conditions [AI] - [A?] of Section 3.2 and [A9]
Theorem 4.2.1.
hold.
Then for each pair (u,t), the decomposition (3.5.22) holds true.
Moreover, under the probability measure Ps the finite-dimensional diso
tributions of the process (An,k (t) eu): UEIR i , tE[O,I]} converge to
n
those of {A(u,t): UEIR', tE[O,I]} where
Proof.
We shall derive the expansion (3.5.22) under our alternative set
of assumptions.
The rest of the proof is along conventional lines.
Now,
from our definitions
where
s~ =
So
Wn,T , (4.2.3)
n
+
Aun
Cffi1
-k
2,
IA/ < 1.
In view of our definition of the process
be reWL'itten
(4.2.4)
Our proof will be complete once we show 0n(SO,u,t) -+ 0 in P -probability
eo
for each pair (u,t).
Now for each eEe and k,
l~k~n,
102
••
(k)
P e G,
,n)
(4.2.5)
Thus
-1..
In
PeCk.
::; n
-1
(kn (t))
kn(t)
\'
L
. 1
1=
,n) - n
-1
00
(~
Pe
(1~(t))
o
,n)1
, ••
f (Z.)
e
1
(4.2.6)
By virtue of [A9] (i) and the strong laws of large numbers (SLLN)
n- l
n
I
su~
i=l /e-eOI<E
If~(x.) - f e (x.) I
1
0 1
-1:
E [k(X,E)]. (4.2.'7)
1 e0
All convergences are valid under the fixed probability measure Pe .
o
Now if [A9] holds one obtains, for each k, l::;k::;n
n
(n-k)h(Zk,E)::;
I
i=k+l
n
h(Zi,E)::;
I
i=l
h(Xi,E) ,
and thus, once again from the SLLN, we get
luTISup Ee (n- 1 max (n-i)h(Zj,E)) ::; E (h(X,E)) .
e0
n+oo
a l~i::;n
If instead of [A9] (ii) the alternative condition [A9] (iii) holds, note
that H(x,E) is nondecreasing in x ffild so once again from the SLLN,
limsup Ee (n- 1 max (n-i)h(Z.,E))::; E (H(X,E)).
ea
n+oo
a l::;i~n
1
(4.2.9)
Hence from (4.2.6) through (4.2.9) we have proved
(4.2.10)
103
since the right hand side of (4.2.7), (4.2.8) or (4.2.9) can be made
arbitrarily small by choosing
E:
to be so.
Hence it only remains to
verify
(4.2.11)
in Pe -probability, and our theorem will be proved.
o
Recall (4.2.5). Fix e at eo and surrnnon Result 2 of the Appendix.
-1
at in Pe -probability, where at is given
o
So we have
From Lemma 3.3.3 n kn(t)
by J
at
(eO) = tJ (eo)'
a
n
+
k (t)
-1 n \' .•
L
i=l
f
e0 (Z.)
1
->L
1
Furthermore, by the continuity of G~ (x)
o
in Pe -probability.
Therefore one has
o
n-1··
Pe
-1
(k (t))
(~
o
n
,n)
+
Fe (at)
J0
{
••
f
e0 (x)dF ea(x)
-00
+
(l-at)G~ (F~l(at))}
o
0
(4.2.12)
in Pe -probability. That the right hand side of (4.2.12) actually
o
reduces to ~tJa(eO) is shown in Section 4 of the Appendix. This completes
0
the proof of Theorem 4.2.1.
Remark 4.2.1.
In Theorem 4.2.1 we replaced condition [A8] by [A9].
weak convergence of the process u
The
n,r n (u) in C(JR ') was obtained in
Theorem 3.5.2 under the assumptions [AI] through [AS]. (See Remark
3.5.5.)
+
A
It will be interesting to examine whether our
104
alternative set of assumptions here suffice to ensure this weak convergence.
Recalling (4.2.4) we shall need to demonstrate that, for each
L > 0
lo~(eO?u)1 ->-
sup
lul~L
IU
=-'2
2
n
~probability,
(4.2.13)
0
-1,·
p
"81
{r1
0, in Pe
(Z"
(T
n
)
n)
?"
J ex. (8)}
O?
+
a.~d
1 ~ 8 + Aun -k2
8n
O'
11
IAI<
Now we may replace (4.2.10) by
1.
-1..
sup In P8'(~
lul~L
(Tn)
,n) -
-1
(Tn)
,n)1
a
n
-k
0 •
P8 (~
n
-1
0
1
-k
since 181-801 ~ luJn 2 s Ln 2 < €, provided lui ~ Land n sufficiently
n
-1..
(Tn)
large. Of course n P8 G,
,n)·~ -Jex.(8 ) in P '-probability,
Hence
0
e
o
(4.Z.l3) holds.
h
>
0
Therefore if ul,uZEIR ' such that luil s L (i=I,Z) and lul-u Z'
0, we have
sup
IU1-u21~h
Ilog
s 2 sup
I -,
I
IU SL
I
h,
A
(u )
n,T n l
IU·lsL
1
-7;'
~
•
Y-
16*n
(8 ,u) I + hn -~J~
an
I
7-'"
9"
'
T
n
(e 0){/o
/J~ (8"}
"'n T . n, T '. 0)
' n
n
1
V~?5
Since n 2J 2 (8) ->- J 2(8 ) and L [~'1 r.n, Tn 0
ex. 0
1 j"n
obtain for arbitrary € > 0,
2
/j"1- r_ (8 )
", L n
0
iP8
] ->-
a
1-.1"r
+ lLL.. ex.
(
eo'
)
N(O,lL we
lim lim P8 [
sUI? Ilog An T (U l ) - log An T (u Z) I >
, n
h+O n~
0 !ul-uzish
'n"
€]
=
°.
lu·lsL
1
Summoning the results from Section 5 of the Appendix, it follows that
the process
U ->-
An,T (u), UEIR I converges in C(IR) to a process
I
n
Z
A(u) = exp{~(80)s - ~u Jex.(8 0)} where s is a (O,l)-Gaussian variable.
U ->-
105
4.3.
An example.
In this section we retain the basic assumptions on the underlying
variables {X.; i2l}.
1
Thus {X.; i21} is a sequence of iid random var1
iables with a common distribution function Fe and probability density
(with respect to Lebesgue measure on the real line) f e . Here e is a
parameter, eE8c lR', and we assume 8 to be an open subset of lR. In
the context of clinical trials and life-testing, the X's are nonnegative
and so Fe is concentrated on the half line [0,00), for each eE8.
Further-
more, it is customary to call Ge(x) = 1 - Fe(x), the survival function
and re(x) = fe(x)/Ge(x) ,the hazard rate (or force of mortality or intensity
rate).
The function Re on [0,00) given by
x
=
is called the hazard function.
f r e (z) d]l (z)
(4.3.1)
,
°
It is then easy to verify that
(4.3.2)
As mentioned before, in the typical situation encountered in life-testing
and clinical trials the observable random variables are the order statistics Zl<",<Zn corresponding to a random sample of size n.
In our
notation the conditional probability density of Zi given ZI"",Zi-1
is
defined for z > Z.1- l'
where z > Z.1- l'
From (4.3.2) this can be rewritten
106
We now restrict attention to the class of survival distributions
for which
Y8(x) = h(x)/Q(8)
(4.3.4)
?
where Q(8) is a nonnegative fUl1ction of 8 only and hex) a function of x
only.
Appropriate analytic restrictions will be imposed on Q and h
later in this discussion.
Now in view of (4.3.4) we have for each i?
