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\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\input{tcilatex}

\begin{document}


\[
\mathbf{Estimating\ a\ Life\ Distribution\ by\ Sampling\ from\ the} 
\]

\[
\mathbf{Obituaries\ and\ Related\ Problems\ } 
\]
\vspace{0.5cm}

\[
\text{By Mark D Rothmann}
\]

\[
\text{Department of Statistics and Actuarial Science}
\]

\[
\text{The University of Iowa}
\]

\[
\text{Iowa City, IA 52242}
\]
\vspace{0.5cm}

\textbf{Abstract}\vspace{0.5cm}

Consider a system where units enter according to a nonhomogeneous Poisson
process, \vspace{0.5cm}

\noindent have independent and identically distributed lengths of stay and
then depart the system. This \vspace{0.5cm}

\noindent paper relates the units' underlying life distribution with the
distribution of the ages of units in \vspace{0.5cm}

\noindent the system, length of stay of units that departed the system and
the most recent ages at death. \vspace{0.5cm}

\noindent Results can be used to estimate the underlying life
distribution(s) or truncated version of \vspace{0.5cm}

\noindent such based on the ages and/or most recent ages at death in both
one sample and two sample \vspace{0.5cm}

\noindent situations. Results include a complete characterization of the
possible distribution of ages of \vspace{0.5cm}

\noindent those units in the system, how to estimate the underlying life
distribution from the most \vspace{0.5cm}

\noindent recent ages at death, and how to test for an underlying monotone
failure rate function based \vspace{0.5cm}

\noindent on independent samples from the ages and most recent ages at
death. Two sample inferences \vspace{0.5cm}

\noindent that involve a likelihood ratio ordering make use of the results
in Robertson, Kochar and \vspace{0.5cm}

\noindent Dykstra (1995), which provides the maximum likelihood estimators
and a likelihood ratio \vspace{0.5cm}

\noindent test when the two distribution satisfy a likelihood ratio
ordering. For the ages of the active \vspace{0.5cm}

\noindent units and the ages at death among the departed units, limits for
their distributions and strong \vspace{0.5cm}

\noindent limiting results for their empirical distributions will be
provided.\vspace{0.5cm}

Key words: Life distributions, Likelihood ratio ordering, Nonhomogeneous
Poisson \vspace{0.5cm}

\noindent processes, Weighted distributions\vspace{0.5cm}

Current mailing address: Biologics Therapeutic Statistical Staff, U.S. Food
and Drug Administration, 5600\ FishersLane, Room 15B45, HFD-711, Rockville,
MD 20855.

E-mail address: rothmannm@cder.fda.gov

\newpage

\noindent \textbf{1. Introduction}\medskip

Consider a system where units arrive according to a nonhomogeneous Poisson
process, \vspace{0.5cm}

\noindent remain for a random period and then depart. Units can be thought
of as being born, living \vspace{0.5cm}

\noindent and then dying, or entering, having a length of stay and then
graduating or departing. At any \vspace{0.5cm}

\noindent point in time only a portion of those units which have entered the
system are still present or\vspace{0.5cm}

\noindent\ ``active''. This paper will consider the problem of estimating
the life distribution based on \vspace{0.5cm}

\noindent those ages of units present in this system and/or those ages at
death of the most recent \vspace{0.5cm}

\noindent departed units. To do so requires relating the underlying life
distribution, with the theoretical \vspace{0.5cm}

\noindent distribution for the ages and/or most recent ages at death. These
theoretical distributions \vspace{0.5cm}

\noindent depend on the entrance rate and the underlying life distribution.\
For a given underlying life \vspace{0.5cm}

\noindent distribution, both the distribution of the ages and the
distribution for the most recent ages at \vspace{0.5cm}

\noindent death will be larger for a constant or a slow growing intensity
function than for an \vspace{0.5cm}

\noindent exponential or faster growing intensity function.\vspace{0.5cm}

Ways of estimating the underlying life distribution(s) or a truncated
version of such \vspace{0.5cm}

\noindent based on the ages and/or most recent ages at death in both one
sample and two sample \vspace{0.5cm}

\noindent situations will be given. The two sample cases that are considered
are cases where there is \vspace{0.5cm}

\noindent a special ordering between the underlying life distributions. It
will be proven that all possible \vspace{0.5cm}

\noindent age distributions are those which have bounded densities near zero
that are smaller with \vspace{0.5cm}

\noindent respect to a likelihood ratio ordering to the distribution of the
time back to a random arrival. \vspace{0.5cm}

\noindent This result can be used to estimate the underlying life
distribution from the distribution of \vspace{0.5cm}

\noindent ages. We will also give an example of estimating an underlying
life distribution by sampling \vspace{0.5cm}

\noindent from the most recent ages at death, i.e., sampling from the
obituaries, and show how \vspace{0.5cm}

\noindent independent samples from the ages and most recent ages at death
can be used to test for a \vspace{0.5cm}

\noindent monotone failure rate for the underlying life distribution.\vspace{%
0.5cm}

Many of the distributional results involve a likelihood ratio ordering. Two
sample \vspace{0.5cm}

\noindent inferences make use of the results in Robertson, Kochar and
Dykstra (1995), which provides \vspace{0.5cm}

\noindent the maximum likelihood estimators and a likelihood ratio test when
the two distribution \vspace{0.5cm}

\noindent satisfy a likelihood ratio ordering. Limiting results for the ages
and ages at death will also be \vspace{0.5cm}

\noindent provided.\vspace{0.5cm}

\[
\mathbf{2.\ Main\ results} 
\]
\vspace{0.5cm}

Since various results involve different types of stochastic orderings for
distributions, we \vspace{0.5cm}

\noindent will first introduce these orderings. One result is that the
distributions of ages satisfy an \vspace{0.5cm}

\noindent uniform stochastic ordering if and only if the underlying life
distributions satisfy a likelihood \vspace{0.5cm}

\noindent ratio ordering. Also presented will be the distributions for the
ages, the distribution for the \vspace{0.5cm}

\noindent ages at death and the distribution for the ``most recent'' ages at
death and all possible age \vspace{0.5cm}

\noindent distributions. An example is given in estimating the life by
sampling from obituaries. The \vspace{0.5cm}

\noindent distribution of the most recent ages at death is shown to be a
weighted distribution of the \vspace{0.5cm}

\noindent distribution of ages where the weight function is the failure rate
function. This gives a way of \vspace{0.5cm}

\noindent testing for a monotone failure rate function by using independent
samples from the ages and \vspace{0.5cm}

\noindent most recent ages at death. We finish by discussing some more
stochastic ordering relations \vspace{0.5cm}

\noindent between the various distributions and give limiting results for
age and age at death \vspace{0.5cm}

\noindent distributions and almost sure limits for the empirical
distributions of the ages and of the ages \vspace{0.5cm}

\noindent at death.\vspace{0.5cm}

For the ages of the active units and the ages at death among the departed
units, limits for \vspace{0.5cm}

\noindent their distributions and strong limiting results for their
empirical distributions will be \vspace{0.5cm}

\noindent provided.\vspace{0.5cm}

\[
\mathbf{2.1}\text{ }\mathbf{Stochastic}\text{ }\mathbf{Orders} 
\]
\vspace{0.5cm}

We will discuss some results related to various stochastic orders. \ Some
relevant \vspace{0.5cm}

\noindent orderings with respect to the results of this paper are given in
this subsection. \ Let $W$ and $Y$ \vspace{0.5cm}

\noindent be random variables having respective distribution functions, $F$
and $G$ with corresponding \vspace{0.5cm}

\noindent densities $f$ and$\ g$, corresponding survival functions $%
\overline{F}$ and $\overline{G}$ and corresponding failure rate \vspace{0.5cm%
}

\noindent functions $r_{F\text{ }}$and $r_{G}.\;\;$In all the orderings
below $F$ is the ``larger'' distribution.\vspace{0.5cm}

\textit{Proportional Hazards Order:}\ \ \ If $1\leq k=r_{G}(u)/r_{F}(u)$,
for all $u$ \TEXTsymbol{>} 0, then we say that $W$ \vspace{0.5cm}

\noindent ($F$) is larger than$\ Y$ ($G$) with respect to a proportional
hazards order. We denote this by \vspace{0.5cm}

\noindent $W\geq _{ph}Y\ $or $F\geq _{ph}G$.\vspace{0.5cm}

\textit{Likelihood Ratio Order:} \ If\ $f(u)/g(u)$ is nondecreasing in $u$
over the union of the supports \vspace{0.5cm}

\noindent of $Y$ and $W$, then we say that $W$ ($F$) is larger than$\ Y$ ($G$%
) with respect to a likelihood ratio \vspace{0.5cm}

\noindent order. We denote this by $W\geq _{lr}Y\ $or $F\geq _{lr}G$. Note
that if $F$ can be expressed by \vspace{0.5cm}

\noindent $F(x)=\int_{-\infty }^{x}w(y)dG(y)$ for all $-\infty <x<\infty $,
and some nondecreasing (nonincreasing), \vspace{0.5cm}

