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\noindent{\gr A NOTE ON NONPARAMETRIC ESTIMATIONS}
\medskip \noindent
{\it P\'eter Major$^1$ and L\'{\i}dia Rejt\H{o},$^{1,2}$ }
\medskip\noindent
$^1$ Mathematical Institute of the Hungarian
Academy of Sciences, Budapest, Hungary \smallskip\noindent
$^2$ Department of Mathematical Sciences,
University of Delaware, \newline
\hphantom{$^2$ }Newark, Delaware, USA
\medskip\noindent
{\it Dedicated
to Mikl\'os Cs\"org\H o on the occasion of his 65-th birthday}
\medskip{\narrower\noindent {\it Abstract:}\/ We give an informal
explanation with the help of a Taylor expansion about the most important
properties of the maximum likelihood estimate in the parametric case.
Then an analogous estimate in two nonparametric models, in the
estimate of the empirical distribution function from censored data
and in the Cox model is investigated. It is shown that an argument very
similar to the proof in the parametric case yields analogous
properties of the estimates in these cases too. There is an
important non-trivial step in the proofs which is discussed in more
detail. A double stochastic integral with respect to a standardized
empirical process has to be estimated. This corresponds
to the estimate of the second term of the Taylor expansion in the
parametric case. We think that the method explained in this paper is
applicable in several other models. \par}
 
\medskip \noindent{\bf 1. Some general remarks}
\medskip\noindent
An important problem of statistics is to estimate an unknown parameter
or distribution by means of a sample of size $n$, i.e.\ by a sequence
of independent and identically distributed random variables
$\xi_1,\dots,\xi_n$ with some unknown distribution $F$. If this unknown
distribution $F$ belongs to a class of distribution functions
$F(x,\vartheta)$, where $\vartheta$ is a real number or more
generally the element of a finite dimensional vector space, and we are
interested in the value $\vartheta$ or a function $g(\vartheta)$ of it,
then we speak of parametric estimation. In the other case when the set
of possible distributions where the sample can come from is an
``infinite dimensional space" we speak of nonparametric
estimation.
 
The case of parametric estimation is considerably simpler. In this case
there is a powerful technique, the maximum likelihood method which has
the following two nice properties:
 
 
\item{i.)} It supplies a method for a large class of problems.
\item{ii.)} Under general conditions it is asymptotically optimal.
 
 
In case of nonparametric estimation problems no such good and general
method is available. Nevertheless, there are some special cases where
such a good estimate can be given as in the parametric case. The
investigation of both question, i.e.
\vsize= 23truecm
 
\vfill\eject
 
 
\item{i.)} to find a general principle which enables us to
give a good estimate in the nonparametric case
\item{ii.)} to show that the estimate is as good as the estimate
in the parametric case
 
\noindent
are challenging problems. There exists a large literature on
nonparametric maximum likelihood estimation 
(see e.g.~[bhhw],~[gil1],~[gil2], ~[lc]).
Some special nonparametric models will be considered, and we
prove that the estimates proposed in these cases are as good as the
maximum likelihood estimate in parametric models. The structure of the
proof is similar to that in the parametric case, but some additional
technical problems appear.  These problems and their solutions deserve
special attention.  To explain them first we present a short informal
explanation about the limit behaviour of the maximum likelihood method
in the parametric case.  Here we assume that the distributions satisfy
some natural smoothness conditions which we do not formulate explicitly.
 
Let us consider the simplest parametric problem when a parameter
$\vartheta_0\in {\bold R}^1$ has to be estimated from a class of
distribution functions $F(x,\vartheta)$, $\vartheta\in {\bold R}^1$,
by means of a sequence of independent random variables
$\xi_1(\omega),\dots,\xi_n(\omega)$ with distribution
$F(x,\vartheta_0)$.
 
