\documentclass{article} \usepackage{amssymb} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %TCIDATA{OutputFilter=LATEX.DLL} %TCIDATA{Created=Sat Feb 20 20:49:40 1999} %TCIDATA{LastRevised=Thu Jun 03 20:20:41 2004} %TCIDATA{} %TCIDATA{} %TCIDATA{Language=American English} %TCIDATA{CSTFile=Exam.cst} %TCIDATA{PageSetup=90,90,90,90,0} %TCIDATA{AllPages= %F=7,\PARA{038

\hfill \QTR{small}{\thepage}} %} \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}[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 $-\infty s+t|W>t)\geq$\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{ }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$0t.$\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} 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 $00,\;$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