\noindent nonnegative function $w$, then $W$ is larger (smaller) than $Y$
with respect to a likelihood ratio \vspace{0.5cm}

\noindent ordering. For life distributions replace $-\infty $ with $0$ in
the integrand.\vspace{0.5cm}

To further elaborate on the meaning of this ordering, suppose one of the two
random \vspace{0.5cm}

\noindent lives $W$ and $Y$ is randomly selected with probabilities \textit{p%
} and 1-\textit{p}, respectively. The \vspace{0.5cm}

\noindent distribution of $W$ is larger than that of $Y$ with respect to a
likelihood ratio ordering, if and \vspace{0.5cm}

\noindent only if the larger the observed randomly selected value, \textit{x}%
, the greater the probability that $W$ \vspace{0.5cm}

\noindent was selected. This follows from:\vspace{0.5cm}

pr($W$ selected \TEXTsymbol{\vert} random value = \textit{x}) = lim$%
_{h\rightarrow 0^{+}}$ pr($W$ selected \TEXTsymbol{\vert} random value in [%
\textit{x}, \textit{x} + \textit{h}]) \vspace{0.5cm}

\noindent =$\frac{pf(x)}{pf(x)+\left( 1-p\right) g(x)}=\frac{p}{%
p+\{1-p\}\{g(x)/f(x)\}}.$\vspace{0.5cm}

\textit{Reversed Hazard Rate Order:} $\ $If $F(u)/G(u)$ is nondecreasing in $%
u$, then we say that $W$ ($F$) \vspace{0.5cm}

\noindent is larger than$\ Y$ ($G$) with respect to a reversed hazard rate
order (denoted $W\geq _{rhr}Y\ $or \vspace{0.5cm}

\noindent $F\geq _{rhr}G$).\ This is equivalent to $f(u)/F(u)\geq g(u)/G(u)$
for all $u\in (-\infty ,\infty )$.\vspace{0.5cm}

\textit{Uniform Stochastic Order/Hazard Rate Order:}\ $\ $If $%
pr(W>s+t|W>t)\geq $\vspace{0.5cm}

\noindent $pr(Y>s+t|Y>t)$ for all $s\geq 0$ and all $t\in (-\infty ,\infty )$%
\ (or equivalently $\overline{F}(u)$ $/$ $\overline{G}(u)\;$is \vspace{0.5cm}

\noindent nondecreasing in $u$ $\in (-\infty ,G^{-1}(1))$ or i.e., for life
distributions, $r_{G}(u)\geq r_{F}(u)$ for all $u>0$), \vspace{0.5cm}

\noindent then we say that $W$ ($F$) is larger than$\ Y$ ($G$) with respect
to an uniform stochastic order \vspace{0.5cm}

\noindent (denoted $W\geq _{us}Y\ $or $F\geq _{us}G$).\vspace{0.5cm}

To further elaborate on the meaning of this ordering, consider two units
born at time 0 \vspace{0.5cm}

\noindent with random lives $W$ and $Y$. At time $t$, one of the two units
is randomly selected with \vspace{0.5cm}

\noindent probabilities $p$ and 1-$p$ and the unit is noted as being alive
at time $t$. The distribution of $W$ is \vspace{0.5cm}

\noindent larger than that of $Y$ with respect to an uniform stochastic
ordering, if and only if the larger \vspace{0.5cm}

\noindent the value of $t$, the greater the probability that $W$ was
selected. This follows from:\vspace{0.5cm}

pr(unit with life $W$ selected \TEXTsymbol{\vert} life \TEXTsymbol{>} $t$) = 
$\frac{p\overline{F}(t)}{p\overline{F}(t)+(1-p)\overline{G}(t)}=\frac{p}{%
p+\{1-p\}(\overline{\{G}(t)/\overline{F}(t)\}}$.\vspace{0.5cm}

\textit{Regular Stochastic Order:} \ If\ \ $\overline{G}(u)=pr(Y>u)$ $\leq $ 
$\overline{F}(u)=pr(W>u)$ for all \vspace{0.5cm}

\noindent $u\in (-\infty ,\infty )$, then we say that $W$ ($F$) is
stochastically larger than$\ Y$ ($G$) (denoted $W\geq _{st}Y\ $or \vspace{%
0.5cm}

\noindent $F\geq _{st}G$).\vspace{0.5cm}

\textit{Laplace Transform Order:}\ \ If $M_{F}(s)=E(\exp
(-sW))=\int_{0}^{\infty }\exp (-sx)dF(x)\leq $\vspace{0.5cm}

\noindent $\int_{0}^{\infty }\exp (-sx)dG(x)=M_{G}(s)=E(\exp (-sY))$ for all 
$s>0$, then we say that $W$ ($F$) is larger \vspace{0.5cm}

\noindent than$\ Y$ ($G$) with respect to a Laplace transform order (denoted 
$W\geq _{Lt}Y$ \ or $F\geq _{Lt}G$).\vspace{0.5cm}

\textit{Reverse} \textit{Laplace Transform Ratio Order:}\ \ If\ $%
\{1-M_{F}(s)\}/\{1-M_{G}(s)\}\;$is decreasing in \vspace{0.5cm}

\noindent $s>0$, then we say that $W$ ($F$) is larger than$\ Y$ ($G$) with
respect to a reverse Laplace transform \vspace{0.5cm}

\noindent ratio order (denoted $W\geq _{rLt}Y$ or $F\geq _{rLt}G$).\vspace{%
0.5cm}

To elaborate on some consequences of the above two orderings suppose that
the units \vspace{0.5cm}

\noindent consist of \textit{n} different types, where the chance that any
given unit is type \textit{i} with probability \textit{p}$_{i}$. \vspace{%
0.5cm}

\noindent For rates of entry that are slower than exponential, of an
exponential order and faster than \vspace{0.5cm}

\noindent exponential orders, Rothmann and Russo (1997) found for a system
with fixed points of \vspace{0.5cm}

\noindent entry that when the life distribution for type units becomes
larger with respect to a Laplace \vspace{0.5cm}

\noindent transform ordering as \textit{i} increases, then the limiting
proportion of types among the active \vspace{0.5cm}

\noindent units, $\pi _{i}$, \textit{i} = 1, \textit{n}, is greater than the
underlying probabilities, $p_{i}$, \textit{i} = 1, \textit{n}, with respect
to a \vspace{0.5cm}

\noindent regular stochastic ordering (i.e. $\sum_{1\leq i\leq k}p_{i}\geq
\sum_{1\leq i\leq k}\pi _{i}$ for all $1\leq k\leq n$). It was also shown 
\vspace{0.5cm}

\noindent that if the \textit{n} unit types have life distributions that are
ordered with respect to a reverse \vspace{0.5cm}

\noindent Laplace transform ratio ordering, then the limiting distribution
of types among the active \vspace{0.5cm}

\noindent units, $\mathbf{\pi }$, becomes larger with respect to a
likelihood ratio ordering as the ``order'' of the \vspace{0.5cm}

\noindent exponential rate of arrivals becomes smaller.\vspace{0.5cm}

Some of these orderings imply other orderings. A proportional hazard order
implies a \vspace{0.5cm}

\noindent likelihood ratio ordering, which in turn implies a uniform
stochastic ordering and also a \vspace{0.5cm}

\noindent reverse hazard ratio ordering, each of which in turn implies a
regular stochastic ordering, \vspace{0.5cm}

\noindent which in turn implies a Laplace transform ordering. An uniform
stochastic order implies a \vspace{0.5cm}

\noindent reversed Laplace transform order, proven independently by Shaked
and Wong (1997) and \vspace{0.5cm}

\noindent Rothmann and Russo (1997), which implies a Laplace transform
order. For a broader \vspace{0.5cm}

\noindent discussion of these and other orders, see Shaked and Shantikumar
(1994), and Shaked and \vspace{0.5cm}

\noindent Wong (1997).\vspace{0.5cm}

\[
\mathbf{2.2\ Finite\ time\ behavior} 
\]
\vspace{0.5cm}

Suppose units arrive into a system according to a NHPP, $\{N(t),t>0\}$, that
has the \vspace{0.5cm}

\noindent positive intensity function $\lambda $, remain for a random period
and then depart. \ The k-th entering \vspace{0.5cm}

\noindent unit, enters at time $S_{k}$ and has life $L_{k}$ where $%
L,L_{1},...$ is a seuence of independent, identically \vspace{0.5cm}

\noindent distributed nonnegative random variables having distribution $F$
satisfying $F(0)=0$.\vspace{0.5cm}

Define for all t \TEXTsymbol{>} 0,

\[
D(t)=\sum_{k\geq 1}I(S_{k}+L_{k}\leq t)\;\;\;\;\;\;\;\;A(t)=N(t)-D(t) 
\]
\vspace{0.5cm}

\noindent $D(t)$ is the number of departed units by time $t$ and $A(t)$ is
the number of active units at time $t$.\vspace{0.5cm}