Obviously we require h to be Lebesgue integrable over every interval
[a,b], O<a<b<oo.
Let us define a new sequence of variables YO,Y , ... ,Yn ,
1
n~O.
Z.1
(n-i+1)
J
h(x)d]J(X); l=:;i=:;n .
Z.1- 1
Then from (4.3.5), the Y.'s,
1
(i~l)
are iid random variables with the
simple exponential distribution with mean Q(8).
So
From (4.3.5)
~n* ,l' n
= ~*
. (8) ~ Q8(Z.1 IB11,1. 1)
,l
= Q' (8)
Q2(8)
(Y. -Q(8)) .
1
So E8(~*n,l·18n,l. I) = 0 for all i; further, for each i,
(4.3.6)
107
Thus vn, k(e) =
(Q~~~~)2k, lsksn and also J n,T (e) = Ee(Vn, Tn (e)) =
n
(e) 2
(Q'Q(8))
E8 (T n)·
We defined the sequence of integer-valued functions
{~(t): tE[D,l]} in (3.2.9) by
8
k (t) = max{k: V k(e) s tJ
On
n,
n,T (e)}.
n
In the current situation this simplifies to ~(t) = [tEe(T n)]. (For any
real number a ~ 0, [a] denotes the largest integer s a.) Our fundamental
assumption on the sequence {Tn;
n
for each tE[D,l].
-1
~(t) +
n~l}
is [A7].
[at] , as n
+
This entails
(4.3.7)
00
The process {We. (t): tE[D,l]} reduces as follows.
n, [n
k (t)
n
1
I
E,* . /12
(e)
i=l n,l n,T n
(4.3.8)
Observe that for fixed GEe, the variables
(Yi~Q(e))/Q(e),
lsksn,
~ne
iid with mean zero and variance 1. Therefore, the process
[tEe (Tn)]
-h
{(an) 2 L
(Y.-Q(8))/Qce); tE(O,l]}
. 1
1=
1
has sample paths in D[O,l] and converges weakly to the Brownian Motion
process Win D[O,l] endowed with the Skorohod topology.
-1
n Ee(T n)
+
a as n
+
00,
it follows from (4.3.8) that Wn'T
Since
n
-~
Win
108
D[O,l].
This demonstrates Theorem 3.4.1.
e was held fixed in 8. We now fix e at
Up to this stage
80 and
consider a sequence of values en given by
We proceed directly with the analysis of the likelihood ratio process
A k (t) (u) of (3.2.14).
n, n
e
= 80')
(Note the J<n(t) here is given by (3·.2.9) with
Now
and so
(4.3.9)
To proceed further we make the following assumption on Q.
Suppose Q
admits a first order Taylor expansion in the neighborhood of 8 ,
0
-k
Q(e ) = Q(e ) + un 2Qi (8:")
n
0
il
where 18*-8
n 0I
We thus have
~
luln
-k:
2.
Then
~
From the continuity of Q and Q' we have both
in view of (4.3.8).
Now
+
=
Q'(8*) 2
2-l[
~ n
Q(8 n ) J
0
+
-1
o(n )
)J 2 + 0 (1) ] (-~ + 0 (l))u 2n --1
(I + 0(1)) [(Q'(8
Q(8 0T
0
2 -1
= - ~ n
[Q'Q(8(80)) ] 2 +
0
Hence we get finally
From (4.3.9) it follows that
0
(n
-1
)
110
-
~
2
tJ (8 ) + 0 }
n
a 0
Qi (8 ) 2
0
-probability and J (eO) - a(cr(
e
80
n
0
a
strates Theorem 3,5,1,
where 0
4,4,
-+
r)
0 in P
Asymptotic
<distY'ib~tiOl~_5LLth2·li~el.Jl~q(L_!:~lt·jo
This demon-
function when the
observat"j ons are g"1 ven 'in t~~_ :)~ri.§~.s scheme,
We shall present here an outline of how the previous analysis on
the progressively censored likelihood ratio statistics A
1
.n,K
n
«_)
(u)
t-
can
be adapted for the derivation of a local asymptotic expansion when the
underlying observations are neither independent nor identically distri·buted.
l~k~n}
Suppose {X
n, k:
measurable space
is a double array of random variables on a
For each k, l~k~n~ write ~n, k
(X,A).
= (Xn, l""'Xn, k)
and denote by Bn, k the a-field generated by L
0 = Xn, 0
. ·u, k' Let X
"1l,
=
O.
Suppose there exists a family of probability measures {Pe: eEe} on
(X,A) , indexed by the real parameter GEe.
of 1R I .
The proj ection of Peon Bn,k is the probability measure
We assume that the:ce exis ts
9.
)Toduct measure
.,
'I'k .k.,
CarteSIan
prOduce space (7\ ,A )
Po (~
"1,
Assume e to be an open subset
~
. . , ..)l1k
tna'c " '
0
on
~X1JX. , , x~
«~k
Sl.Kll
P~ ,K
on the
d"
an w.crce
k; n) for a specified veTsion of the probability density function
involved.
Here x~n,K
1
=
(xn, l"···xn,K
.,) ~ l~ksn.
If qe (xn, 1( IBn; k-l) is the
conditional probability density of ~,k given Bn,k-l we obtain
Pe (L,'K;n)
. 'L
==
k
l1
(X . lB. -,)
i=l 0 n~l n,l-l
°
L
,
l~k~n.
(4.4.1)
Let us make the following assumption.
[Bl]
The functiorsPe(x
1
"1l,K
;n) are differentiable with respect to e on 8,
III
for all x"'Il , k'
l~k~n.
We may then define the entities
and
for each k,
l~k~n
and all eEG.
We assume further
° < Ee(~;~k)
[B2]
For each n~l and all k, l~k~n,
eEG.
Expectations are evaluated under the measure Pee
< 00, independently of
We may set
and write J n, k(e) =EeVn, k(e), l~k~n.
otherwise arbitrary element of G.
In what follows
e 1S
a fixed but
Introduce a sequence of integer valued, nondecreasing right continuous functions on [0,1] by
~(t) = kn8 (t) = min{k: Vn, k(8)
where {Tn;
n~l}
>
tJn T (6)} ,
, n
(4.4.2)
is a general sequence of stopping variables, such that
We now define the function
for each n~l, Tn is adapted to
8 (t) on rO,l] by
t + W
Il,T
n
~,kn (t) (8)
(4.4.3)
8
8
Then the process Wn,T = Wn,T
= {Wn,T
(t): tE[O,l]} has paths in DIO,I].
n
n
n
We shall discuss the weak convergence of Wn,T in D[O,I]. To this end
n
we make the following additional assumptions.
e
H2
[B3]
J qe(xn, k lBn, k-l)dp
For each n~l and all k, l~k~n, let
X
be
differentiable Wlder the integral sign, for alI SEG.
[B4]
Lindeberg condition:
For each
E
>
o~
tErO,l]
k (t)
J-
l
n, 'Tn
n
(e)
L
i::: 1
sli.n , 'Tn(8))1Sn ~ 1. -1 )
E (s*2. I (IS:*·1 >
en, 1
n? 1
Wlder the probability measure Pe, as n
Theorem 4.4.1.
00,
e fixed
Suppose conditions IBI] - [B4] hold, then with
G and W
defined by (4.4.3), Wn,T
n,Tn
' TI
topology.
Proof.
+
-'w~ W
in
in DIO,I] in the Skorohod
We only need to verify the conditions (a), (b), (c) of Result 5
of the Appendix, for the array {l;n
/J
1:;;i$n}. In view of [B2] and
n
[B3] {sn,k; ~n,k}~=l is a zero-mean martLigale, so that (c) holds trivially.