We also assume that the distribution functions $F(x,\vartheta)$
have a density function $f(x,\vartheta)$ with respect to a measure
$\mu$ on the real line. The maximum likelihood method
suggests to choose the estimate $\hat \vartheta_n
= \hat\vartheta_n(\xi_1,\dots,\xi_n)$ of the parameter
$\vartheta_0$ as the number where the density function of the random
vector $(\xi_1,\dots,\xi_n)$ (with respect to the product measure
$\underbrace{\mu\times\cdots\times\mu}_{n \;{{\text{times}}}}$), i.e.\ the
product
$$
\prod_{k= 1}^n f(\xi_k,\vartheta)= \exp\left\{\sum_{k= 1}^n\log
f(\xi_k,\vartheta)\right\}
$$
takes its maximum. This point can be found as the solution of the
equation
$$
\sum_{k= 1}^n\frac{\partial}{\partial\vartheta}\log
f(\xi_k,\vartheta)= 0\;. \tag1.1
$$
We are interested in the asymptotic behaviour of the random variable
$\hat\vartheta_n-\vartheta_0$, where $\hat\vartheta_n$ is the
(appropriate) solution of the equation~(1.1).
Let us take Taylor expansion of the expression at the left hand
side of~(1.1) around the point $\vartheta_0$. We get
$$
\aligned
\sum_{k= 1}^n\frac{\partial}{\partial\vartheta}\log
f(\xi_k,\hat\vartheta_n)&=
\sum_{k= 1}^n\frac{\frac{\partial}{\partial\vartheta}
f(\xi_k,\vartheta_0)}{f(\xi_k,\vartheta_0)}\\
&\qquad+(\hat\vartheta_n-\vartheta_0)
\(\sum_{k= 1}^n\(\frac{\frac{\partial^2}{\partial\vartheta^2}
f(\xi_k,\vartheta_0)}{f(\xi_k,\vartheta_0)}-
\frac{\(\frac{\partial}{\partial\vartheta}
f(\xi_k,\vartheta_0)\)^2}{f^2(\xi_k,\bar\vartheta_0)} \)\)\\
&\qquad\qquad+O\(n(\hat\vartheta_n-\vartheta_0)^2\) \\
&= \sum_{k= 1}^n\(\eta_k+\zeta_k(\hat\vartheta_n-\vartheta_0)\)
+O\(n(\hat\vartheta_n-\vartheta_0)^2\)\;, \endaligned \tag1.2
$$
where
$$
\eta_k= \frac{\frac{\partial}{\partial\vartheta}
f(\xi_k,\vartheta_0)}{f(\xi_k,\vartheta_0)}\quad \text{and}\quad
\zeta_k=
\frac{\frac{\partial^2}{\partial\vartheta^2}
f(\xi_k,\vartheta_0)}{f(\xi_k,\vartheta_0)}-
\frac{ \left( \frac{\partial}{\partial\vartheta}
f( \xi_k,\vartheta_0)\right)^2}{f^2(\xi_k,\bar\vartheta_0)}
$$
for $k= 1,\dots,n$. We want to understand the asymptotic behaviour of
the (random) expression on the right-hand side of~(1.2). The
relation
$$
E\eta_k= \int\frac{\frac{\partial}{\partial\vartheta}
f(x,\vartheta_0)}{f(x,\vartheta_0)}f(x,\vartheta_0)\,d\mu(x)
= \frac{\partial}{\partial\vartheta}\int
f(x,\vartheta_0)\,d\mu(x)= 0
$$
holds, since $\int f(x,\vartheta)\,d\mu(x)= 1$ for all $\vartheta$, and
differentiating this relation we get the last identity. Similarly,
$E\eta^2_k= -E\zeta_k= \int\frac{\(\frac{\partial}{\partial\vartheta}
f(x,\vartheta_0)\right)^2}{f(x,\vartheta_0)}\,d\mu(x)>0$, \
$k= 1,\dots,n$. Hence by the central limit theorem
$$
\chi_n= \frac{1}{\sqrt n}\summ_{k= 1}^n\eta_k
$$
is asymptotically normal with expectation zero and variance
$$
I^2= \int\frac{\left(\frac{\partial}{\partial\vartheta}
f(x,\vartheta_0)\right)^2}{f(x,\vartheta_0)}\,d\mu(x)>0\;.
$$
In the statistics literature this number $I$ is called the Fisher
information. By the laws of large numbers
$\frac{1}{n}\summ_{k= 1}^n\zeta_k\sim -I^2$.
Hence it follows from relation~(1.2) that
$$
\frac{1}{\sqrt n} \sum_{k= 1}^n\frac{\partial}{\partial\vartheta}\log
f(\xi_k,\hat\vartheta_n)= \chi_n-\sqrt n(\hat
\vartheta_n-\vartheta_0)\,I^2 +\text{negligible error.} \tag1.3
$$
Formulas~(1.1) and~(1.3) imply that
$$
\sqrt n(\hat \vartheta_n-\vartheta_0)= I^{-2}\,\chi_n
+\text{negligible error,}\tag1.4
$$
which is asymptotically a normal random variable with expectation zero and
variance~$I^{-2}$. Another result of the mathematical statistics, the
so-called Cramer--Rao inequality states that this result is
asymptotically optimal, since under general conditions all unbiased
estimates (such estimates whose expectation equals the estimated
parameter) have a variance which multiplied by $\sqrt n$
cannot be smaller than $I^{-2}$.
 
We are also interested in the magnitude of the ``negligible error"
term in formula (1.4) and want to compare it with the error term
appearing in the analogous formulas proved for the nonparametric
estimates discussed in Sections~2 and~3. We shall show that the
(normalized) error can be approximated by (normalized) sums of
independent random variables in the parametric and by a linear
functional of the standardized empirical distribution function in the
nonparametric case. We give an informal explanation that the error of
this approximation is of order $O(n^{-1/2})$ in both cases.
 
In this section we discuss the parametric case. Put
$$
\e_n = \sqrt n \left( \sqrt n(\hat\vartheta_n-\vartheta_0)
-I^{-2}\chi_n\right)
$$
and
$$
\omega_n= n^{-1/2}\sum_{k= 1}^n(\zeta_k+I^2)\;,
$$
and express $\sqrt n(\hat\vartheta_n-\vartheta_0)$ and
$\summ_{k= 1}^n\zeta_k$ through $\e_n$ and $\omega_n$. Then
relations~(1.1) and~(1.2) imply that
$$
\sqrt n\chi_n+(n^{-1/2}\e_n+I^{-2}\chi_n)(\omega_n-\sqrt n I^2)
+O\left( (n^{-1/2}\e_n+I^{-2}\chi_n)^2\right)= 0\;,
$$
or equivalently
$$
\omega_n\chi_n I^{-2}-I^2 \e_n+n^{-1/2}\omega_n\e_n+
O\left( (n^{-1/2}\e_n+I^{-2}\chi_n)^2\right)= 0\;.
$$
Since the random variables $\chi_n$ and $\omega_n$ are stochastically
bounded, the last relation implies that $\e_n= \sqrt n\left(\sqrt
n(\hat\vartheta_n-\vartheta_0)-I^{-2}\chi_n\right)$
is stochastically
bounded. This is equivalent to saying that the normalized error $\sqrt
n(\hat\vartheta_n-\vartheta_0)$ of the maximum likelihood estimate
can be approximated by an appropriate normalized sum of independent
random variables (by $I^{-2}\chi_n$) with a (random) error of order
$n^{-1/2}$.\/ A longer Taylor expansion in formula~(1.2) and an
Edgeworth expansion for sums of independent random variables yields a
better approximation, an Edgeworth type expansion for $\sqrt
n(\vartheta-\vartheta_0)$.
 
We are interested in what can be preserved from the methods
and results of the parametric estimates when  nonparametric estimation
problems are considered. Since in nonparametric models
typically the set of all probability measures in the model cannot have a
density function with respect to a fixed measure, the maximum likelihood
principle cannot be applied. Nevertheless, in several
interesting models a good estimate can be found by an appropriate
modification of the maximum likelihood argument. We shall consider such
models and show that these estimates are good.
 