Let $Y_{1},...,Y_{D(t)}$ and $Z_{1},...,Z_{A(t)}$ denote respectively, a
random permutation of the \vspace{0.5cm}

\noindent collection of ages of death among the departed units and a random
permutation of the \vspace{0.5cm}

\noindent collection of ages among active units at some fixed time $t$. \ We
have the following theorem \vspace{0.5cm}

\noindent for the distribution of the randomly ordered collection of ages at
death at time $t$ and for those \vspace{0.5cm}

\noindent ages among the active units at time $t$:\vspace{0.5cm}

\noindent \textbf{Theorem 2.1. } $Y_{1},...,Y_{D(t)}$, \textit{is a random
sample of random size} $D(t)$ \textit{from the distribution }$G_{t}$\textit{%
\ }\vspace{0.5cm}

\noindent \textit{given by} \vspace{0.5cm}

\[
G_{t}(y)=\int_{[0,y]}w(x)dF(x)/\int_{[0,\infty )}w(x)dF(x) 
\]
\vspace{0.5cm}

\noindent \textit{where }$w(x)=\int_{0}^{t-x}\lambda (s)ds,$\textit{\ if }$%
0\leq x\leq t$\textit{\ and }$w(x)=0$\textit{, if }$x>t$\textit{, and is
independent of the} \vspace{0.5cm}

\noindent $Z_{1},...,Z_{A(t)}$\textit{\ a random sample of random size} $%
A(t) $ \textit{from the distribution} $H_{t}$ \textit{given by} \vspace{0.5cm%
}

\begin{eqnarray*}
H_{t}(y) &=&\int_{0}^{y}\lambda (t-s)\{1-F(s)\}ds/\int_{0}^{t}\lambda
(t-s)\{1-F(s)\}ds\text{ }\vspace{0.5cm} \\
&=&\int_{0}^{y}w_{F}\left( x\right) dF(x)/\int_{0}^{\infty }w_{F}\left(
x\right) dF(x)\;\text{for }0\text{ }<y\leq t\;\text{where }w_{F}(x)=\lambda
(t-x)/r_{F}(x).
\end{eqnarray*}
\vspace{0.5cm}

The proof is provided in section 3. Note that $w$ is a nonincreasing
function and thus, this \vspace{0.5cm}

\noindent age at departure distribution, $G_{t}$ is smaller than $F$ with
respect to a likelihood ratio ordering. \vspace{0.5cm}

\noindent For $\lambda (s)\equiv \lambda $, $w(\cdot )$ is constant then $%
G_{t}(y)=F(y)/F(t)$ for $0\leq y\leq t.$ Note that the \vspace{0.5cm}

\noindent distribution of the ages is a continuous distribution with a
density, even if the underlying life \vspace{0.5cm}

\noindent distribution is discrete. Also, $H_{t}$ $\leq _{lr}J_{t}$, where $%
J_{t}(y)=\int_{0}^{y}\lambda (t-s)ds/\int_{0}^{t}\lambda (t-s)ds$ for $0$ 
\vspace{0.5cm}

\noindent $<y\leq t$. The relationship between these distributions can be
given by $H_{t}(y)=\int_{0}^{y}w(x)dJ_{t}(x)$, \vspace{0.5cm}

\noindent where $w(x)=c_{t}(1-F(x))$ and $c_{t}=\int_{0}^{t}\lambda
(t-s)ds/\int_{0}^{t}\lambda (t-s)\{1-F(s)\}ds$. This is more \vspace{0.5cm}

\noindent easily seen by looking at the densities and will be elaborated
further.\vspace{0.5cm}

\textit{Density for the ages at time t: \ }From Theorem 2.1 we obtain the
following density for the \vspace{0.5cm}

\noindent ages at time t.\vspace{0.5cm}

\begin{eqnarray}
h_{t}(y) &=&\frac{\lambda (t-y)\{1-F(y)\}}{\int_{0}^{t}\lambda
(t-s)\{1-F(s)\}ds}\;\;,0\leq y\leq t\vspace{0.5cm}  \TCItag{(1)} \\
&\leq &_{lr}\text{ the density }j_{t}(y)=\lambda (t-y)/\int_{0}^{t}\lambda
(s)ds\;\;,0\leq y\leq t  \nonumber
\end{eqnarray}
\vspace{0.5cm}

Note that the density for the ages would be the same if $\lambda (\cdot )$
were replaced by $c\lambda (\cdot )$ for \vspace{0.5cm}

\noindent any $c>0$. Also, note that there is a one-to-one mapping between
those truncated life \vspace{0.5cm}

\noindent distribution $F$ at time \textit{t} that satisfy $F(0)=0$ and
those possible age distributions $h_{t}$. Further \vspace{0.5cm}

\noindent note that the value of $j_{t}(0)/h_{t}(0)$ $=\lim_{x\rightarrow
0^{+}}j_{t}(x)/h_{t}(x)$ is the probability that a randomly \vspace{0.5cm}

\noindent selected unit that entered the system in $(0,t]$ remains in the
system, or equivalently the \vspace{0.5cm}

\noindent fraction of the expected number of units in the system at time $t$
to the expected number of \vspace{0.5cm}

\noindent units to enter the system in $(0,t]$. If one has an impression of
this proportion, then $j_{t}(0)/h_{t}(0)$ \vspace{0.5cm}

\noindent can be estimated or approximated.\vspace{0.5cm}

\textit{Characterizing the possible age distributions: }The density $j_{t}$
is the density for the \vspace{0.5cm}

\noindent distribution of the time from time $t$ back to a random arrival
(given at least one arrival), \vspace{0.5cm}

\noindent which is also what the distribution of the ages would be if units
never departed from the \vspace{0.5cm}

\noindent system (lived forever). This distribution is also the censoring
distribution for a randomly \vspace{0.5cm}

\noindent selected unit that entered the system by time \textit{t}.\textit{\ 
}Without loss of generality, we will assume \vspace{0.5cm}

\noindent that $\lambda (\cdot )$ is a continuous function. Then we have
that $0<h_{t}(0)=\lim_{x\rightarrow 0^{+}}h_{t}(x)<\infty $. The \vspace{%
0.5cm}

\noindent result in (1) is that all possible\ age distributions are among
those distributions smaller with \vspace{0.5cm}

\noindent respect to a likelihood ratio ordering than the distribution with
density $j_{t}$ that satisfy \vspace{0.5cm}

\noindent $0<h_{t}(0)=\lim_{x\rightarrow 0^{+}}h_{t}(x)<\infty $. It will
now be shown that this likelihood ratio order result \vspace{0.5cm}

\noindent with $0<h_{t}(0)=\lim_{x\rightarrow 0^{+}}h_{t}(x)<\infty $
completely characterizes the possible age distributions \vspace{0.5cm}

\noindent (for $\lambda (\cdot )\;$positive valued). The arguments are given
below:\vspace{0.5cm}

Let $h^{\ast }$ denote the density for a continuous distribution $H^{\ast }$%
, where $H^{\ast }\leq _{lr}J_{t}$ and \vspace{0.5cm}

\noindent $0<h_{t}(0)=\lim_{x\rightarrow 0^{+}}h^{\ast }(x)<\infty $. Then
there exists a nonincreasing, positive function $w^{\ast }(\cdot )$\ on 
\vspace{0.5cm}

\noindent \lbrack 0, t] such that \vspace{0.5cm}

\[
h^{\ast }(y)=j_{t}(y)w^{\ast }(y),\text{ }0\leq y\leq t\text{.} 
\]
\vspace{0.5cm}

\noindent Note that $0<w^{\ast }(0)=\lim_{x\rightarrow 0^{+}}w^{\ast
}(x)<\infty $, since $0<h_{t}(0)=\lim_{x\rightarrow 0^{+}}h^{\ast
}(x)<\infty ,$ and \vspace{0.5cm}

\noindent $0\leq w^{\ast }(y)/w^{\ast }(0)\leq 1$ is nonincreasing, since ($%
H^{\ast }\leq _{lr}J_{t}$). Define $F^{\ast }\;$by $1-F^{\ast }(y)=$\vspace{%
0.5cm}

\noindent $w^{\ast }(y)/w^{\ast }(0),\;0\leq y\leq t.\;$Then, if the life
distribution were $F^{\ast }$, the age distribution at time $t$ \vspace{0.5cm%
}

\noindent has density \vspace{0.5cm}

\[
\frac{\lambda (t-y)w^{\ast }(y)}{\int_{0}^{t}\lambda (t-s)w^{\ast }(s)ds}%
,0\leq y\leq t. 
\]
\vspace{0.5cm}

\noindent This density is proportional to $j_{t}(y)w^{\ast }(y)=h^{\ast
}(y).\;$Since the above and $h^{\ast }$ are both densities \vspace{0.5cm}

\noindent and thus, integrate to 1 over the support and these densities are
proportional, the above \vspace{0.5cm}

\noindent density is $h^{\ast }$.\vspace{0.5cm}

\textit{Estimating the underlying life distribution from the ages: }Dykstra,
Kochar and Robertson \vspace{0.5cm}