,
1
1,
T ;
Condition (a) is a direct consequence of IB4].
We are left with verifying
(b) which, in this case, reduces to
V
n,kn Ct)
/J11, Tn
~.
in
t
Pe~probability
(4.404)
Now for each tdO, 1] we have f,:om (/L ii·, 2)
and V·TI.K1 C'~)" 1
'rJ
.. n,T
_<
'Il
Hence, for arbitrall'
€
n
> 0
::;
J-l~
k (-,-)IB
n, nEe(s.*2
il,on
ln, kn (t)-l)
::;
2
E
L
k (t)
"n
2
+ J-~
E (s*
n,T
n
.I _
1=1
n,l.I(!r*:1
~,l
e
1
> EP
Yl T
)1 B .
, n
n ' 1-1
)
'
113
and thus (4.4.4) follows from fB4].
This completes the proof.
0
Theorem 4.4.1 parallels Theorem 3.4.1 of Chapter III.
Remark 4.4.1.
Note that the condition on Tn , fA?] is not needed here since the behavior of J
(8) with n is not utilized in the proof. We may recall
n,Tn
that the convergence n-1J
(8) + J (8), under fA?] was instrumental
n,T n
a
in deriving the Lindeberg condition which we have included as an assumption Ll Theorem 4.4.1.
To proceed further, we introduce the likelihood ratio processes
(4.4.5)
where 8n
= 8
0
8
+
uJ
-k
2
(8) .
n,T·
0
eo is a fixed element of 8 and
·n
by
n* (u) =
n,i
The following
[B5]
now
Let us define the entities n*n, 1" (u),
stands for k OCt) given in (4.4.2).
l~i~n
~ (t]
n
addi~ional
For each L >
(q8 (Xn,l·18n,l. 1)1~
n
- 1 .
l
qe eX
o
i1~
·1 B . 1) J
1 n,l-
assumption will be made.
0
Under assumptions [Bl] through [BS] we can establish results corresponding to Lemmata 3.5.1 through 3.5.5.
The proof of these results follows
114
along exactly the same lines and tllUS i,'1 the jnterest of brevity we shall
not pursue the pedestrian details here,
The final consequence is, of
course, a theorem paralleLing Theorem 3,5,1,
We can establish
Theorem 4.4.2.
Suppose conditions rBI] ,- [BS] are satisfied and the
functions (u, t)
-+
An k (t) (u) are defined by (4,4,5), fUf each 'CE 10 ,1]
, It
and UEIR' such that e EG.
n
A k (t)(u)
n, n
=
lBen A, k (L)(U) can be exvressed TIl the form
n, n L
eO
2
exp{uWn T (t) - 1zu t
' n
\..mere, for each pair (u, t), on
(u,t)
-+
+
+
on} ,
° in Pe-probability.
Moreover, if
Xn,~(t)(u) is the continuous extension of An,kn(t)(u) to all
UEIR', tE[O,l], the finite-dlinensional distributions of (u,t) -+ An k (t)(u)
, n - ,
converge weakly, under Pe ' to those of (u,t) -+ A(u,t), where
o
A(u,t) = exp{uW(t) - 1zu
2
t} .
The question of the weak crnlvergence of the process U -+ An,T (u)'
n
can also be dealt with along the same Jines as in the discussion following
Lemma 3.5.6.
Indeed, for the continuous extension U -+ An T (u), uEIR'
, n
of the process u -)- AnT eu), we have the following theorem corresponding
1
Ii
to 1neorem 3.5.2.
TI1eorem 4.4.3.
Suppose
condi~ions
[BI]
~
[BS] are satisfied.
Then the
process U -+Xn,T_ (u) converges weakly to the process U ~h(u) in C(IR), in
11
the topology of uniform convergence on compacta, wnere J\(u) is given by
A(u)
and
z;;
=:
exp{uz;; ~ ~u2} ,
is a (O,l)-Gaussian random variable.
CHAPTER V
SOME APPLICATIONS AND SUGGESTIONS FOR fURTHER RESEARCH
5.1.
Intnoduction.
This final chapter is devoted to a brief outline of some applica-
tions of the asymptotic expaIJ.sion and. asymptot:ic distribution of the
likelihood ratio fWlctions that we dealt with III the previous chapters,
These applications generally pertain to large sample properties of
estimators and of statistical tests.
OUT
discussion here ,,\lill be mainly
confined to some aspects of asymptotic efficiency of estimators.
We
shall make a few comments on asymptot.ic hypothesis testlllg towards the
end of this chapter.
Throughout this chapter we shall work within the general .framework
of the mathematical model introduced in Section 1.2 of Chapter I.
Accord-
ingly, let {X.; i21} be a sequence of ll1dependent ffild identically distri1
buted (iid) random variables (IV) with values in the real line (JR
be the probability distribution of the X.,.L ~ where
e
is dominated by Lebesgue me&sure II in
em
1
B)
Q
is a paramete:c
The parameter space 8 is assumed to be an open subset of 1R 1.
each ve
1 J
0
Suppose
JB) and write f
e
for
a specified version of the probability density function (pdf) and let
Fe
denote the corresponding distTibution fLIDction (df) on IR I .
We also
00
denote by (X' ,A) the infinite product space 11. 1 (IR! ~B.) where (JR: ~B.) ~
J=
J ]
J J
j 2 1 are copies of the real 1 me, and write Pe for the product probability
measure in (X' ,A) induced by v e.
with respect to this measure.
Ee will denote the expectation evaluated
116
We remarked earlier that in the framework of clinical trials and
life testing it seems more reasonable to formulate statistical procedures
in terms of the ordered observations Xn, I ~ ... ,Xn~ n corresponding to a
sample XI, ..• ,Xn , rather than directly on the original variables which
are in this context usually unobservable.
Recal1iAg the notation of
Section 3.2 we are thus led to the study of statistics of the general
fonn Tn,k
=
Tn,kG.(k)), l:o;k:o;n.
Furthennore, limitations on time and cost
often force restrictions on the duration of experimentation.
This leads,
quite naturally, to the introduction of a general class of stopping
variables {Tn;
n~l}
into our investigation.
For each
n~l,
Tn is
adapted to the a-fields B k = a(Z(k)), l:o;k:o;n. Hence the consideration
n,
of the "stopped" statistics 1.
, nd.
Chapter II was concerned with
11, Tn
deriving the asymptotic nonnality of Tn,T and of Tn when the explicit
n
fonn of these variables is given in terms of certain linear combinations
of the observables Xn, 1""'X
-n,n .
Now if we are to utilize these statistics in a wide variety of
practical situations, it becomes necessary to examine carefully the
delicate question of the efficiency of statistical procedures based on
them.
More specifically we may enquire what considerations of optimality
should lead us to advocate the use of a particular sequential plan
(T*n,T * , T*)
over all others within the class of statistics {Tn,K,: l:o;k:o;n}
n n"
and stopping rules Tn' Furthermore, given certain criteria for optimality,
do our notions lead to an optimal procedure (T*n;I * , T~)?
11
n
5.2.
Lower bound for the variance of Tn,T .
n
We shall adhere to the basic assumptions [Al-] - [AS] listed in
Sections 3.2 and 3.5 of Chapter III.
Our first result is a simple state-
ment of the classical Cramer-Hao-Wald-Wolfowitz lower bound for the
117
variance of a sequential estimator ll1 the particular case of the
statistics T
.
n,Tn
In the sequel it is assumed that the T k are
n,
measurable with respect to u-fields 8 k
n,
= a(~(k)),
l:::;k$;n.