The simplest nonparametric estimation problem is the estimation of an
unknown distribution function $F(x)$ by means of a sequence of
independent $F$ distributed random variables $\xi_1,\dots,\xi_n$.
Define the empirical distribution function
$$
F_n(x)= \frac{1}{n}\#\{k: \; \xi_k\le x\}\;.
$$
By a classical result of probability theory the processes
$\sqrt n (F_n(x)-F(x))$ converge weakly to a Gaussian process $Z(x)$,
$-\infty<x<\infty$, with expectation zero an covariance function \quad
$EZ(x)Z(y)= \min\{F(x),F(y)\}-F(x)F(y)$.\/ We want to show that a similar
limit theorem holds for \
$\sqrt n\times \text{(the estimate $-$ the real value)}$ \
for a good estimate in other estimation problems. Moreover, we want to
investigate the distance of this expression for fixed $n$ from the
limit process. In the parametric case this expression can be
approximated by the normalized sum of independent random variables, or
what is equivalent to this, by a normal random variable with an
accuracy of order $n^{-1/2}$. Actually, it would demand a more detailed
explanation of how we measure the accuracy of approximation in the last
statement. But we shall not do this, we only explain the
content of this statement in the nonparametric models  discussed in
the sequel.
 
We shall discuss two models, the so-called random censorship and the
Cox model and write down an estimate in both models and show that they
are good. The difference of the estimate and the real distribution
multiplied by $\sqrt n$ converges to a Gaussian process. Moreover, this
difference can be approximated by a linear functional of a normalized
empirical process with an error of order $n^{-1/2}$. More explicitly,
the error is a random process, and if it is multiplied by $\sqrt n$
then the probability that the supremum of this process is larger than
some $x>0$ is smaller than $C_1e^{-\lambda x}$ with some constants $C$
and $\lambda$ independent of $n$ and $x$.
 
The linear functional of a standardized empirical function is the
natural infinite dimensional counterpart of sums of independent random
variables which appears in the parametric case. In the nonparametric
estimates considered in this paper the main contribution to the error
(multiplied by $\sqrt n$) of the estimate is expressed in such a form.
It converges to a Gaussian process as the sample size $n\to\infty$.
The difference of the estimate and the real distribution multiplied
by $\sqrt n$ can also be approximated by an appropriate Gaussian
process. This appears as the limit of the linear functionals of the
empirical processes that yield the main contribution to the (normalized)
error of the estimate. The goodness of this Gaussian approximation
can be determined with the help of the bound on the linear functional
approximation of the error and a result of Koml\'os, Major and
Tusn\'ady~[kmt] (see in~[csr] for more details) about the
approximation of the empirical process by a Brownian bridge.
 
The proof of these results is based on an expansion which can be
considered as an adaptation of the investigation of the maximum
likelihood estimate in the parametric case. One point of the
proof deserves special attention. If we want to adapt the method
of the parametric case, then an expression analogous to formula~(1.2)
has to be studied. A term which corresponds to the error
term $O\left(n(\hat\vartheta_n-\vartheta)^2\right)$
of the Taylor expansion in~(1.2) has to be well estimated. In the
nonparametric case this problem can be solved if the distribution of
certain non-linear functionals of the empirical distribution function
of the sample $(\xi_1,\dots,\xi_n)$ is well bounded.
 
Hence the following problem has to be studied. Let us
consider the normalized empirical distribution of a sample, the
$s$-fold direct product of their (random) measure with itself, and let
us estimate the integral of a bounded function of $s$ variables with
respect to this product measure. If $s\ge 2$, then this integral is a
non-linear functional of the empirical distribution. We are mainly
interested in the special case $s= 2$, but the results will be
formulated for general~$s$. The proof of these results can be found
in~{[mp]}, {[mr]}, and ~{[mrsi]}. First we formulate the result in a
 slightly
restrictive form, when the random variables whose empirical
distribution is considered are uniformly distributed in the interval
$[0,1]$. This result is sufficient for instance in the investigation of
the random censorship model.
 
\noindent
{\bf Theorem A.} {\it Let $\xi_1(\omega),\dots,\xi_n(\omega)$
be independent uniformly distributed random variables on $[0,1]$,
$F_n(u)= F_n(u,\omega)= \frac{1}{n}\#\{k:\; 1\le k\le n,\; \xi_k\le u\}$,
$0\le u\le1$, their empirical distribution function and $\mu_n(u)= \sqrt
n(F_n(u)-u)$ the standardization of this empirical distribution
function. Let $f(u_1,\dots,u_s)$ be a function on $[0,1]^s$ such that
$\sup\limits_{u_1,\dots,u_s}|f(u_1,\dots,u_s)|\le1$, and
$f(u_1,\dots,u_s)= 0$ if $u_j= u_k$ with some $1\le j<k\le s$.
There exist some universal constants $C_s$ and $\alpha_s$ depending
only on the dimension $s$ in such a way that
$$
P\left(\sup_{0\leq t \leq 1}\left|\int_0^t \int_0^1 \cdots
\int_0^1 f(u_1,\dots,u_s)d\mu_n(u_1)\dots d\mu_n(u_s)
\right|\ge x \right) \leq C_{s} \exp\left\{-\alpha_s x^{2/s}\right\}
$$
for all $x>0$, and function $f$ with the above properties.}
\medskip
In other cases, like in the Cox model,  the integral with respect to the
product measure of an  empirical distribution in a higher dimensional
Euclidean space is needed.  The proof of such results (actually the
reduction of such results to the case described in Theorem~A) is not
harder in the case of general separable metric spaces, hence we
formulate the result in such a form. To do this we introduce some notations.
 