\noindent (1995) obtained the maximum likelihood estimates in the two sample
problem where the two \vspace{0.5cm}

\noindent distributions satisfy a likelihood ratio order. The form of the
solution to the one sample \vspace{0.5cm}

\noindent problem may be different. The density \emph{j}$_{t}$ can be
determined based on data involving the \vspace{0.5cm}

\noindent arrivals. If this data is regarded as a random sample then the
distribution of the ages can be \vspace{0.5cm}

\noindent determine according to Dykstra, et. al. (1995). If the
distribution of $j_{t}$ is regarded as fixed \vspace{0.5cm}

\noindent and known, then the distribution of ages should be estimated with
respect to the likelihood \vspace{0.5cm}

\noindent ratio ordering to a known distribution. In either case, for a
random sample of $n\;$ages at time \vspace{0.5cm}

\noindent $t,$ the density of the age distribution at time $t,$ $h_{t}$ can
be estimated by $\widehat{h}_{n}\;$with respect to the \vspace{0.5cm}

\noindent likelihood ratio ordering with $j_{t}$. Let $\widehat{j_{t}}$\
denote either the known or estimated distribution for \vspace{0.5cm}

\noindent the time back from time $t$ to a random arrival.\ Then $F$
truncated at $t$ is then estimated by$\;\widehat{F}_{n}$ \vspace{0.5cm}

\noindent given by\vspace{0.5cm}

\[
1-\widehat{F}_{n}(y)=\frac{\widehat{h}_{n}(y)/\widehat{j_{t}}(y)}{\widehat{h}%
_{n}(0)/\widehat{j_{t}}\left( 0\right) },\;0\leq y\leq t\text{.} 
\]
\vspace{0.5cm}

\noindent The value of $\widehat{j}_{t}(0)/\widehat{h}_{n}(0)$ is some
estimate of the proportion of all units to enter the system in \vspace{0.5cm}

\noindent $(0,t]$ that still live in the system\ or for cases with observed
ages near zero, $\widehat{j}_{t}(0)/\widehat{h}_{n}(0)$ can \vspace{0.5cm}

\noindent be estimated by the $\widehat{j}_{t}(y)/\widehat{h}_{n}(y)$ as $y$
goes down to 0.\vspace{0.5cm}

Let $y_{1},...,y_{n}$ be a random sample from the distribution having
density $h_{t}$. If $j$ is regarded \vspace{0.5cm}

\noindent as known, $h_{t}$ can also be estimated by first determining an
unrestricted estimator, $\widetilde{h}_{n}$, that has \vspace{0.5cm}

\noindent $0<\widetilde{h}_{n}(0)<\infty $. This unrestricted estimator
would ignore the likelihood ratio requirement to \vspace{0.5cm}

\noindent the density $j_{t}$. Then making use of a technique of Hengartner
and Rojo in a work in progress, \vspace{0.5cm}

\noindent define $\widehat{h}_{n}$ for some $0\leq \alpha \leq 1$ by\vspace{%
0.5cm}

\[
\widehat{h}_{n}(y)=\alpha \frac{\stackunder{s\leq y}{\inf }\frac{\widetilde{h%
}_{n}(s)}{j_{t}\left( s\right) }j_{t}(y)}{\int_{0}^{t}\stackunder{s\leq x}{%
\inf }\frac{\widetilde{h}_{n}(s)}{j_{t}\left( s\right) }j_{t}(x)dx}%
+(1-\alpha )\frac{\stackunder{y\leq s\leq t}{\sup }\frac{\widetilde{h}_{n}(s)%
}{j_{t}\left( s\right) }j_{t}(y)}{\int_{0}^{t}\stackunder{x\leq s\leq t}{%
\sup }\frac{\widetilde{h}_{n}(s)}{j_{t}\left( s\right) }j_{t}(x)dx} 
\]
\vspace{0.5cm}

\noindent or\vspace{0.5cm}

\[
\widehat{h}_{n}(y)=\frac{\alpha \stackunder{s\leq y}{\inf }\frac{\widetilde{h%
}_{n}(s)}{j_{t}\left( s\right) }+(1-\alpha )\stackunder{y\leq s\leq t}{\sup }%
\frac{\widetilde{h}_{n}(s)}{j_{t}\left( s\right) }}{\int_{0}^{t}\{\alpha 
\stackunder{s\leq x}{\inf }\frac{\widetilde{h}_{n}(s)}{j_{t}\left( s\right) }%
+(1-\alpha )\stackunder{x\leq s\leq t}{\sup }\frac{\widetilde{h}_{n}(s)}{%
j_{t}\left( s\right) }\}j_{t}(x)dx}j_{t}(y). 
\]
\vspace{0.5cm}

\noindent In either case, it is easy to show that if $\widetilde{h}_{n}$ is
either a weakly or strongly consistent estimator, \vspace{0.5cm}

\noindent then $\widehat{h}_{n}$ is a weakly or strongly consistent
estimator.\vspace{0.5cm}

\noindent \textbf{2.2.2} \textbf{Theoretical distribution for the age at
departure for a departure at time }$t$\vspace{0.5cm}

The listing in the obituaries, involve a sample of recent deaths. So, to
estimate the \vspace{0.5cm}

\noindent underlying life distribution from the recent deaths, it is
necessary to determine the form of \vspace{0.5cm}

\noindent the distribution of the age at death at specific points in time
and how such distributions relate \vspace{0.5cm}

\noindent to the underlying distribution.\vspace{0.5cm}

\noindent \textbf{Theorem 2.2 }\textit{The probability a departure at time }$%
t$\textit{\ had an age }$\leq y$\textit{\ is given by}\vspace{0.5cm}

\[
K_{t}(y)=\int_{[0,y]}w(x)dF(x)/\int_{[0,\infty )}w(x)dF(x) 
\]
\vspace{0.5cm}

\noindent \textit{where }$w(x)=\lambda (t-x),$ $x\leq t$\textit{\ and }$%
w(x)=0,$ $x>t.$\vspace{0.5cm}

The proof will be given in section 3. The distribution $G_{t}$\ is a
``weighted average'' of those \vspace{0.5cm}

\noindent distributions $K_{s}$ $0<s\leq t$. We have that $%
G_{t}(y)=\int_{0}^{t}\lambda _{D}(s)K_{s}(y)ds/\int_{0}^{t}\lambda _{D}(s)ds$
where $\lambda _{D}(s)=$\vspace{0.5cm}

\noindent $\int_{0}^{s}\lambda (s-u)dF(u)$ is the intensity function for the
departure process.\vspace{0.5cm}

We have the following consequences of Theorem 2.2.\vspace{0.5cm}

\textit{(i)} If $\lambda $ is nondecreasing (nonincreasing), then $w$ is
nonincreasing (nondecreasing) and \vspace{0.5cm}

$K_{t}\leq _{lr}F$ ($K_{t}\geq _{lr}F$).\vspace{0.5cm}

\textit{(ii)} For $\lambda (s)\equiv \lambda $, $w(\cdot )$ constant then $%
K_{t}(y)=F(y)/F(t)$ for $0\leq y\leq t.$\vspace{0.5cm}

\textit{(iii)} If $\lambda (\cdot )$ is continuous, then $\stackunder{%
\varepsilon \rightarrow 0}{\lim }$ pr(age at departure $\leq y|$departure in 
$(t-\varepsilon ,t]$)$=K_{t}(y).$\vspace{0.5cm}

\noindent \textbf{2.3 Weighted Distributions and Estimation}\vspace{0.5cm}

The distribution $K$ is a called a weighted distribution of $F$ if \vspace{%
0.5cm}

\[
K(x)=(1/W)\int_{-\infty }^{x}w(y)dF(y), 
\]
\vspace{0.5cm}

\noindent where $0<W=\int_{-\infty }^{\infty }w(y)dF(y)<\infty \ $and $w$ is
a nonnegative function. When the densities \vspace{0.5cm}

\noindent exist, we have $k(x)\propto w(x)f(x)$.\vspace{0.5cm}

The distributions $G_{t}$ and $K_{t}$ are weighted distributions of the
underlying life distribution \vspace{0.5cm}

\noindent \textit{F }and also of \textit{F} truncated at \textit{t}. The
distribution $H_{t}$ can be whown to be a weighted \vspace{0.5cm}

\noindent distribution of $J_{t}$.\vspace{0.5cm}

Several authors have dealt with finding the nonparametric maximum likelihood 
\vspace{0.5cm}

\noindent estimator of $F$ and $K$ and its asymptotic properties based on
independent random samples \vspace{0.5cm}

\noindent from $F$ and/or $K$ when either the weight function is known or
the form of the weight function \vspace{0.5cm}

\noindent is known. If $w$ is unknown, but believed to be either
nondecreasing or nonincreasing, then $F$ \vspace{0.5cm}

\noindent and $K$ are related by a likelihood ratio ordering and thus, can
be estimated from the results of \vspace{0.5cm}