The restrie-
tion of the probability measure P8 to B , k is denoted P~' k
Thl) stopp:ing
n
variable T is adapted to (8 k) I i'_1 - - the measurable set in (JR k, Sk)
n
n, k -~
corresponding to [Tn=k] win be denoted by
on (lRk,Bk ),
Then
Theorem 5, 2 , 1.
~,k/dlJk
ren
'I'
~i(
lJ}- is Lebesgue measure
...
P8(~(k)n) is the pdf of l(k).
=
Suppose that for each n:2:J, and all k, l::>k$11
T
f
n,k
:IE
(z (k))1) {z (k) n)dll (z (k))
1: e\.~
,
k ~
N
n,k
exists and is differentiable with respect to e through the integral sign,
for each eEG.
'Then for each n2':l and all eEG
Vare (T
n,T n
Proof.
) ) 2/3
) 2': ( dedE e (Tn T
n T (e) .
' n
' n
(5.2.1)
Now
n
E (Tn
e
T )
'n
LIT
'"'
k=1 f'rn=k] n,k
n
"" I
k=l
)r
IE
.
~~, k
e
( k)
(k)
(k)
T,
)p eN'
(7..
,n)dlJ k (z
)
n '/, (z'
~
~
1-"
n,k
Therefore
d Ee(Tn,T )
de
n
=
I
k=l ill
J
N
n,k
where ~
n,k
T
(z(k))~,
(z(k) e)p (z(k) n)dlJ (zO())
n,k
n,K ~
,
e~ ,
k ~
(z(k) e) = d~ (log Pe(k(k) ,n)), l:::;k:::;n, eEG,
N
,
The right hand side of the above equation is simply E (T
~,
),
.
e n,T n n;rn
Moreover, Ee (~n
)::: 0, by virtue of Corollary 3.3.1. Hence
,Tn
) = E [(T
-E (T
)) c
]
aed Ee (Tn 'n
T e n Ten T
':>-11 T
'n
'n
'n
'
(5.2,2)
118
and an application of the Cauchy-Schwarz inequality in (5.2.2) yields
= Ee (s2).
n,T
the desired result (5.2.1), since I n
(e)
,Tn
Remark 5.2.1.
When
T
n
=n
0
n
the inequality (5.2.1) gives the classical
Cramer-Rao lower bound for the variance of the statistic Tn, n
=
Tn,n(Xl"",Xn ), based on the iid sequence XI, ... ,Xn , Here In,n(e) =
nICe), where I(e) is the Fisher information function corresponding to a
single observation.
Suppose now the equality holds in (5.2.1) for all
eEe and n sufficiently large.
Then one has
(5.2.3)
where an(e) is a constant (independent of the observations) and
An(S) = Ee(Tn,T ).
n
Of course this yields
and so in conjunction with (5.2.1) we obtain
(5.2.4)
Let us write (5.2.3) in the fonn
We have seen earlier that if
-1
Therefore, since n J
Corollary 5.2.1.
n,T n
-+
-+ J
a
(8) as n
-+
00, we have shown
Suppose the equality holds :in (5.2.1) for all eEe
and n sufficiently large.
as n
(e)
e obtains
If for each eEe A~(e)
00, then when e obtains
=
~e Ee(Tn,T )
n
-+
Aa (e)
(5.2.5)
Remark 5.2.2.
The asymptotic conver.gence (5.2.5) is a consequence of
(5.2.1) when the equality holds therein for all n sufficiently large.
It is interesting to examine what can be stated when this equality obtains
for some fixed n.
Of course (5.2.3) crnlLD1ues to hold.
2
Suppose Ee(X i ) = e and Vare(X i ) := 0 . Let Wk "" If=lXi~ k~l
and suppose for each k ~ 1, Wk is a sufficient statistic for estimating
e from Xl'" . ,Xk .
each
k~l.
Then Tk = E(X1IWk) is em unbiased estjJUator of e for
If N is a stopping rule adapted to {0(X1, ... ,Xk); k~l}, then
it can be shown that under certain conditions the variance of the sequential
estimator TN satisfies
(5.2.6)
Blackwell and Girshick (1947) proceed to show that equality holds in
(5.2.6) if and only if Pe(N=n O) = 1, where nO = ffiDl{n:
Hence the stopping rule N is degenerate in this case.
Pe(N~)
I OJ.
It will be interesting 1::0 decermine to what extent a parallel
result can be obtained in
pursue this study here.
OUT
specific dependent model.
We shall not
We now tum LO the generalization of a result
of Bahadur (1964).
5.3.
Lower bound for tIle asymptot'jc variance of T'1
.
,Tn
We continue with our investigation of (5.2.1), but now in a differ-
ent direction.
r
In view of the convergence (5.2.5) let
lY5
consider the
class of sequential plans (Tn,T ,Tn) for which there exists functions
2
2
n
lla(e) and vale) > 0 such that when e obtains
120
(5.3.1)
Note that we are always working within the framework of the assumptions
enunciated in Section 3.2 and 3.5.
relates to assumption [A?]: n-IT
e obtains.
In particular, the constant aE(O,l]
+ ex.
n
in Pe-probability as n
+
00,
when
It is our intention in this section to demonstrate that the
inequality
holds for almost all eEG, with respect to Lebesgue measure.
To this end,
let us first derive the following ])reliminary results.
Let An k = log(Pe (~(k) ,n)/Pe (k(k) ,n)) , l~k$n, where
,
n
0
-k
en = e0 + un 2 EG. Then with u fixed in lR'
2 2
(5.3.2)
L[A
IP e ] + N(-~a,a)
Lemma 5.3.1.
n,Tn
0
and
LPn TIP e
'n
n
(5.3.3)
]
0th a2 = U2J (e )'
Wl
a O
Proof.
The convergence (5.3.2) has been demonstrated in Corollary 3.5.1.
We only need to show (5.3.3).
Let us proceed directly.
Let z be a
fixed real number.
=
=
[A
J
~zJ
n,T
n
exp(A
n,Tn
)dP e
0
z
=
J
-00
exp(y)~(y)
(5.3.5)
121
where
= Ps [An
~(y)
o
lim H (yJ,
n
n-+oo
,Tn
= JY
-00
~y].
In view of (5.3.2), for each yEIRI
1
exp (- 1
-Z(x +
2
12'ITa
20' •
~a
2) 2)dx
It follows that for each ZEIR' >
z
lim
n-+oo
J
exp(y)~(y)
z
1
122
= J exp (y }-~~~~-exp(- '=-r(Y
-00
l2'ITa 2
-00
z
20'
1
122
= J - --2 e:xp( - -(y lZ'ITa
_00
+ ~a ) ) dy
20'2
~a )
)dy .
(5.3.5)
The right hand side of (5.3.5) represents the normal distribution function
with mean ~a2 and variance 0'2.
Thus from (5.3.4) the desired result
0
follows.
Remark 5.3.1.
The result proved here is a direct consequence of the
contiguity of certain sequences of probability measures.
We do not
intend to pursue this line of enquiry here.
Lemma 5.3.2.
Suppose the sequence {Tn T } satisfies the condition
, n
2
where \)o.(SO) > 0, and in addition the -restriction
liminf P
1 [T
<
n-+oo e +n -'2 n, Tn
o
-1
~ Jo. (SO) •
Proof.
Cn
Let s > 0 be arbitrary.
= [An,T
n
>s], Dn
= [Tn
T
' n
~
Set
en ]
en = 80
for all
+
n~l.
n
-k
2
and define the sets
Then
122
= 1 - Pen [Tn,T
n
< eo + n
-~
n~l.