Let a probability space $(\Omega,{\Cal A},P)$ and a separable metric
space $(X, {\Cal X})$ be given.  Let $\xi\: \Omega \to X$ be an
$X$ valued random variable on $(\Omega,{\Cal A}, P)$. Let $\mu$ denote
the distribution of the random variable $\xi$, i.e.\ let
$$
\mu(B)= P(\xi \in B)= P(\xi^{-1}(B))\qquad \forall \; B \in {\Cal X}\;.
$$
Suppose that $\xi_1,\xi_2,\dots,\xi_n$ are independent, identically
distributed random variables on $(\Omega, {\Cal A}, P)$ with values
on the space $(X, {\Cal X})$ and distribution $\mu$. We introduce the
empirical measure
$$
\bar{\mu}_{n}(B) =  \frac{1}{n}\sum_{i= 1}^n I(\xi_i \in B) \qquad
\forall\; B\in {\Cal X}\;,
$$
and its standardization
$$
\mu_n (B)= \sqrt{n}\left(\bar\mu_n(B)-\mu(B)\right) \qquad
\forall \; B\in {\Cal X}\;.
$$
Let $X_t$, $0\leq t\leq1$ be a system of sets in ${\Cal X}$
with the following property:
 
\medskip\noindent
{\bf Property (i)} $X_s \subseteq X_t$ for all $s\le t$,
$X_0= \emptyset$, $X_1= X$, $\mu(X_t)= t$.
 
\medskip
Let us consider the product space $\underbrace{X\times
\cdots\times X}_{s \; {\text{times}}}= X^s$
with product measure $\mu^{(s)}(\cdot)$ and the diagonal set
$A\in X^s$ is defined as
$$
A= \{(x_1,\dots x_s)\: x_i= x_j \quad \text{for some }i\neq j \}
$$
Let ${\Cal F}$ denote the set of the real valued measurable
functions $f(u_1,\dots,u_s)$ defined on the space $X^s$
whose absolute value is less than 1, and which disappear on the
diagonal set $A$, i.e.\ let
$$
{\Cal F}= \{f(u_1,\dots,u_s)\: |f|\le1, \quad f(u_1,\dots,u_s)= 0\;
\forall \;  (u_1,\dots,u_s)\in A\}\;.
$$
Then
 
\noindent {\bf Theorem B.} {\it There exist some universal constants
$C_s$ and $\alpha_s$ depending only on the dimension $s$ in such a
way that
$$
P\left(\sup_{0\leq t \leq 1}\left|\int_{X_{t}} \int_{X} \dots
\int_{X} f(u_1,\dots,u_s) d\mu_n(u_1) \dots d\mu_n(u_s)
\right|\ge x \right) \le C_{s} \exp\left\{-\alpha_{s} x^{2/s}\right\}
$$
for all  $f\in {\Cal F}$ and $x>0$, where the sets $X_t$, $0\leq t\leq1$
satisfy Property~(i).}
 
\medskip\noindent
Theorem~B shows certain analogy with respect to multiple  stochastic
integrals with respect to a Gaussian process.  Here  the underlying
process, the empirical distribution function is not Gaussian, but it is
almost Gaussian. In both cases the diagonal is cut from the domain of
integration. The (random) measures of disjoint intervals are almost
independent. In Theorem~B actually we investigate how strong
cancellation is caused by this almost independence. The upper bound
$C\exp\left\{-\lambda x^{2/s}\right\}$ given for the tail distribution
of an $s$-fold integral is sharp. It expresses the fact that the tail
behaviour of an $s$-fold stochastic integral is similar to the tail
behaviour of the distribution of the $s$-th power of a Gaussian random
variable.
 
In the statistical problems we discuss below, we first write up
the statistics under investigation as a multiple integral with respect to
an empirical measure plus some possible additional terms we can handle.
Then we have to handle an expression of the form
$$
Z_n(t)= \int_{X_t}\int_X
g(u_1,u_2)\,d\bar\mu_n(u_1)\,d\bar\mu_n(u_2)\;, \tag1.5
$$
where $\bar\mu_n$ denotes the empirical distribution (without
normalization). With the help of simple algebra we get that
$$
\align
d\bar\mu_n(u_1)\,d\bar\mu_n(u_2) &=  d\mu(u_1)\,d\mu(u_2)+
\frac1{\sqrt n}d\mu(u_1)d\mu_n(u_2)\\
&\qquad+\frac1{\sqrt
n}d\mu(u_1)d\mu_n(u_2)+ \frac{1}{n}d\mu_n(u_1)\,d\mu_n(u_2)\;,
\endalign
$$
where $\mu_n(\cdot)= \sqrt n(\bar\mu_n(\cdot)-\mu(\cdot))$.
Then we can write
$$
Z_n(t)= D(t) + n^{-1/2}U_n(t)+n^{-1}V_n(t)\;,\tag1.6
$$
where
$$
D(t)= \int_{X_t}\int_X g(u_1,u_2)\,d\mu(u_1)\,d\mu(u_2)
$$
is a deterministic function,
$$
U_n(t)= \int_{X_t}H_1(u)\,d\mu_n(u)+\int_X H_2(u)\,d\mu_n(u)
$$
with $H_1(u)= \int_X g(u,u_2)\,d\mu(u_2)$ and
$H_2(u)= \int_{X_t}g(u_1,u)d\mu(u_1)$ is a linear functional of the
empirical distribution $\mu_n$ and the term $V_n(t)$
is a non-linear function of the measure~$\mu_n$.
Because of Theorem~B we have a good bound on the distribution of
$\sup\limits_{0\le t\le 1}|V_n(t)|$.\/ Hence we can handle the expression
$$
\sqrt n(Z_n(t)-D(t))=  U_n(t)+\frac1{\sqrt n}V_n(t)\;.
$$
In the following examples we shall see why this observation is useful in
the study of certain nonparametric estimates.
 
 
\beginsection 2. The Kaplan-Meier product limit estimator
 
In this section the following problem is considered. Let $(X_i,C_i)$,
$i= 1,\dots,n$, be a sequence of independent, identically distributed
random vectors such that the components $X_i$ and $C_i$ are also
independent with distribution functions $F(x)$ and $G(x)$. We want to
estimate the distribution function $F$ of the random variables $X_i$,
but we cannot observe the variables $X_i$, only the random variables
$Y_i= \min(X_i,C_i)$ and $\delta_i= I(X_i\leq C_i)$. For the sake of
simplicity we assume that both distributions $F$ and $G$ have no
atom. In other words, we want to solve the following problem. There are
certain objects whose lifetime $X_i$ are independent and $F$
distributed. But we cannot observe  this lifetime $X_i$, because
after a time $C_i$  the observation must be stopped. We also know
whether the real lifetime $X_i$ or the censoring variable $C_i$ was
observed. We make $n$ independent experiments and want to estimate with
their help the distribution function~$F$.
 