\noindent Dykstra, Kochar and Robertson (1995). Using a specific
nonincreasing or nondecreasing \vspace{0.5cm}

\noindent weight function or form for a weighted function creates a more
specific relation or \vspace{0.5cm}

\noindent ``ordering'' than a likelihood ratio ordering.\vspace{0.5cm}

When $w$ is a known positive valued function over the support of $F$, $F$
can be estimated \vspace{0.5cm}

\noindent based on a random sample from $K$. The above will then represent a
one to one mapping from \vspace{0.5cm}

\noindent $F$ to $K$ and we will have \vspace{0.5cm}

\[
F(x)=W\int_{-\infty }^{x}\frac{1}{w(y)}dK(y). 
\]
\vspace{0.5cm}

\noindent In our case of interest, $K_{t}$ is a weighted distribution of $F$
truncated at $t$ having a positive \vspace{0.5cm}

\noindent weight function over $(0,$ $t)$. If $y_{1},y_{2},...,y_{n}$ are
the values from a random sample from $K$, then \vspace{0.5cm}

\noindent the nonparametric maximum likelihood estimate of $K(x)$ is \vspace{%
0.5cm}

\[
\widehat{K}_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}I(y_{i}\leq x). 
\]
\vspace{0.5cm}

\noindent By the maximum likelihood invariance principle, the nonparametric
maximum likelihood \vspace{0.5cm}

\noindent estimate of $F(x)$ is 
\[
\widehat{F}_{n}(x)=\sum_{i=1}^{n}\frac{1}{w(y_{i})}I(y_{i}\leq x)\ /\
\sum_{i=1}^{n}\frac{1}{w(y_{i})}. 
\]
\vspace{0.5cm}

\noindent $\widehat{F}_{n}(x)\ $can be shown to be a consistent estimate of $%
F(x)$ and also, $\sqrt{n}(\widehat{F}_{n}-F)$ can be shown \vspace{0.5cm}

\noindent to converge to a pinned Gaussian process. These follow from
results in Vardi (1981) and \vspace{0.5cm}

\noindent El Barmi and Rothmann (1998).\vspace{0.5cm}

\noindent \textit{Estimating }$F$\textit{\ by sampling from the obituaries: }%
We will give an application of estimating a \vspace{0.5cm}

\noindent human life distribution by sampling from the obituaries in the
Cedar Rapids Gazette during \vspace{0.5cm}

\noindent Oct. 10-12, 1997. In order to get an impression of the underlying
life distribution it will be \vspace{0.5cm}

\noindent necessary to adjust for the rate of births. We will assume that
births occur according to a \vspace{0.5cm}

\noindent NHPP with intensity function $\lambda $ where $\lambda $ is
constant within any calendar year, i.e., $\widehat{\lambda }(s)=$ \vspace{%
0.5cm}

\noindent number of births in year $\left[ s\right] $, where $[\cdot ]$
represents the greatest integer function. A graph of the \vspace{0.5cm}

\noindent number of births in the United States from 1910 to 1995 is given
in Figure 1 below. The \vspace{0.5cm}

\noindent source of the data is the National Center for Health Statistics.%
\vspace{0.5cm}

For a random sample of recent ages at departure, $y_{1},...,y_{n},$ the
nonparametric maximum \vspace{0.5cm}

\noindent likelihood estimate of $F(x)$ is\vspace{0.5cm}

\begin{equation}
\widehat{F}_{n}(y)=\frac{\sum_{i=1}^{n}(\frac{1}{\widehat{\lambda }%
(1997-y_{i})})I(y_{i}\leq y)}{\sum_{i=1}^{n}(\frac{1}{\widehat{\lambda }%
(1997-y_{i})})}.  \tag{(2)}
\end{equation}
\vspace{0.5cm}

For this application use a random sample of 20 obituaries in the Cedar
Rapids Gazette \vspace{0.5cm}

\noindent (Oct. 10-12, 1997). Figure 2 gives the standard empirical
distribution function and $\widehat{F}_{n}$\ as in \vspace{0.5cm}

\noindent (2). The two distribution functions are not terribly different.
This is because for this sample \vspace{0.5cm}

\noindent of obituaries, the number of births in the years the deceased were
born varied slightly, \vspace{0.5cm}

\noindent ranging from 2.4 million to 3 million.\vspace{0.5cm}

Had this sample had taken place in 2017, there would have been a much
greater \vspace{0.5cm}

\noindent difference between the empirical distribution function and the
distribution function which \vspace{0.5cm}

\noindent adjusted for the number of births in a year. The number of births
in the years the deceased \vspace{0.5cm}

\noindent were born would have ranged from 2.4 million to 4.3 million. Had
this sample of obituaries \vspace{0.5cm}

\noindent been sampled in 2017 instead of 1997 the larger range in the
number of births in the years \vspace{0.5cm}

\noindent the deceased were born would lead to a larger range for the masses
of probability for the \vspace{0.5cm}

\noindent estimate of $F$. In this particular case, the estimate of $F$
would mostly have been a larger \vspace{0.5cm}

\noindent distribution had these ages at death been sampled in 2017 instead
of 1997. Figure 3 gives $\widehat{F}_{n}$\ \vspace{0.5cm}

\noindent as in (2) and $\widehat{F}_{n}$ with 2017 replacing 1997. There is
a notable early difference between the \vspace{0.5cm}

\noindent two distribution functions.\vspace{0.5cm}

Table 1 gives a summary of the range of probabilities (mass) assigned at
each \vspace{0.5cm}

\noindent observation, the quantiles and the mean. Notice the larger range
of probabilities for the 2017 \vspace{0.5cm}

\noindent adjusted estimate of $F.$\vspace{0.5cm}

\noindent \textbf{2.4 Testing for a nonincreasing hazard rate function}%
\vspace{0.5cm}

We have for $0<y\leq t$ from theorems 2.1 and 2.2 that\vspace{0.5cm}

\begin{eqnarray}
K_{t}(y) &=&\int_{0}^{y}\lambda (t-x)dF(x)/\int_{0}^{t}\lambda (t-x)dF(x)%
\vspace{0.5cm}  \TCItag{(3)} \\
&&\int_{0}^{y}r_{F}(x)\{\lambda
(t-x)/r_{F}(x)\}dF(x)/\int_{0}^{t}r_{F}(x)\{\lambda (t-x)/r_{F}(x)\}dF(x)%
\vspace{0.5cm}  \nonumber \\
&=&\int_{0}^{y}r_{F}(x)dH_{t}(x)/\int_{0}^{t}r_{F}(x)dH_{t}(x)  \nonumber
\end{eqnarray}
\vspace{0.5cm}

\noindent and thus, $k_{t}(y)/h_{t}(y)\varpropto r_{F}(y)$. The probability
at time $t$ that a specific living unit will be that \vspace{0.5cm}

\noindent given departure at time $t$\ is proportional to its current
failure rate. If L has an exponential \vspace{0.5cm}

\noindent distribution then $K_{t}=H_{t}$. This is because if each living
unit has the same underlying \vspace{0.5cm}

\noindent exponential distribution for its life, then the remaining life
distrtibution is the same for all the \vspace{0.5cm}

\noindent living units, and thus, each living unit has an equal probability
of being the next departure. It \vspace{0.5cm}

\noindent follows from (3) that the failure rate for the underlying life
distribution is increasing \vspace{0.5cm}

\noindent (decreasing), if and only if, the distribution for the age at
death for a death at time $t$\ is greater \vspace{0.5cm}

\noindent (smaller) than the distribution for the ages at time $t$\ with
respect to a likelihood ratio \vspace{0.5cm}

\noindent ordering. Dykstra, et. al. give a likelihood ratio test for
testing whether two independent \vspace{0.5cm}

\noindent random samples come from distributions that satisfy a likelihood
ratio ordering. So, testing \vspace{0.5cm}

\noindent whether the distribution of the ages and the distribution for the
most recent ages at death \vspace{0.5cm}

\noindent satisfy a likelihood ratio ordering is equivalent to testing
whether the underlying life \vspace{0.5cm}

\noindent distributions has a nonincreasing or nondecreasing failure rate.%
\vspace{0.5cm}

\noindent \textbf{2.5 Some ordering results comparing two underlying life
distributions}\vspace{0.5cm}

Suppose there is a constant probability $p$ that an object is ``type 1''
(e.g., female), and \vspace{0.5cm}

\noindent probability $1-p$, that an object is ``type 2'' (e.g., male). So,
the two processes - one process \vspace{0.5cm}

\noindent of only type 1 units and the other process of only type 2 units -
have proportional intensity \vspace{0.5cm}

\noindent functions. The type 1 units have a common life distribution, as do
the type 2 units. Note that \vspace{0.5cm}

\noindent if the two underlying life distributions have proportional
hazards, then\vspace{0.5cm}

\[
\frac{f_{1}\left( y\right) }{f_{2}\left( y\right) }\varpropto \frac{%
1-F_{1}\left( y\right) }{1-F_{2}\left( y\right) }. 
\]
\vspace{0.5cm}