],
By our assumptions, therefore,
limsup Pe (Dn )
n-t<X)
n
Also
en (Cn ) = Pen fA n,T n
P
(5.3.6)
~ ~
> s]
= 1 - Pe [An T
n
'n
~
s]
= 1 - Pe [(An T _~02)/0 ~ (£-~02)/0J
n
' n
where
0
2
= Ja(e O)'
Summoning Lerrnna 5.3.1 we obtain
lim P (C ) = 1 _
n-t<X) en n
where
£
>
~
~((£_~02))
,
is the standard normal distribution ftmction.
~02, (5.3.6) and
(5.3.7) lead
limsup Pe (D )
n~
n n
~ ~ >
(5.3.7)
If we select
to the inequalities
IBn Pe (Cn )
n~"
n
Thus, for infinitely mm1Y n,
(5.3.8)
Now by the generalized Neyman-Pearson Lemma, (see Eisenberg, Ghosh
and Simons (1976)) the test based on An T is the most powerful test of
, n
its size among all tests whose stopping variable is Ln'
from (5.3.8)
So we have
123
(5.3.9)
for infinitely many n.
But
Pe (D )
==
1 - Pe [Tn
Pe (C )
o n
==
1 - Pe
o
T
nO? n
and
a[~n,T. n
< eO
-1:
-I-
n 2]
s e]
Proceeding to the limit via a subsequence satisfying (5.3.9), we obtain,
in view of LelIDla 5.3.1 and our hypotheses
That is,
1
,f,
'¥
2
(e+~0)
0
~
holds for all e > ~02.
M (
'¥
1
'Uet- (e a ))
Hence 'U~(eo) ~ J~l(eo) and our Lerrnna is proved.
o
Lennna 5.3.3.
The function e
measurable for each
Proof.
PerTn,T
n
<e]
-+
IPe[Tn
T
, n
<
eJ - ~i I Ie on
JR
I
is Borel-
n~l.
=
(5.3.10)
124
Here IE n , k is a measurable set in elR k, Bk ) corresponding to {Tn=k] ,
l~~n.
Define, for n~l, l~k~n
If C denotes the Borel field in 8, it
easy to show that Bn, k is
C measurable for each nd and k, h~. Then (5.3.10) may be
B k-
1S
n, x
written
%[Tn
n
T
, n
<8]
L fk
=
k=l. lR
4
IB
n,k
(f(k) ,e)P8(f(k) ,n)d~k(f(k)) .
Each term I B
(z(k)
(z(k) , n) is product measurable since p 8~
(z(k) , n)
~
, 8)p 8~
n,k
is continuous in 8 and measurable ,.nth respect to Bn,k for each k. It
follows that Pe [Tn,T <8] is C-measurable.
n
This terminates the proof.
o
We can now establish the following
Theorem 5.3.1.
Suppose the sequence {T
n,Tn
for some function u2 (8) > 0 ffild foY all 0E8.
ex.
n~l}
satisfies
Then for ahnost all
eE8
(5.3.12)
Proof.
n~l.
Define ~(8) = !P8[Tn,T <8] -~IIo(e),
CJ
By Lemma 5.3.3 &n
n
is Borel-measurable for each
n~l.
Now fran our assumption (5.3.11) we
obtain
lim PerT _ <e]
n4<O
n,l n
Hence lim
n4<O
~(8)
= 0 and 0
=~,
for all 8EG .
~ ~(8) ~ ~,
for each eEG
. Also
125
+00
f
1
~(8+n-~)d¢(8)
-00
::: J fU (8)
-
1
-!-,;
!2n
-00
2
ex-pC -~(8-n 2) )d8 ,
Therefore, by the dominated convergence theorem,
co
lim
n-+co
J ~l (8+n -!z)d¢(8)
=:
a,
00
-!-,;.
and it follows that &11 (8<j'11 2)
+
0 as n -;..
•
<Xl
In CP~measure,
Thus, there
exists a subsequence {n) for which
1
~1 (e+n-~) ~ 0
as
v ~
00
v
The ¢-measure in JRV(the measure induced in
almost everywhere-¢.
(lR I ,B) by ¢) is equivalent to Lebesgue measure 1.1.
eEG
So for almost all
(with respect to Lebesgue measure) we have liminf
n-w>
~(e+n-~l ::: 0.
and thus also liminf P
1 [T
<8] ~ ~, for almost all eEG.
n-+co
e+n-~
n,Tn
conjunction with Lernna 5.3.2 leads to (5.3.12).
Corollary 5.3.1.
Suppose there exists functions
1)2
(8) > 0 and ]JaI (8)
a
]Jc/e1
and
Thi.s in
1)~ce)
0
with
~ 0 and contir,uous on 8:; such that for all SEG ,
Then 1) 2 (8)
a
, for almost all 8EG.
We remarked earlier that the results which we have obtained here will
be useful in dealing with the question of asymptotic efficiency of estima""
tors of the type Tn T . In order to provide some notion of efficiency, let
, n
us consider a class C* of estimators (of 8) of the fonn T
with
n,T n
126
Tn,k
= Tn,k (Z(k))
l~k~n , a~d
~,
earlier.
For each
n~l
T
n
the stopping variable described
and eEG consider the following collection of
intervals
and
where, in the second case the inteliTals may depend on the choice of
sequence
h n;
~I}
T
n,Tn
in C *.
Then for a given sequence of stopping variables
we say the sequence of estimators {V
} is asymptoticaZly
n,Tn
efficient with respect to the class C* if lim Pe[V
exists for all t ,t
l 2
>
a
d (e,tl't 2)]
n-+co
n, Tn n
and each 8Ee and the inequality
holds for all {Tn,T }EC *, every tl't 2 >
n
a and
almost all eEe.
This
definition is modelled after the concept of efficiency introduced by
Wolfowitz (1965) in cOIll1ection with the maximum likelihood estimator in
the iid case.
Our treatment here is only an initial attempt in arriving
at a versatile theory relating to the efficiency of the Tn T in which
, n
the choice of sequence {Tn;
~l}
shou.ld playa more prominent role.
Following Roussas (1972) we may fuI'Tlish upper bounds for the left hand
side of (5.3.13) in the case of se,reral classes C*.
In Theorem 5.3.2
we have provided such a bound for the class
C*=: J{T
l
n , Tn
}:
1
L[n~(T
11,
n
-8) IPe] ~ N(O,U~(e))
for each eEe} .
Note that the discussion of Chapter II ensures that there exists a
very wide and potentially quite useful collection of statistics within
this class C*.
e
~27
5.4.
Concluding remarks.
There is much left to be done in providing suitable criteria for
judging the efficiency of the statistics Tn
T'
} n
Our discussion in the
previous sections of this chapter made no mention of the "optimal"
selection of Tn.
This
question~
of great practical value,
apart from mere academic interest, is
For a given problem we would wish to know
which sequences {T } of stopping rules and statistics {T k;
n
11,
l~k~n}
will
yield the "most efficient" statistical procedure via Tn,T for the
n
problem at hand. From the large sample point of view we may investigate
how the concept of asymptotically pointwise optimal (APO) rules may be
introduced into our general infrastructure.
A preliminary study of this
question may be initiated along the arguments of Bickel and Yahav.
The same questions arise in relation to detennining the appropriate
statistical tests for a wide varie'ty of problems which can be posed within
our framework.
Once again several results can be derived for asymptotically
miformly most powerful (AUMP) and asymptotically uniformly most powerful
unbiased (AUMPU) tests.
See
JO~lson
and Roussas (1968, 1970) and
ROL~sas
(1965) .
