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\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 ~~} 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~~