It is not easy to find the right estimate of the distribution function
$F$ on the basis of the above observations.  Kaplan and Meier in~[km]
proposed the so-called product limit estimator to estimate the unknown
survival function $S= 1 - F$ on the basis of the above observations.
They proposed the following estimator $S_n$:
$$
1-F_n(u)= S_n(u)= \left\{
\alignedat2
&\prod_{i= 1}^n\left(\frac{N(Y_i)}{N(Y_i)+1}\right)^{I(Y_i\leq u,
\delta_i= 1)} && \text{ if }u\leq\max(Y_1,\dots,Y_n)\\
&0&& \text{ if } u\geq\max(Y_1,\dots,Y_n),\;\delta_n = 1,\\
&\text{undefined} &&\text{ if }u\geq\max(Y_1,\dots,Y_n),\;\delta_n
= 0, \endalignedat\right. \tag2.1
$$
where
$$
N(t)= \#\{Y_i,\;\;Y_i>t,\;1\le i \le n\}= \sum_{i= 1}^n I(Y_i>t)\;.
$$
 
One would like to understand how to find the estimate (2.1) and why it
is good. We do not discuss the first question, we only
refer to some papers where the answer to this question is explained.
The estimator (2.1) is a generalized maximum likelihood estimator
(GMLE) of the unknown distribution function. The meaning of
GMLE needs an explanation because in the nonparametric case we are
searching for a measure in a non-dominated family of probability
measures. Kiefer and Wolfowitz~[kw] gave a definition for GMLE
(see~[rt] in this volume), and Johansen~[joh] proved that the product
limit estimator is GMLE in this sense.
 
We want to show that the estimate (2.1) is really good.
This expression in its original form is rather
complicated and hard to study. But it can be rewritten in a form more
appropriate for our purposes. We briefly explain how this can be done.
Our calculation leading to a better representation of the
expression~(2.1) closely follows the paper~[mr] where the details are
worked out. We give an expansion of the random variable
$S_n$ defined in (2.1). The leading term of this expansion is $1-F(u)$,
the quantity we wanted to estimate. The second term is $n^{-1/2}$
times a linear functional of a standardized empirical distribution
function, and the remaining error term can be bounded by $n^{-1}$ times
a random variable with finite moment generating function. In such a
way a result can be proved which shows that the Kaplan--Meier estimate
behaves very similarly to the maximum likelihood estimate in the
parametric case. The method of the proof deserves special attention
since it is also applicable in case of other nonparametric GMLE-s.
 
In the calculations, similarly to the study of the
maximum likelihood estimate, appropriate Taylor expansions can be made,
and the error term of these expansions can be well bounded. Most steps
are routine, but there is a step which deserves special attention.
During our calculations we have to estimate a quadratic functional of a
standardized empirical distribution function, and this estimate is
non-trivial. This corresponds to the estimate of the second term of the
Taylor expansion in the maximum likelihood estimate in the parametric
case, and such an expression can be well bounded by the result
formulated in Theorem~B of this paper.
 
First we introduce some notations. Put
$$
\aligned
H(u) &= P(Y_i\leq u)= 1-\bar H(u),\\
\tilde H(u) &= P(Y_i\leq u,\,\delta_i= 1),\quad
\tilde{\tilde H}(u)= P(Y_i\leq u,\,\delta_i = 0),
\endaligned \tag2.2
$$
and
$$
\aligned
H_n(u) &= \frac{1}{n} \sum_{i= 1}^n I( Y_i \leq u),\\
\tilde H_n(u) &= \frac1n \sum_{i= 1}^n I(Y_i\leq u,\, \delta_i
= 1), \quad \tilde{\tilde H}_n(u)= \frac{1}{n}\sum_{i= 1}^n I( Y_i
\leq u, \, \delta_i= 0)\;.
\endaligned\tag2.3
$$
Clearly $H(u)= \tilde H(u)+\tilde{\tilde H}(u)$  and
$ H_n(u) =  \tilde H_n(u)+\tilde{\tilde H}_n(u)$\;.
We consider $F_n(u)-F(u)$ on an interval $(-\infty, T]$, where
$$
1-H(T)>\delta \quad \text{with some fixed } \delta>0\;.\tag2.4
$$
We introduce the so-called cumulative hazard function and its empirical
version
$$
\Lambda(u)= -\log(1-F(u)), \quad \Lambda_n(u)= -\log(1-F_n(u))\;.\tag2.5
$$
From~(2.1) it is obvious that
$$
\Lambda_n(u)= -\sum_{i= 1}^n I(Y_i\leq u, \, \delta_i= 1)
\log\left(1-\frac{1}{1+N(Y_i)}\right).
$$
 
Since $F_n(u)-F(u)= \exp(-\Lambda(u))\(1-\exp(\Lambda(u)-\Lambda_n(u))\)$
a simple Taylor expansion yields
$$
F_n(u)-F(u)= (1-F(u))\left(\Lambda_n(u)-\Lambda(u)\right)+R_1(u)\;,\tag2.6
$$
and it is easy to see that $R_1(u)= O\(\Lambda(u)-\Lambda_n(u))^2\)$.
 
It follows from the subsequent estimations that
$ \sup\limits_{u\le T}\sqrt{n}|\Lambda(u)-\Lambda_n(u)|$ has
exponential tail, thus the same is true for
$\sup\limits_{u\le T} n|R_1(u)|$. Hence it is enough to investigate
the term
$\Lambda_n(u) - \Lambda(u)$.
 