We have the following results.\vspace{0.5cm}

1. Fix \textit{t} \TEXTsymbol{>} 0. \ For proportional intensity functions,
the two distributions for the ages \vspace{0.5cm}

\noindent satisfy a likelihood ratio ordering if and only if the underlying
life distributions satisfy an \vspace{0.5cm}

\noindent uniform stochastic order. \ In fact,\vspace{0.5cm}

\[
\frac{h_{1,t}\left( y\right) }{h_{2,t}\left( y\right) }\varpropto \frac{%
1-F_{1}\left( y\right) }{1-F_{2}\left( y\right) }. 
\]
\vspace{0.5cm}

\noindent So testing whether underlying life distributions satisfy an
uniform stochastic ordering is the \vspace{0.5cm}

\noindent same as testing whether the age distributions satisfy a likelihood
ratio ordering. Based on \vspace{0.5cm}

\noindent independent samples of ages for two, their respective theoretical
age distributions can be \vspace{0.5cm}

\noindent tested for satisfying a likelihood ratio ordering, according to
Dykstra, et. al. The test \vspace{0.5cm}

\noindent automatically tests the respective underlying life distributions
satisfying an uniform \vspace{0.5cm}

\noindent stochastic ordering.\vspace{0.5cm}

If in addition, the underlying life distributions have proportional hazards,
then the ratio \vspace{0.5cm}

\noindent of densities for the ages is proportional to the ratio of
densities for the underlying \vspace{0.5cm}

\noindent distributions. That is, we have,\vspace{0.5cm}

\[
\frac{h_{1,t}\left( y\right) }{h_{2,t}\left( y\right) }\varpropto \frac{%
f_{1}\left( y\right) }{f_{2}\left( y\right) }. 
\]
\vspace{0.5cm}

Both of these results follow from\vspace{0.5cm}

\[
\frac{h_{1,t}\left( y\right) }{h_{2,t}\left( y\right) }=\frac{1-F_{1}\left(
y\right) }{1-F_{2}\left( y\right) }\cdot \frac{\lambda _{1}\left( t-y\right) 
}{\lambda _{2}\left( t-y\right) }\cdot \frac{\int_{0}^{t}\lambda _{2}\left(
t-x\right) dF_{2}\left( x\right) }{\int_{0}^{t}\lambda _{1}\left( t-x\right)
dF_{1}\left( x\right) }\text{.} 
\]
\vspace{0.5cm}

\noindent For proportional intensity functions, ($\lambda _{1}\left(
x\right) /\lambda _{2}\left( x\right) =k>0$, for $0<x\leq t$), we have, 
\vspace{0.5cm}

\[
\frac{h_{1,t}\left( y\right) }{h_{2,t}\left( y\right) }\varpropto \frac{%
1-F_{1}\left( y\right) }{1-F_{2}\left( y\right) }=\frac{r_{F_{2}}(y)}{%
r_{F_{1}}\left( y\right) }\cdot \frac{f_{1}\left( y\right) }{f_{2}\left(
y\right) }. 
\]
\vspace{0.5cm}

2. When the intensity functions are proportional, the two age at death
distributions satisfy \vspace{0.5cm}

\noindent a likelihood ratio ordering if and only if the underlying life
distributions satisfy a likelihood \vspace{0.5cm}

\noindent ratio ordering. This holds also for the two age at death at time $%
t $ distributions.\ In fact, the \vspace{0.5cm}

\noindent ratio of age at death densities are proportional to the ratio of
underlying densities. That is, we \vspace{0.5cm}

\noindent have,\vspace{0.5cm}

\[
\frac{g_{1,t}\left( y\right) }{g_{2,t}\left( y\right) }\varpropto \frac{%
f_{1}\left( y\right) }{f_{2}\left( y\right) }\varpropto \frac{k_{1,t}\left(
y\right) }{k_{2,t}\left( y\right) }. 
\]
\vspace{0.5cm}

\noindent So testing whether underlying life distributions satisfy a
likelihood ratio ordering is \vspace{0.5cm}

\noindent equivalent to both testing whether the age at death distributions
satisfy a likelihood ratio \vspace{0.5cm}

\noindent ordering and also to testing whether the age at death at time $t$
distributions satisfy a \vspace{0.5cm}

\noindent likelihood ratio ordering\ .\ In addition, when the underlying
life distributions have \vspace{0.5cm}

\noindent proportional hazards, then the ratio of densities for the age at
death distributions (ratio of \vspace{0.5cm}

\noindent densities for the age at death at time $t$ distributions)\ is
proportional to the ratio of the \vspace{0.5cm}

\noindent underlying survival functions That is, we have,\vspace{0.5cm}

\[
\frac{g_{1,t}\left( y\right) }{g_{2,t}\left( y\right) }\varpropto \frac{%
1-F_{1}\left( y\right) }{1-F_{2}\left( y\right) }\varpropto \frac{%
k_{1,t}\left( y\right) }{k_{2,t}\left( y\right) }. 
\]
\vspace{0.5cm}

Both of these results follow from (analogous for the age at death at time $t$
distribution \vspace{0.5cm}

\noindent results)\ \vspace{0.5cm}

\[
\frac{g_{1,t}\left( y\right) }{g_{2,t}\left( y\right) }=\frac{f_{1}\left(
y\right) }{f_{2}\left( y\right) }\cdot \frac{\Lambda _{1}\left( t-y\right) }{%
\Lambda _{2}\left( t-y\right) }\cdot \frac{\int_{0}^{t}\Lambda _{1}\left(
t-x\right) dF_{2}\left( x\right) }{\int_{0}^{t}\Lambda _{2}\left( t-x\right)
dF_{1}\left( x\right) }\varpropto \frac{f_{1}\left( y\right) }{f_{2}\left(
y\right) }=\frac{r_{F_{1}}(y)}{r_{F_{2}}\left( y\right) }\cdot \frac{%
1-F_{1}\left( y\right) }{1-F_{2}\left( y\right) } 
\]
\vspace{0.5cm}

\noindent where $\Lambda _{1}$\ and $\Lambda _{2}$\ are the cumulative
intensity functions.\vspace{0.5cm}

3. Note that\vspace{0.5cm}

\[
\frac{h_{t}(y)}{H_{t}(y)}=\frac{\lambda (t-y)\{1-F(y)\}}{\int_{0}^{y}\lambda
(t-s)\{1-F(s)\}ds}=\frac{\lambda (t-y)f(y)/r_{F}(y)}{\int_{0}^{y}\lambda
(t-s)f(s)/r_{F}(s)ds}. 
\]
\vspace{0.5cm}

\noindent Thus, if $\lambda (t-x)/r_{F}(x)$ is constant in $x$ for some
fixed \textit{t}, then $h_{t}(y)/H_{t}(y)=f(y)/F(y)$. When \vspace{0.5cm}

\noindent $\lambda _{i}(t-x)/r_{F_{i}}(x)$ is constant for $i=1,2$, the
underlying life distributions satisfy a reverse \vspace{0.5cm}

\noindent hazard rate ordering if and only if the two age at death
distributions satisfy a reverse hazard \vspace{0.5cm}

\noindent rate ordering.\vspace{0.5cm}

\noindent \textit{Proportion and limiting proportion of type 1 units. }The
odds that a randomly selected unit \vspace{0.5cm}

\noindent that is in the system at some \ fixed time \textit{t} (``active''
not ``departed'') is a type 1 unit opposed \vspace{0.5cm}

\noindent to a type 2 unit is \vspace{0.5cm}

\[
\frac{p}{1-p}\cdot \frac{\int_{0}^{t}\lambda (t-s)\{1-F_{1}(s)\}ds}{%
\int_{0}^{t}\lambda (t-s)\{1-F_{2}(s)\}ds}. 
\]
\vspace{0.5cm}

\noindent Assuming $\int_{0}^{t}\lambda (s)ds\rightarrow \infty $, we have
the following results.\vspace{0.5cm}

(i) If $\lambda (t-s)/\lambda (t)\rightarrow 0$ for all $s>0,\;$then the
limiting odds that a randomly selected \vspace{0.5cm}

\noindent active unit is type 1 is $\frac{p}{1-p}.$\vspace{0.5cm}

(ii) If $\lambda (t-s)/\lambda (t)\rightarrow \exp (-s/c)$ for all $s>0,\;$%
then the limiting odds that a randomly \vspace{0.5cm}

\noindent selected active unit is type 1 is \vspace{0.5cm}

\[
\frac{p}{1-p}\cdot \frac{1-\int_{0}^{\infty }\exp (-s/c)dF_{1}(s)}{%
1-\int_{0}^{\infty }\exp (-s/c)dF_{2}(s)}=\frac{p}{1-p}\cdot \frac{%
1-M_{1}(1/c)}{1-M_{2}(1/c)}. 
\]
\vspace{0.5cm}

(iii) If $\lambda (t-s)/\lambda (t)\rightarrow 1$ for all $s>0,\;$and the
mean lifetimes are finite, then the limiting \vspace{0.5cm}