A few remarks on the sequences {Tn, k; l$k~n} are in order. Suppose
we wish to obtain a suitable estimator for an unhl0wn parameter e based
on the ordered observations Z1"'" Zn'
We are confronted inunediately with
the problem of identifiability - - two different probability distributions
may give rise to the same vector of order statistics, say ~(k), and so
using Tn, k = Tn, k(~(k)) alone will not distinguish them. However, in our
formulation of the problem, the explicit parametric form of the joint
probability density Pe(·,n) of ~(k) will be assumed to be available.
128
In this situation, let us, for example, define for each k,
en, k to
l~k~n,
be a solution of the equation
Under appropriate condition (see Schmetterer (1966)) the
selected to be properly measurable and the estimator
A
en, k can be
en,T
may be
n
A
behavior of en,T .
n
The entire question of the as~totic properties of maximum likelihood
defined.
We are then interested in the
as~nptotic
and of Bayes' estimators has been elegantly investigated by Ibragimov
and Khas'minskii (1973, 1975b).
These authors treat both the iid and
independent not necessarily identically distributed cases by utilizing
the weak convergence of associated likelihood ratio processes.
For our
specific dependent model involving ordered observations, the technical
machinery has been set up in Chapter I I I .
Indeed, we can prove, under
A
certain restrictions, that en,T is a consistent estimator of
n
8 obtains
e and when
Other investigations of the maximum, likelihood estimator are available.
See Roussas (1968), Philippou and Roussas (1975) and Basawa, Feigin
and Heyde (1976).
The latter trea':::ment is appropriate to the generali-
zations of Section 4.4.
As~)totic
optllllil testing is treated
same framework in Basawa and Scott (1977).
L~
this
Our current research is con-
cerned with utilizing these basic ideas in the context of our mathematical
model.
e
APPENDIX
1.
We collect here the few basic theorems which were repeatedly referred
to in the preceding chapters.
The first result is based on a theorem of
Sen (1961) on the convergence of moments of extreme values from a distribution with finite end points.
We shaJ.l treat here only the situation
that we have encountered.
Result 1.
i~l}
Let {Y.;
1
be a sequence of iid nonnegative r.v. 's with
a continuous pdf h(y), yER + and h(O)
g: lR +
-+
>
O.
be continuous in some neighborhood of the
lR
origin: g'(O)
=
lim g'(y)
yi-O
: Elg(y)!a <
Then, if Yn,1.
=
min{Y.;
1
00,
for some a > O.
l~i~n},
lim E{nalg(Yn l)--g(O)l a }
n->oo
Proof.
'
Let H(y)
= P[Yl~Y]'
f(a+l){lg'(o)l/h(O)}a.
+
ydR . 1ne pdf of Yn, 1 is nh(y){l_H(y)}n-l.
Therefore,
00
=
j nalg(x)-g(O) lanh(x) (l-H(x))n-l dx
o
x
=
a+1 n,O
a,
n I
n
[f
Ig(x)-g(O) I n(x)(I-H(x)) - dx
o
f
+
x
00
Ig(x)-g(O)I~(x)(l-H(x))n-ldx]
n,O
where xn, OE(O,oo) will be selected later.
(A.I)
130
Now {l-H(x)} is a decreasing function of x and so in [xn,O,oo) the
maxiJm.nn of the function is attained at xn, 0'
Hence
00
J
n,
2 = na+l
f g(x)-g(O)
i l a n(x) (l-H(x)) n-l dx
x
n,O
00
J
x
~
Ig(x)-g(O)I~(x)dx
11,0
na+l {l-H(Xn,O)} n-l EIg(Yl)-g(O) ,a
Select xn, a such that H(xn, 0) = cn
a+l
n
n {l-H(xn, a)}
-0
,where c >
=
a+l
-0 n
n
(l-cn)
~
na+l {exp(-cn -0 )} n
° and ° < 0 < 1.
Then
e
1-0
a+l
= n exp (-cn ).
Hence lim na+l{l-H(Xn O)}n = 0; also lim{l-H(x O)}= 1.
n~
,
n~
n,
obtain lim J 2 = O.
n~
n,
We therefore
To handle the first integral in (A.I) we proceed as follows.
g has
a first order Taylor expansion in a right neighborhood of zero; we write,
for XE: (0 ,xn, 0)
g(x) - gee) = Xg'(AX)
Let
where O<A<l .
x
J
n,l
a+l n,O
=n
J Ig(x) -g(O) Ian(x){l-H(x)} n-l dx.
°
Then
(A. 2)
131
Note that A depends on n through xn,O'
There
IS
a point Xn,oE(O,Xn,o)
such that
° °
°
°
Now xn, -+ as n + 00 and for any E: > there exists nee:) > such that
/g' (y) - g'(O) I < E for < y < n(E). Also for all n ~ some no(E) ,
xn,
°
< neE).
°
Hence
(A.4)
It is shown in Sen (1961) that
H(x 0)
n)
nf' x£+r-l(l_ x )n-rdx
r (r
=
r(rr
+n
(-£-1) .
n -£ r r o
°
Set r = 1,
Q, =
a.
Combining (A.2), (A.3) and (A.4) we obtain
lim] 1
n+OO n,
=
r(a+l){lgY(O)!/hCO)}a
and this terminates the proof.
2.
o
Our next result is a variant of a result of Hoeffding'(1953).
Let {X.;
1
i~l}
bution F on R'.
be a sequence of iid r,v. 's with a continuous distri-
Let Xn, l""'Xn,n denote the ordered observations of
the sample Xl"",Xn ' For each n ~ 1 let L n be an integer-valued r.v.
taking values in {l, ... ,n}.
132
Result 2.
Let g: JR
+
be a (Borel) measurable function,
JR
EIg(X)1 < 00, and
n-1 Tn ->pr aE(O,l] as n
+
00
•
Then
Let Pn denote the empirical distribution function of X1, ... ,Xn .
Proof.
We note that
T
n-
1
n
L
g(X 1
n,
0
i=l
)
==
-co
Therefore
p-l(a)
J
g(x)dF(x)
-00
== J
n,l" + J l'1,2 say.
CA.5)
We first consider the case a < 1.
J
n,
2 = {n- 1
Now
n
L g(Xo)I(Xo<p-l(a))
i==l
1
p-l(a)
J
-
1
g(x)dP(x)},
-00
and thus from the strong law of large numbers (SLl.l'J) we obtain
J n, 2 ->L 0,
as:n
+
00
(A.6)
•
1
Our problems are only with J
n,
1; we must show J
n,
1 ->L
1
0 as n
+
00.
..
•
133
Suppose
(D,~,P)
is the underlying probability space.
En, l' En, 2cR x D as fol1aws.
Por arbitrary
Then Gn ,En, l' En, 2 are measurable sets.
Define GncD,
> (),
E:
Note that Xn,T
n
=
-1
-1
Pn (n Tn)·
IJ 1 1 :0;
fuE Ig(x) II l(x < F--lCn-IT.))
- l(x < p- 1 (a)) IdPn(x) ,
n,
E
n
n
n,l n,2
for each WED
But
00
=
E(lG (w)n-
1 :n
l.