We approximate $\Lambda_n(u)$ with the help of the relation
$-\log(1-x)= x+O(x^2)$ for small $x$.  We get
$$
\Lambda_n(u)= \sum_{i= 1}^n \frac{I(Y_i\leq u, \,\delta_i= 1)}{N(Y_i)}
+R_2(u)= \tilde{\Lambda}_n(u)+R_2(u)\;,\tag2.7
$$
and the error term $nR_2(u)$ has also exponential tail (e.g. [mr] for the
details).
 
The expression $\tilde{\Lambda}_n(u)$  is still not appropriate for
our purposes. Since the denominators $N(Y_i)= \summ_{j= 1}^n I(Y_j>Y_i)$
are dependent on different $i$'s, we cannot see directly the limiting
behaviour of $\tilde{\Lambda}_n(u)$.
 
By exploiting the fact that the conditional
distribution of $N(Y_i)$ given $Y_i$ is a binomial distribution
with parameters $n-1$ and $1-H(Y_i)$, we can rewrite
$\tilde{\Lambda}_n(u)$ in a more appropriate form. We shall approximate
it by an expression which can be handled better.
By writing
$$
N(Y_i)= \sum_{j= 1}^n I(Y_j>Y_i)= n\bar H(Y_i) \(1+
\frac{\summ_{j= 1}^nI(Y_j>Y_i)-n\bar H(Y_i)}{n\bar H(Y_i)}\),
$$
and applying the inequality $\left|\dfrac{1}{1+z}-1+z\right|<2z^2$,
for $|z|<\dfrac{1}{2}$, with the choice
$z= \dfrac{\summ_{j= 1}^nI(Y_j>Y_i)-n\bar H(Y_i)}{n\bar H(Y_i)}$,
we obtain
$$
\aligned
\tilde{\Lambda}_n(u)&= \sum_{i= 1}^n\frac{I(Y_i\leq u,\,\delta_i= 1)}
{n\bar H(Y_i)}\(1-\frac{\summ_{j= 1}^n I(Y_j>Y_i)-n\bar H(Y_i)}
{n\bar H(Y_i)}\)+R_3(u)\\
&=  2A(u)-B(u)+R_3(u),
\endaligned\tag2.8
$$
where
$$
A(u)= A(n,u)= \sum_{i= 1}^n\frac{I(Y_i\leq u,\,\delta_i= 1)}
{n\bar H(Y_i)}
$$
and
$$
B(u)= B(n,u)= \sum_{i= 1}^n \sum_{j= 1}^n\frac
{I(Y_i\leq u,\,\delta_i= 1)I(Y_j>Y_i)}{n^2\bar H^2(Y_i)}\;.
$$
Again the reader is referred to [mr] for the tail behaviour of $nR_3(u)$.
Thus ~(2.7) and~(2.8) together yield
$$
\Lambda_n(u)= 2A(u)-B(u)+\text{negligible error},\tag2.9
$$
and the sums $A$ and $B$ can be rewritten as stochastic integrals
in the same way as in [mr]. Finally one obtains
 
$$
\aligned
\sqrt n\(\Lambda_n(u)-\Lambda(u)\)
&= \frac{\sqrt n\(\tilde H_n(u)-\tilde H(u)\)}{1-H(u)}-\int_{-\infty}^u
\frac{\sqrt n(\tilde H_n(y)-\tilde H(y))}{\(1-H(y)\)^2}\,dH(y)\\
&\qquad+\int_{-\infty}^u\frac{\sqrt n\(H_n(y)-H(y)\)}{\(1-H(y)\)^2}
\,d\tilde H(y)\\
&\qquad-\sqrt n B_1(u)+\text{negligible error},
\endaligned\tag2.10
$$
where
$$
B_1(u) = \frac 1n \int_{-\infty}^u\int_{-\infty}^{+\infty}
\frac{I(x>y)}{\(1-H(y)\)^2}\,d\(\sqrt n\(H_n(x)-H(x)\)\)\,
d\(\sqrt n(\tilde H_n(y)-\tilde H(y)) \)\;.
$$
 