\noindent odds that a randomly selected active unit is type 1 is \vspace{%
0.5cm}

\[
\frac{p}{1-p}\cdot \frac{\int_{0}^{\infty }\{1-F_{1}(s)\}ds}{%
\int_{0}^{\infty }\{1-F_{2}(s)\}ds}=\frac{p}{1-p}\cdot \frac{\mu _{1}}{\mu
_{2}}. 
\]
\vspace{0.5cm}

>From lemma 3a of Rothmann and Russo (2000), we see that these limits above
also \vspace{0.5cm}

\noindent represent the almost sure limits for the empirical proportion of
active units that are type 1. \vspace{0.5cm}

\noindent Iin the case where $\lambda (t-s)/\lambda (t)\rightarrow 1$, $%
\lambda (t)/\log t\rightarrow \infty $ is needed for almost sure convergence.%
\vspace{0.5cm}

If the common life distribution of a type 1 unit\ is greater than the common
life \vspace{0.5cm}

\noindent distribution of a type 2 unit with respect to a Laplace transform
ordering then for all \vspace{0.5cm}

\noindent $0\leq c\leq \infty ,$ the limiting odds (probability) that a
randomly selected unit is a type 1 unit is \vspace{0.5cm}

\noindent greater than the odds (probability) that a random ``birth'' is a
type 1 unit. If type 1 units have \vspace{0.5cm}

\noindent a larger life distribution\ than type 2 units with respect to a
reverse Laplace transform ratio \vspace{0.5cm}

\noindent ordering the limiting odds (probability) that a randomly selected
unit is a type 1 unit is \vspace{0.5cm}

\noindent nondecreasing in $c$.\vspace{0.5cm}

\textbf{2.6 Limiting Behavior}\vspace{0.5cm}

Let $Z_{1},Z_{2},...,Z_{A(t)}$ denote a random permutation of the collection
of ages among active \vspace{0.5cm}

\noindent units at time $t$. Define for $y>0$, \vspace{0.5cm}

\[
\widehat{H}_{t}(y)=\frac{\sum_{i=1}^{A(t)}I(Z_{i}\leq y)}{A(t)}. 
\]
\vspace{0.5cm}

$\widehat{H}_{t}(y)$ is the empirical proportion of the ages which is less
than or equal to $y$.\vspace{0.5cm}

We have the following corollary to theorems in Rothmann and Russo (2000) and 
\vspace{0.5cm}

\noindent Rothmann and El Barmi (2001)\vspace{0.5cm}

\textbf{Corollary 2.1}\vspace{0.5cm}

Suppose that $\int_{0}^{t}\lambda (s)ds\rightarrow \infty .$\vspace{0.5cm}

(a) Suppose $\lambda \;$is nondecreasing. If $\lambda (s)/\log s\rightarrow
\infty ,\;$or if for some positive $\alpha $, we have \vspace{0.5cm}

\noindent $0\leq \lambda (t)\leq \alpha +t^{\alpha }$ for all $t\geq 0$ with 
$(\lambda (t)/\log t)\int_{0}^{t}pr(L>s)ds\rightarrow \infty ,$ then\vspace{%
0.5cm}

\[
\left| \widehat{H}_{t}(y)-\int_{0}^{y}\lambda
(t-s)\{1-F(s)\}ds/\int_{0}^{t}\lambda (t-s)\{1-F(s)\}ds\right| \rightarrow 0%
\text{\textit{almost surely}} 
\]
\vspace{0.5cm}

(b) If $\lim_{t\rightarrow \infty }\int_{(-\infty ,x]}\frac{%
\int_{0}^{t}\lambda (t-s)F_{y}(s)ds}{\int_{0}^{t}\lambda (t-s)F(s)ds}%
dG(y)=\int_{(-\infty ,x]}\lim_{t\rightarrow \infty }\frac{%
\int_{0}^{t}\lambda (t-s)F_{y}(s)ds}{\int_{0}^{t}\lambda (t-s)F(s)ds}dG(y)$,
then\vspace{0.5cm}

\[
\lim_{t\rightarrow \infty }G_{t}(x)=\int_{(-\infty ,x]}\delta (u)dG(u)%
\mathit{\ }\text{\textit{almost surely,}} 
\]
\vspace{0.5cm}

\noindent where $\delta (x)=\lim_{t\rightarrow \infty }\frac{%
\int_{y}^{t}\lambda (t-s)ds}{\int_{0}^{t}\lambda (t-s)F(s)ds}.$\vspace{0.5cm}

The first part of Corollary 2.1follows from Theorem 2 of Rothmann and Russo
(2000) \vspace{0.5cm}

\noindent with $V\equiv y$ and the second part follows from Theorem 2.3 of
Rothmann and El Barmi (2001) \vspace{0.5cm}

\noindent with $X=L$. According to analogous results in Rothmann and Russo
(2000), the assumption \vspace{0.5cm}

\noindent of $\lambda \;$nondecreasing can be weakened.\vspace{0.5cm}

Suppose $\lambda \;$is nondecreasing and$\;\lambda (t-s)/\lambda
(t)\rightarrow \alpha (s)$ for all $s>0.\;$By\ Corollary 2.1, the \vspace{%
0.5cm}

\noindent following results hold with probability one:\vspace{0.5cm}

(iv) If $\alpha (s)\equiv 0,$ then$\ \widehat{H}_{t}(y)\rightarrow 1$ and $%
\widehat{G}_{t}(y)\rightarrow 0$ for all $y.$\vspace{0.5cm}

(v) If $\alpha (s)=\exp \left( -s/c\right) ,$ then \vspace{0.5cm}

\noindent $\widehat{H}_{t}(y)\rightarrow \int_{0}^{y}\exp
(-x/c)\{1-F(x)\}dx/\int_{0}^{\infty }\exp (-x/c)\{1-F(x)\}dx$ and \vspace{%
0.5cm}

\noindent $\widehat{G}_{t}(y)\rightarrow \int_{0}^{y}\exp
(-x/c)dF(x)/\int_{0}^{\infty }\exp (-x/c)dF(x)$ for all $y.$\vspace{0.5cm}

(vi) If $\alpha (s)\equiv 1,\ \widehat{G}_{t}(y)\rightarrow F(y)$ for all $y$%
.\vspace{0.5cm}

(vii) If $\alpha (s)\equiv 1,$ $\lambda (t)/\log t\rightarrow \infty ,$ and$%
\;EL<\infty ,$ then $\widehat{H}_{t}(y)\rightarrow
\int_{0}^{y}\{1-F(x)\}dx/EL.$\vspace{0.5cm}

(viii) If $\alpha (s)\equiv 1,$ $\lambda (t)/\log t\rightarrow b\in
(0,\infty ),\;EL=\infty ,$ then $\widehat{H}_{t}(y)\rightarrow 0.$\vspace{%
0.5cm}

(ix) If $\lambda (t)\equiv \lambda ,\;t\{pr(L>t)\}\rightarrow \infty ,$ then 
$\widehat{H}_{t}(y)\rightarrow 0.$\vspace{0.5cm}

Note that when c increases in $[0,\infty ]$, the respective limiting
distributions becomes larger \vspace{0.5cm}

\noindent with respect to a likelihood ratio ordering. These limits also
represent limits for $G_{t}(y),$ $H_{t}(y)$ \vspace{0.5cm}

\noindent and $K_{t}(y).$ The limits of $K_{t}(y)$ are the same as the
limits of $G_{t}(y)$.\vspace{0.5cm}

\textbf{3. Proofs}\vspace{0.5cm}

\noindent \textit{Proof of Theorem 2.1. }Note that the points $\left\{
(S_{k},L_{k})\right\} $ arise according to a Poisson point \vspace{0.5cm}

\noindent process in $[0,\infty )\times \lbrack 0,\infty )$ with intensity
measure $\lambda (t)dtF_{x}(dy)$. \ Let $(S_{1}^{\ast },L_{1}^{\ast }),$ $%
...,$ $(S_{A(t)}^{\ast },L_{A(t)}^{\ast })$ \vspace{0.5cm}

\noindent denote a random permutation of $\{(S_{k},L_{k}):$ $S_{k}+L_{k}>t\}$
and $(S_{1}^{\prime },Y_{1}),$ $...,$ $(S_{D(t)}^{\prime },Y_{D(t)})$ denote 
\vspace{0.5cm}

\noindent a random permutation of $\{(S_{k},L_{k}):S_{k}+L_{k}\leq t\}$. It
follows Resnick (1992; page 359) that \vspace{0.5cm}

\noindent $(S_{1}^{\ast },L_{1}^{\ast }),$ $...,$ $(S_{A(t)}^{\ast
},L_{A(t)}^{\ast })\ $is a random sample of random vectors and $%
(S_{1}^{\prime },Y_{1}),$ $...,$ \vspace{0.5cm}

\noindent $(S_{D(t)}^{\prime },Y_{D(t)}),$ is a random sample of random
vectors. Let $Z_{j}=t-L_{j}^{\ast }$ for $j=1,...,A(t)$. The \vspace{0.5cm}