- 1
1=
n
Ig(x·)I)
(A. 7)
1
By the SLUJ,
n
E(IG (w))
=
n
IXn,T
n
-1 n
L Ig(x.) I ->
i=l
1
L1
Elg(X) I, as n
PClxn,T -P- 1 (a)1 > s)
n
- P-1 (a)
I
:0;
:0;
IF-1 (n -1 Tn)
n
sup
(0,1)
+
° as n
+
- F-1 (n -1 Tn)
,p~l(x) - P-l(x) I
n
+
00.
in view of the inequalities,
00,
I
+
IP-1 (n -1Tn)
+
Ip-l(n-IT ) - p-l(a)!
n
I
n
and the continuity of P; thus CA.7) vanishes in the limit.
with the quantity ECIJ 11 I c),
n,
G
- P-1 (a)
We are left
134
= J Ig(x) II CdF (x)
E
n,l
+
G n
n
f
A
-1
For (x,w)EEn, l' wiGn we have x ; : F (ex) and x
For (x,w)EEn, 2' wiGn we have x
(A. 8)
Ig(x) II cdFn(x) .
G
n
E
n,2
-1 -1
1
Fn (n Tn) ::; F- (ex) + E
<
> ]:";-1 (ex ) - E
F-1 (ex) and x ; : Fn-1 (n -1 T)
n
<
.l
.
Therefore
E
f
Ig(x) II
n,l
GC
dF (x)
n
f
=
n
E
n,l
n (R xG c )
[F
f
(ex) ,F
again, by the SLLN.
E
f
-1
(ex)+E)
Ig(x)/ -(
1
-1
(ex) +E
If
F- (ex)
Ig(x) IdF(x) , as
n
-+
00
Similarly
Ig(x) II cdFn(x) =
G
n
n,2
n
n
F
~1
Ig(x) IdF (x)
E
n,
f
c
2n (IR xG )
Ig(x) IdFn(x)
n
F- 1 (ex)
::;
iF
-1
f
(ex) -s,F
-1
(ex))
Ig(x) jdF (x) -(
n
1 F
-If
(ex)-E
Ig(x)ldF(x)
asn-+
y
Since Ejg(x)1 <
00,
the indefinite integral
J Ig(x) IdF(x)
is continuous in y
-00
and thus since
E
> 0 is arbitrary i t follows from (A.8) and the above argu-
ments that
EIJ 1,1 I
n,
GC
-+
0 as n
n
This establishes Result 2 for the case ex < 1.
If ex = 1 Result 2 claims that
-+
00.
oo •
135
n
T
n
-1
.r g(~,i)
1=1
+00
-11_ J g(x)dF(x)
.
(A. 9)
00
To show this, observe that (A.s) is still valid with the proper interpretation and indeed (A.6) is trivially true. To handle In,llet us define
for arbitrary E > 0 and n ~ 1
and
Then both En' Hn are measurable sets.
in the form
Furthermore, we can re -express I n ?1.
+00
J n, 1
f
=
-00
g(x){I E (x)-l}dFn (x) .
n
1 n
Then IJn III H ~ I H n- .L !g(Xi ) I· Since E(Hh) = P(Hn) + 0 as n + 00, we
,
n
n
1=1
have from the SLLN, J lI H -L:> O. Consider (x,w)dR'xn with w,{H and
n,
n
1
n
(x,w)/En . Then n-lT (w) > l-E and ~. (w) ~ x. Hence the inequalities
n
,Tn
x ~ Xn,[n(l-E)]'
Now
+00
/In,lII Hc ~-L Ig(x)II(X~~?[n(1_E)J)d2n(X)
n
-roo
=-L
Ig(x)!n - I(x<XIij[n(l_E)])}d}~(x)
-1 n .
= n i~lg(Xi)
1-
_l[nU-E)]
i~l
n
/g(Xn,i) I
(A.IO)
.
From the first part of our proof and the SLLN the right hand side of (A.IO)
converges in Ll to
+00
f
-00
!g(x) IdF(x)-
p-l CI -E )
J
Ig(x) IdF(x)
_00
which can be made arbitrarily small by letting E be so.
Hence (A.9).
0
136
3.
We now state a few results taken from McLeish (1974) on central limit
We cons ider an array of
k
n
random variables {~n,1": l~i~kOn } on a probability space (~,G,P). (Bn ,i)i=O
theorems for sums of dependent random variables.
is a sequence of sub-a-fields of G such that, for each i, Bn , 1"-1 c Bn,1".
Suppose each ~n,1"is Bn,1. measurable. Let J be an interval, either of the
form [O,T], for some T <
00
or the half line [0,00) and let D(J) denote the
class of right-continuous real valued functions on J with left hand limits
endowed with the usual Skorohod topology.
We wish to discuss weak conver-
gence of random functions with paths in this space.
For compact J the
topology in question is described in Billingsley (1968) Chapter 3.
For
semi-infinite J, it follows from Stone (1963) or Lindvall (1973), (see
also Whitt (1975)) that it suffices to consider weak convergence with the
functions restricted to every finite interval [O,T], T > O.
Let kn(t) , tEJ be a sequence of integer-valued nondecreasing right
continuous functions on J such that
~ (0)
= a for all n
~
1.
Define the
process {Wn(t): tEJ} = W
n , by
Then Wen
has paths in DCJ) and Wn (0) :::: 00 We state below several sets of
1
conditions under which the weak convergence of W
n to Brownian Motion Win
D(J) can be realized.
As usual -> , -> denote respectively convergence
pr w
in probability and weak convergence.
We call
{.~
":
n,1
l~i~k
n
} a martingale difference array whenever
"1) = 0 a. s. for all i and n.
n,1. IBn,lis Result 5.
E (~
The main theorem for our purposes
~
137
Result 3.
(a)
Suppose
{~n
, i}
is a martingale difference array satisfying:
max
I~ I -> 0, and
l:;;;i:;;;~(t) n,l
0
k (t)
n
(b)
~2. _> t, for each tEJ .
.I..~ "'n
1
pr
1=1
'
Then Wn ->w W in D(J).
Result 4.
Suppose {cn,lo} is
from 0 and and
ffil
array of positive constants bounded away
00
(a)
(b)
e
max
l:;;;i::s:k (t)
n
k (t)
n
I~
·1
n,l
pr 0
->
: 1}1 ->pr
i~l IE{~,iI(I~n,il :; ; cn,l.) IBn,l-
k (t)
n
(c)
L
~2
1=1 n ' i
o
->
pr
t
' for each tEJ.
Then Wn ->w W in D(J).
Result 5.
(a)
Let
{~
.} be an array satisfying
n,l
The conditional Lllldeberg:
For each s > 0
knCt)
2
ECsn
C
I C1 C • I > s)
1=1
' 1 sn , 1
o
\'
L
<
lB.
.)
n,1-1
knCt)
(b)
L EC~2.
18 . ) ->
n,l n,1-1 pr
i=l
k (t)
n
(c)
t
,
and
i~l IEC~n,iI8n,i-l)1 -pr 0 .
Then Wn ->w Win DCJ).
pr 0,
->
0 , and
138
Remark 3.1
(i)
The conditions (a), (b), (c) of Result 5 imply those of
Result 4.
(ii)
Results 3,4,5 remain valid if
piIig time adapted to
k
''11
{B
.; l:-:;i~n}.
i E(~2 ·18
i::: 1
is, for each n and t, a stop-
For example if we take
n,l
(t) ::: inf{j;
~(t)
n,1
.
1)
n,1 -
>
t}
(A,l1)
then condition (b) of Result 5 is a consequence of the conditional LindeThus if {~n,l.; Bn,l·}~.l
1= is a martingale difference array and
kn(t) is given by (A.ll), then Result 5 obtains under the conditional
berg (a).
Lindeberg (a).
4.
Alternative
expressions for J (e).
----------"'-------.-,0.We define J (e) as in (3.2.17) - for each aE(O,l] and eEe by
a
(A.12)
We are retaining here the asswrrptions of Sectirnl 3.2.