 
This formula is the analogous one of ~(1.6). To prove this we still have to
show that the term $B_1(u)$ is also small. Theorem~A suggests such an
estimate. However, this result cannot be applied directly in the present
case, since in the integral defining $B_1$, one has to integrate with
respect to two different processes in the variables $x$ and $y$. In the
paper~[mr] we could overcome this difficulty by rewriting
$B_1$ as a double integral of an appropriate kernel function with respect
to a
standardized empirical
process which contains all information on $H_n(\cdot)$ and
$\tilde H_n(\cdot)$. Here we choose a different argument. We deduce the
needed estimate directly from Theorem~B. The advantage of this argument is
that it
is more flexible and applicable in other cases too. Instead of using
$H$, $H_n$ and $\tilde H$, $\tilde H_n$, we use the two dimensional
measure $\mu(x_1,x_2)= F(x_1)G(x_2)$ and the empirical measure
$$
\bar\mu_n(x_1,x_2)= \dfrac1n\#\{i\:1\le i\le n,\;
X_i\le x_1,C_i\le x_2\}
$$
on $\bold R^2$. Since they contain all information, the expression
$B_1(u)$ can be rewritten as a double stochastic integral with respect
to the measure $\mu_n= \sqrt n(\bar\mu_n-\mu)$. To see this observe that
$$
\tilde H([a,b])= \mu\(\bold A([a,b])\), \quad
\tilde H_n([a,b])= \bar\mu_n\(\bold A([a,b])\),
$$
$$
\tilde{\tilde H}([a,b])= \mu\(\bold B([a,b])\), \quad
\tilde{\tilde H}_n([a,b])= \bar\mu_n\(\bold B([a,b])\)\;,
$$
where
$$
\bold A([a,b])= \{(x_1,x_2)\: a\le x_1\le b,\; x_1\le x_2\},
$$
and
$$
\bold B([a,b])= \{(x_1,x_2)\: a\le x_2\le b,\; x_2\le x_1\}\;.
$$
Then applying the decompositions $H= \tilde H+\tilde{\tilde H}$ and
$H_n= \tilde H_n+\tilde{\tilde H}_n$, we obtains
$$
\align
B_1(u)&= \frac 1n \intt_{y_1<u}\intt_{{\bold R}^2}
\frac{I(x_1>y_1,x_1\le x_2,y_1\le y_2)}{\(1-H(y_1)\)^2}
\,d\mu_n(x_1,x_2) \,d\mu_n(y_1,y_2) \\
&\qquad+\frac 1n \intt_{y_1<u}\intt_{{\bold R}^2}
\frac{I(x_1>y_1,x_2\le x_1,y_1\le y_2)}{\(1-H(y_1)\)^2}
\,d\mu_n(x_1,x_2)\,d\mu_n(y_1,y_2)\;.
\endalign
$$
Then Theorem~B makes possible to estimate $\supp_{u\le T} |B_1(u)|$ if the
number $T$ satisfies (2.4), since the integrand is bounded in this case.
We omit the details and we only formulate the final result we get in such a
way.
A detailed proof can be found in~[mr].
\medskip\noindent
{\bf Theorem 2.1} {\it Let $T$ be such that $1-H(T)>\delta$ with some
$\delta>0$. Then the process $F_n(u)-F(u)$, $-\infty<u<T$, where
$1-F_n(u)$ is defined in formula {\rm (2.1)} can be represented as
$$
F_n(u)-F(u)= (1-F(u))(U(n,u)+V(n,u))+R(n,u)\;,\quad -\infty<u<T\;,
$$
where
$$
\align
\sqrt n U(n,u)&= \frac{\sqrt n(\tilde H_n(u)-\tilde H(u))}{(1-H(u))}
-\intt_{-\infty}^u \frac{\sqrt n(\tilde H_n(y)-\tilde H(y))}
{(1-H(y))^2}\,dH(y)\\
\sqrt n V(n,u)&= \intt_{-\infty}^u \frac{\sqrt
n(H_n(y)-H(y))}{(1-H(y))^2}\,dH(y)
\endalign
$$
are linear functionals of the empirical processes $\sqrt n(H_n(y)-H(y))$
and $\sqrt n(\tilde H_n(u)-\tilde H(u))$, and the error term $R(n,u)$
can be bounded as
$$
P\(\sup_{u\le T}n|R(n,u)|>x+\frac C\delta\)\le Ke^{-\lambda x\delta^2}
$$
for all $x>0$, where $C>0$, $K>0$ and $\lambda>0$ are universal
constants.}
 
 
 
\beginsection 3. Baseline function estimation in the Cox model
 
The previous section dealt with a model where the observed sample
is $(Y_i, \delta_i),\; i= 1,\ldots,n$ with $Y_i= \min(X_i, C_i)$
and $\delta_i= I(X_i\leq C_i)$, $X_i, C_i$ are two independent
identically distributed (iid.) sequences of random variables.
In this section such a model is considered where an additional
sequence of positive variables $W_i$, $i= 1,\dots,n$, is given together
with the above pairs $(Y_i, \delta_i)$.  The triplets
$(W_i,Y_i,\delta_i)$ are iid.,  the censoring variable  $C_i$ is
independent of the pair $(W_i, X_i)$, and we assume the existence of a
conditional survival function $S_0^{W_i}(t)$ for fixed $X_i$ and $W_i$,
that is we consider a model, where
$$
S_0^{W_i}(t)= P( X_i \geq t \mid W_i)\;,
$$
and $S_0(t)$ is an unknown (deterministic) continuous survival
function. We want to estimate this unknown baseline survival function
$S_0(t)$ based on a sample of the above triplets. Notice, that in the
special case when $W_i= 1$, $i= 1,\dots,n$, this is the censored sample
considered in the previous section.
 
We call this model ``nonparametric Cox model'', because with appropriate
pa\-ram\-etrization it provides the semiparametric Cox model.
Introducing $W_i= \exp(\beta Z_i)$ where $\beta$ is an unknown parameter
and $Z_i$ is the known regressor variable for $i= 1,\dots,n$ we have
the Cox regression model. (See~[rt] for more details.)
 
In paper~[rt] the following GMLE type estimator (see previous section)
of $S_0$ was proposed:
$$
1-F_n(t)= \hat S_n(t)=
\prod_{i= 1}^n\left(\frac{N(Y_i)}{N(Y_i)+W_i}\right)^
{\frac{I(Y_i\leq t, \delta_i= 1)}{W_i}}
\quad   \text{if } t\leq\max(Y_1,\dots,Y_n)\;,
\tag3.1
$$
where
$$
N(t)= \sum_{j= 1}^n W_j \, I( Y_j > t)+
\sum_{j= 1}^n W_j I( Y_j =  t, \delta_j =  0)\;.
$$
The Kaplan--Meier product limit estimator is a special
case of (3.1) when all of the $W_i$-s equal 1.
 
The calculations showing that the above nonparametric likelihood
estimator is as good as a parametric likelihood estimator is very
similar to that given in the previous section. We give an expansion of
$\hat S_n(t)$ defined in (3.1). The leading term of this expansion is
$S_0(t)$, the expression we intend to estimate. The second term is
$n^{-1/2}$ times a linear functional of a standardized empirical
distribution function, and the remaining error term can be bounded by
$n^{-1}$ times a random variable with finite momentum generating
function. In such a way, a result can be proved which shows that the
estimate (3.1) behaves very similarly to the maximum likelihood
estimate in the parametric case. The proof follows the same line as the
one in the Kaplan--Meier case.
 
For the sake of simpler notations  we only deal with complete sample,
but a similar representation can be given for censored
sample.  Thus the given sample is $(X_i, W_i)$,  $i= 1,\dots,n,$ where
$P(X_i>t \mid W_i) =  S_0^{W_i}(t)$.
 