\noindent value of the intensity function for getting an arrival at time $s$
that will be alive at time \textit{t} is \vspace{0.5cm}

\noindent $\lambda (s)\{1-F(t-s)\},$ $0<s<t$. Thus, $pr(Z_{j}\leq y)$ equals
the expected number of such arrivals \vspace{0.5cm}

\noindent in $(t-y,t)$ divided by the expected number of such arrivals in $%
(0,t)$. So we have, \vspace{0.5cm}

\[
pr(Z_{j}\leq y)=\frac{\int_{t-y}^{t}\lambda (s)\{1-F(t-s)\}ds}{%
\int_{0}^{t}\lambda (s)\{1-F(t-s)\}ds}=\frac{\int_{0}^{y}\lambda
(t-s)\{1-F(s)\}ds}{\int_{0}^{t}\lambda (t-s)\{1-F(s)\}ds}=H_{t}(y). 
\]
\vspace{0.5cm}

\noindent Now,

\begin{tabular}[t]{lll}
$pr(Y_{j}\leq y)$ & $=$ & $pr(L_{i}\leq y|S_{i}+L_{i}\leq t)$\vspace{0.5cm}
\\ 
& $=$ & $pr(L_{i}\leq y,S_{i}+L_{i}\leq t)/pr(S_{i}+L_{i}\leq t)$\vspace{%
0.5cm} \\ 
& $=$ & $\int_{0}^{y}pr(S_{i}+L_{i}\leq
t|L_{i}=x)dF(x)/\int_{0}^{t}pr(S_{i}+L_{i}\leq t|L_{i}=x)dF(x)$\vspace{0.5cm}
\\ 
& $=$ & $\int_{0}^{y}\{\int_{0}^{t-x}\lambda (s)ds/\int_{0}^{t}\lambda
(s)ds\}dF(x)/\int_{0}^{t}\{\int_{0}^{t-x}\lambda (s)ds/\int_{0}^{t}\lambda
(s)ds\}dF(x)$\vspace{0.5cm} \\ 
& $=$ & $\int_{0}^{y}\int_{0}^{t-x}\lambda
(s)dsdF(x)/\int_{0}^{y}\int_{0}^{t-x}\lambda (s)dsdF(x)=G_{t}(y).$%
\end{tabular}
\vspace{0.5cm}

Since the $(S_{k}^{\ast },L_{k}^{\ast })$'s and $(S_{j}^{\prime },Y_{j})$'s
arise in disjoint regions of $[0,\infty )\times \lbrack 0,\infty )$, $%
\left\{ (S_{k}^{\ast },L_{k}^{\ast })\right\} $ \vspace{0.5cm}

\noindent and $\{(S_{j}^{\prime },Y_{j})\}$ are independent. Therefore, $%
Z_{1},$ $Z_{2}$ $...,$ $Z_{A(t)}$ and $Y_{1},$ $Y_{2}$ $...,$ $Y_{D(t)}$ are 
\vspace{0.5cm}

\noindent independent random samples.\vspace{0.5cm}

\noindent \textit{Proof of Theorem 2.2.}The departure process forms a NHPP
with intensity function, \vspace{0.5cm}

\[
\lambda _{D}(t)=\frac{d}{dt}E(D(t))=\frac{d}{dt}\int_{(0,t)}\lambda
(x)F(t-x)dx=\int_{[0,t)}\lambda (t-x)dF(x). 
\]
\vspace{0.5cm}

\noindent Likewise, the subprocess of departures that have lifes $\leq y$
forms a NHPP with intensity \vspace{0.5cm}

\noindent function \vspace{0.5cm}

\[
\lambda _{D,y}(t)=\int_{[0,y)}\lambda (t-x)dF(x). 
\]
\vspace{0.5cm}

\noindent Thus, the probability a departure at time $t$ had an age $\leq y$
is given by \vspace{0.5cm}

\[
\int_{\lbrack 0,y)}\lambda (t-x)dF(x)/\int_{[0,t)}\lambda (t-x)dF(x). 
\]
\vspace{0.5cm}

\[
4.\text{ }\mathbf{Acknowledgement} 
\]

This work was done while the author was visiting the Department of
Statistics and \vspace{0.5cm}

\noindent Actuarial Science at the University of Iowa.\vspace{0.5cm}

\[
\mathbf{References} 
\]
\medskip

\noindent \textsc{Dykstra, R., Kochar, S. and Robertson, T.}{\small \ (1995)
``Inference for Likelihood Ratio Ordering in the }\vspace{0.5cm}

\noindent {\small Two-Sample Problem,'' 1034--1040, \textit{J Amer Statist
Assoc} \textbf{90}, No. 431}.\vspace{0.5cm}

\noindent \textsc{Resnick,} \textsc{S.}{\small (1992) \textit{Adventures in
Stochastic Processes} Birkh\"{a}user Boston.}\vspace{0.5cm}

\noindent \textsc{Rothmann,} \textsc{M.} \textsc{D.} \textsc{and} \textsc{El
Barmi,} \textsc{H.} {\small (2001). Stochastic processes involving random
deletion. \textit{J. Appl }}\vspace{0.5cm}

\noindent {\small \textit{Prob} \textbf{38}, 95-107}.\vspace{0.5cm}

\noindent \textsc{Rothmann,} \textsc{M.} \textsc{D.} \textsc{and} \textsc{%
Russo,} \textsc{R.} \textsc{P.} {\small (1997). On the limiting proportion
of types. \textit{J. Theoret. Prob.} \textbf{10}, }\vspace{0.5cm}

\noindent {\small 131-143}.\vspace{0.5cm}

\noindent \textsc{Rothmann,} \textsc{M.} \textsc{D.} \textsc{and} \textsc{%
Russo,} \textsc{R.} \textsc{P.} {\small (2000). Laws of large numbers for
observations that change with }\vspace{0.5cm}

\noindent {\small time. \textit{J. Theoret. Prob.} \textbf{13}, No. 4
1013-1025}.\vspace{0.5cm}

\noindent \textsc{Shaked, M. and Shantikumar, J. G.}{\small \ (1994) \textit{%
Stochastic Orders and Their Applications}, Academic Press, }\vspace{0.5cm}

\noindent {\small New York.}\vspace{0.5cm}

\noindent \textsc{Shaked, M. and Wong, T.}{\small \ (1997) ``Stochastic
Orders Based on Ratio of Laplace Transforms,'' 404-419, }\vspace{0.5cm}

\noindent {\small \textit{J Appl Prob} \textbf{34} No. 2}\vspace{0.5cm}

\noindent \textsc{Vardi,} \textsc{Y.} {\small (1985). Empirical
distributions in selection bias models. \textit{Ann. Statist.} \textbf{13},
178-203.}

\newpage 

\textbf{Table 1. Summary of the estimates of }$F$

\begin{tabular}[t]{cccc}
\hline
& \multicolumn{3}{c}{\textbf{Estimate of }$F$} \\ 
& \textbf{Empirical} & \textbf{1997 adjusted} & \textbf{2017 adjusted} \\ 
\hline
\multicolumn{1}{l}{\textbf{Range of probabilities}} & 0.05 & 0.0465-0.0597 & 
0.0341-0.0620 \\ 
\multicolumn{1}{l}{$\mathbf{25}^{\mathbf{th}}$\textbf{\ percentile}} & 66 & 
66 & 74 \\ 
\multicolumn{1}{l}{\textbf{Median}} & 76 & 76 & 76 \\ 
\multicolumn{1}{l}{$\mathbf{75}^{\mathbf{th}}$\textbf{\ percentile}} & 83 & 
88 & 88 \\ 
\multicolumn{1}{l}{\textbf{Mean}} & 75.7 & 75.57 & 77.28 \\ \hline
\end{tabular}

\bigskip \newpage 

\FRAME{ftbpF}{453.0625pt}{349.75pt}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 453.0625pt;height 349.75pt;depth
0pt;original-width 430.6875pt;original-height 332.0625pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'HYRCMU00.wmf';tempfile-properties "XPR";}}

Fig. 1. Plot of the yearly number of births in the United States.

\bigskip 

\newpage 

\FRAME{ftbpF}{452.25pt}{348.1875pt}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 452.25pt;height 348.1875pt;depth
0pt;original-width 429.9375pt;original-height 330.5625pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'HYRCRR01.wmf';tempfile-properties "XPR";}}

Fig. 2 Plot of the estimates of F.

\bigskip 

\newpage 

\FRAME{ftbpF}{452.25pt}{348.1875pt}{0pt}{}{}{Figure }{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"USEDEF";valid_file "T";width 452.25pt;height 348.1875pt;depth
0pt;original-width 429.9375pt;original-height 330.5625pt;cropleft
"0";croptop "1";cropright "1";cropbottom "0";tempfilename
'HYRCSR02.wmf';tempfile-properties "XPR";}}

Fig. 3. Plot of the estimates of F for the same sample from 2017 as from
1997.

\end{document}