'then if aE(O,l)
we can also write
F~l(a)
:: J
00
f 2 (x)dF e (x)
+
00
and Ee rGe (x)
F~l(a)
Ja(e) ::: -{
f
(A. 13)
F~l(a)
-00
If additionally Eel"fe(x)/ <
J f e(x)dFe (x)}2
(l-a)-l{
I
<
00
,
we also have
.
"fe(x)dFe(x)
+
(l-afGe(F~lca))}
(A. 14)
-00
provided condition (*) of [A9] (see Section 4.2) holds.
(A.13).
we get
Now Ye(x) ::: fe(X) - Ge(x).
Let us first show
So by direct substitution in (A.13)
F~I(a)
Ja(e)
F~I(a)
J f~(X)dFe(x)
=
f
+
-00
-00
139
{G~(x)
2fe (x) Ge (x) }dF'e (x) .
(A. IS)
We need to verify that the second integral in (A. IS) reduces to
(See Remark 3.2.1).
So by an integratirnl by parts? for each ZElR' we have
x
f
fe(x)Ge(z)dFe(x)
=
Ge(z)
-00
f
z
fe(x)dFe(x)
-00
z
+
x
J re(x)~e(x){ J fe(y)dFe(y)}d~(x)
-00
where
~
is Lebesgue measure.
aGe
-ae
(x)
= -
f
Note that for each xdR i
af
x
a~ (y)d~(y)
-00
f
= -
2-+-00
J
fe(y)dFe(y)
fe(x)dFe(x) = O.
We have therefore
-00
2
2
J fe(x)Ge(x)dFe(x)
G~(2)Ge(2) - J Ye(x)Ge(x)dFe(x)
= -
-00
for each 2EIR',
x
-00
·2·
and so lim Ge(z)
,
-00
(A. 16)
-00
By [A4]eb)
-1
Fe (a)
J
00
fe(x)dFe(x)-
_1
-00
f f e ex)dFe (x)
Fe.&.ea )
This yields
00
=
J
el-a)-l{
feex)dFeex)}2
eA.17)
p-l ea )
Note that the asslmlption fe(F~lea)) >
a has been used here. Again by
an integration by parts, for each 2EIR',
140
z 2
J Ge(x)dFe(x)
-00
z
= - G~(Z)Ge(Z) - 2
J ~e(x)Ge(x)dFe(x)
(A. 18)
-00
Therefore from (A.16), (A.I?) and (A.18) we get
00
f fe(x)dFe(x)
"
}2
(I-a) -l{
-1
Fe (a)
This establishes (A.13).
To obtain (A.14) observe that under our additional
assumptions,
"Ge(x)
2
a
=ae2
log G (x) =
= - (G e(x))
e
-1
2 .
-1 a Ge
_2[aG e
(G (x))
--2 (x) - (G (x))
ae (x) ).
e
e
ae
2
x a f
2
2 x afe
J - 2e (z)dJl (z) - (Ge (x)) - ( f 38(z) dJl (z))
-00
dO
_00
So we have
(l-a)G~(F;I(a)) = (Ge(x)G~(X))
-1
x=F
O
(a,)
F;/ (a) a2f
=
f
-00
-1
Fe (a) af
(1-a f1 [
---i(z)dJl(Z)
de
~oo
a~
,2
dfl(Z) J
.
(A.19)
But
((fe (x) )
so that (A.19) entails
-1
af e
ae(x) )
2
+
a
ae
((fo ex))
-1
af e
ae(x))
141
F~l(a)
=-
f
f~(x)dFe(x)
-00
(A.20)
Expression (A. 14) then follows from CA.13) aJId CA.20).
Throughout the above discussion we have assumed a < 1.
We now
verify that
(A.21)
lim (l-a)'Ge(F~l(a)) = a .
a+l-
(A.22)
Then lim
J in (A.B) and (A.14) reduces to the familiar alternative
a+1- a
expressions for the Fisher InfoTInation function.
By the Schwarz inequa1-
ity,
as a
since
Eelf~(x)1
<
00.
Hence (A.2l).
immediately
= 0
For (A.22) we use (A.20).
+
1-
~
We have
142
S.
The space C(lR )
We mention here some of the salient features of the space C ::: Cern.)
of all real-valued continuous functions on JR ::: (-oo,+oo).
~
For each integer j
I, we let p.(x,y)::: sup Ix(t)-y(t)I for any
Itl~j
J
x, YEC and define p: CXC
JR
-+
by
:: I
00
p(x,y)
.
J=l
2- J
p. (x,y)
_~J_ _~
1+p. (x ,y)
J
Then p is a metric in C and
Result 6.
The function space (C,p) is a complete separable metric space
in which lim p(xn,x) :::
n~
° if and only if n-><x>
lim p.(x ,x) ::: 0, for each
n
J
j 2 1.
We are interested in characterizing weak convergence of a sequence
of probability measures in (C,C).
11ere C denotes the minimal a-field
over the open subsets of (C, p) .
Let Pn (n2l) and P denote probability measures in (C,C). We say
{Pn } converges weakZy to P in (C,p) (denoted Pn => P) if and only if
J fdP -+ JfdP for every bo'unded continuous real-valued function f on C.
C
n
C
Let us define the projection mappings in C.
elements
t-l."
t z' ... , tkE m.
'ITt
let 'ITt
. : C -+ :JR k be given by
l' .. 'C k
. (x) '" (x(t,J) ... ,x(ti
l"'~k
For any finite set of
~
~
))
Then, for any probability measure P on (C,C), the collection of measures
-1
PTI t
1'"
t
for all k and all t1, ... ,tkER, is called the finite-dimenk
sional distributions of P.
The finite-dimensional distributions complete-
ly determine the probability measure, in the sense that if pl,P Z are
probability measures in (C,C) such that PlTI- 1
::: PZTI- I
t' for all
t1···t k
t l ", k
e
143
k and all tl,···,tkEIR, then PI :: PZ'
We have the following characterization of weak convergence.
Let Pn (n ~ 1) and P be probability measures in (C,C). Then
Pn => P if and only if
(i) the finite-dimensional distributions of Pn converge weakly to
those of P, and
Result 7.
the sequence {P ;
(ii)
It
n
n~l}
is tight.
is possible to relate weak convergence of probability measures in
(C,C) with weak convergence of certain corresponding measures in (C.,C.),
J
J
where C. :: C[-j,j] and C. is the smallest a-field over the collection of
J
J
open sets in C. generated by the metric p.(x,y):: sup /x(t)-y(t)
J
X,YEC j .
J
It I::;j
To this end let us define
r j (x) (t) :: x(t) ,
Itl::;j, XEC .
Then r.: C -+ C. is a continuous (and hence measurable) function.
J
I
J
The
following simple correspondence can be stated.
Let Pn (n ~ 1) and P be probability measures in (C,C). Then
P => P if and only if P r~l => Pr~l for each j ~ 1. Furthermore, the
Result 8.
n J
n
J
sequence {Pn ; n~l} is tight if and only if the sequence {Pnr -1 ;
j
tight for each j ~ 1.
n~l}
is
Finally, we have the convenient characterization
Let Pn (n ~ 1) and P be probability measures in (C,C). Then
Pn => P if and only if
(i) the finite-dimensional distributions of Pn converge weakly to
Result 9.
those of P, and
(ii)
for each
€ >
0 and j
~
1
144
lim lim P {XEC:
h+O
n+oo
n
sup
Is-tl$h
I s I , I t I $j
Ix(t)-x(s)!
> (} = 0 .
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