We introduce some notations. Put $P(W\leq w)= Q(w)$ and
$$
\aligned
P(X>t)&= \bar G (t) = 1-G(t)= \int_0^{\infty}
S_0^w(t) \,dQ(w),\\
E\( W \,I(X>t) \) &= \bar H (t)=  E(W)-H(t)=
\int_0^{\infty} w S_0^w(t) \,dQ(w)\;,\\
F(x_1, x_2) &=  P( X\leq x_1, W\leq x_2)
\endaligned \tag3.2
$$
Note that
$$
F(x_1,x_2)= P(W\leq x_2)-P(X>x_1,W \leq x_2)= Q(x_2) -
\int_0^{x_2}S_0^w(x_1)\, dQ(w)\;,
\tag3.3
$$
and
$$
1-G(t)= \bar G (t) =  \int_0^{\infty}
\exp\left( w \log(S_0(t) \right)\,dQ(w)=
M_W \left(\log( S_0(t) \right),
\tag3.4
$$
where $M_W$ denotes the moment generating function of $W$.
We introduce the empirical processes
$$
\aligned
G_n(t)&= \frac 1n \summ_{i= 1}^n I( X_i \leq t), \quad
\bar H_n(t)= \frac1n\summ_{i= 1}^n W_i \,I(X_i>t)\;, \\
F_n(x_1,x_2)&= \frac1n\sum_{i= 1}^n I(X_i\leq x_1, W_i\leq x_2).
\endaligned \tag3.5
$$
 
We suppose that the following conditions hold:
 
\item{(i)} $W$ is a bounded positive random variable with
a positive lower bound $\delta_0$, that is
$P(K>W>\delta_0)= 1$ with some constant $K$.
\item{(ii)} On the interval $(-\infty, T]$ there exists a fixed
positive number $\kappa$ such that
$$
1-G(t)>\kappa,\quad \forall \; t\in (-\infty, T]\;.
$$
 
We consider $S_0(t) - \hat S_n(t)$   on the interval  $(-\infty, T]$.
It follows from conditions (i) and (ii) that
$$
\bar H(T)>\delta \quad \text{with some fixed }\delta>0\;.
\tag 3.6
$$
 
Similarly to the case of the product limit estimator we introduce
the cumulative hazard function and its empirical version
$$
\Lambda(t) =  -\log\(S_0(t)\), \quad \Lambda_n(t) =  -\log \(\hat S_n(t)\).
$$
Using (3.1) in the case when there is no
censoring i.e.\ when $\delta_i= 1$ for all~$i$,  we get that
$$
\Lambda_n(t) =  - \summ_{i= 1}^n \frac{I(X_i \leq t)}{W_i}
\log \( 1 - \frac{W_i}{N(X_i) + W_i} \)\;.
$$
 
Using almost the same expansions as in the last section, we also
obtain that
$$
\sqrt n\(\Lambda_n(t)-\Lambda (t)\)
= \sqrt n B_2(t)-\sqrt nB_3(t)-\sqrt n B_4(t)
+ \text{negligible error},
\tag 3.7
$$
where
$$
\align
B_2(t) &=  \frac{1}{\sqrt n}  \int_{y_1\leq t} \, \int_{\bold R^2}
\frac{ x_2\, I( x_1 > y_1) }{\bar H^2(y_1)} \,
dF(x_1, x_2)\, d\mu_n(y_1, y_2)\;,\\
B_3(t) &=  \frac{1}{\sqrt n} \int_{y_1\leq t} \, \int_{\bold R^2}
\frac{ x_2\, I( x_1 > y_1) }{\bar H^2(y_1)} \,
d\mu_n(x_1, x_2)\, dF(y_1, y_2)\;,\\
B_4(t) &=   \frac{1}{n} \int_{y_1\leq t} \, \int_{\bold R^2}
\frac{ x_2\, I( x_1 > y_1) }{\bar H^2(y_1)} \,
d\mu_n(x_1, x_2) \, d\mu_n(y_1, y_2)\;,
\endalign
$$
and where $\mu_n(x_1,x_2)= \sqrt n\(F_n(x_1,x_2)-F(x_1,x_2)\)$.
 
 
This formula is also analogous to~(1.6). Again it remains to
prove that the term
$\supp_{t\leq T} B_4(t)$ is also small. Theorem~B suggests such an
estimate. We have to show that the conditions of Theorem~B hold. This time
we have the integral
of the function $f((x_1,x_2),(y_1, y_2))=
\dfrac{x_2I(x_1>y_1)}{\bar H^2(y_1)}$, and this function equals zero
on the diagonal set. It follows from conditions  (i)---(ii) and (3.6)
that this function is bounded if $y_1\leq T$. This way, Theorem~B is
applicable. We omit the details, and formulate the final result.
\medskip\noindent
{\bf Theorem 3.1} {\it Let $T$ be such that $\bar H(T)>\delta$ with some
$\delta>0$. Then the process $S_0(t)-\hat S_n(t)$, $-\infty<t<T$, where
$\hat S_n(t)$ is defined in formula (3.1) can be represented as
$$
S_0(t)-\hat S_n(t)= S_0(t)\(U(n,t)-V(n,t)\)+R(n,t)\;,\quad -\infty<t<T\;,
$$
where
$$
\align
\sqrt n U(n,t)&= \int_{y_1\leq t}\frac{d\mu_n(y_1,y_2)}{\bar H(y_1)}\\
\sqrt n V(n,t)&= \int_{y_1\leq t}\frac{\sqrt n(\bar H_n(y_1)-\bar
H(y_1))} {\bar H^2(y_1)}\,dF(y_1, y_2)
\endalign
$$
are linear functionals of the empirical processes
$\mu_n(y_1, y_2)= \sqrt n\,(F_n(y_1, y_2)-F(y_1, y_2))$,
$\sqrt n(\bar H_n(y_1)-\bar H(y_1))$, and the error term $R(n,t)$ can
be bounded as
$$
P\(\sup_{t\le T}n\mid R(n,t)\mid>x+\frac C\delta\)\le Ke^{-\lambda
x\delta^2}
$$
for all $x>0$, where $C>0$, $K>0$ and $\lambda>0$ are universal
constants.}
 
\parindent= 36pt
\medskip\noindent
{\it Acknowledgement:}\/ The authors would like to thank the referee
for his help to improve the presentation of the paper.
 
\vfill\eject
 
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\bye
 
 
