7.2Maximum Likelihood Estimators 239
dies in yeari. That is,
λi=P{X=i|X>i− 1 }=P{X=i}
P{X>i− 1 }Also, let
si= 1 −λi=P{X>i}
P{X>i− 1 }be the probability that a newborn who survives her firsti−1 years also survives yeari.
The quantityλiis called thefailure rate, andsiis called thesurvival rate, of an individual
who is entering his or herith year. Now,
s 1 s 2 ···si=P{X> 1 }P{X> 2 }P{X> 3 }
P{X> 1 }P{X> 2 }···P{X>i}
P{X>i− 1 }
=P{X>i}Therefore,
P{X=n}=P{X>n− 1 }λn=s 1 ···sn− 1 (1−sn)Consequently, we can estimate the probability mass function ofX by estimating the
quantitiessi,i=1,...,n. The valuesican be estimated by looking at all individuals
in the population who reached ageione year ago, and then letting the estimateˆsibe
the fraction of them who are alive today. We would then useˆs 1 ˆs 2 ···ˆsn− 1
(
1 −ˆsn)
as the
estimate ofP{X=n}. (Note that although we are using the most recent possible data to
estimate the quantitiessi, our estimate of the probability mass function of the lifetime of
a newborn assumes that the survival rate of the newborn when it reaches ageiwill be the
same as last year’s survival rate of someone of agei.)
The use of the survival rate to estimate a life distribution is also of importance in health
studies with partial information. For instance, consider a study in which a new drug is
given to a random sample of 12 lung cancer patients. Suppose that after some time we
have the following data on the number of months of survival after starting the new drug:
4, 7∗,9,11∗, 12, 3, 14∗,1,8,7,5,3∗wherexmeans that the patient died in monthxafter starting the drug treatment, andx∗
means that the patient has taken the drug forxmonths and is still alive.
LetXequal the number of months of survival after beginning the drug treatment, and
let
si=P{X>i|X>i− 1 }=P{X>i}
P{X>i− 1 }