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Mortality Estimation
Empirical Estimator For Complete Data
Individual data
1
...
The empirical distribution is a distribution based on a study
1
with n observations
...

2
...
m
...
Grouped data is a data set that has a set of intervals and the number of observations in each interval
but does not provide the exact value of each observation
...

2
...
For mortality grouped by age, use of the ogive
to determine survival time is the same as using UDD
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𝑛𝑗
𝑓𝑛 (𝑥) =
𝑛(𝑐𝑗 − 𝑐𝑗−1 )
Where x is in the interval [𝑐𝑗 − 𝑐𝑗−1 ), there are 𝑛𝑗 in the interval and n points altogether
...
We treat a censored
observation as a grouped observation
...
If data
are left censored, we know an observation is below d but not the exact value, then the likelihood
is 𝐹(𝑑)
...
If data are left truncated at d, then the likelihood of x is
𝑓(𝑥)
𝑓(𝑥)
=
Pr(𝑋 > 𝑑) 1 − 𝐹(𝑑)

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For the rarer case of right truncated data, the likelihood of x is
𝑓(𝑥)
𝑓(𝑥)
=
Pr(𝑋 > 𝑑) 𝐹(𝑢)
Where for the expression on the right side we are assuming that X is a continuous random
variable
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Grouped data that are between d and 𝑐𝑗 in the presence of truncation at d has likelihood
𝐹(𝑐𝑗 )−𝐹(𝑑)
1−𝐹(𝑑)


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Formula
𝑗−1

𝑠𝑖
𝑆𝑛 (𝑡) = ∏ (1 − ) ,
𝑟𝑖

𝑦𝑗−1 ≤ 𝑡 ≤ 𝑦𝑗

𝑖=1

-

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𝑆 (𝑦 ) − 𝑆𝑛 (𝑦𝑗 ),
𝑥 = 𝑦𝑗 , 1 ≤ 𝑖 ≤ 𝑘
𝑝(𝑥) = { 𝑛 𝑗−1
0,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Discrete failure rate function: For a discrete random variable that has nonzero probability only
for a set of values 𝑦𝑗 ,
𝑝(𝑦𝑗 )
ℎ(𝑦𝑗 ) = Pr(𝑌 = 𝑦𝑗 |𝑌 ≥ 𝑦𝑗 ) =
,
1≤𝑗≤𝑘
𝑆(𝑦𝑗−1 )
Hazard rate function for a continuous distribution,
𝑓(𝑥)
ℎ(𝑥) =
𝑆(𝑥)
𝑝(𝑦𝑗 ) = 𝑆(𝑦𝑗−1 ) − 𝑆(𝑦𝑗 )
𝑆(𝑦𝑗 )
ℎ(𝑦𝑗 ) = 1 −
𝑆(𝑦𝑗−1 )
𝑗

𝑆(𝑦𝑗 ) = ∏ (1 − ℎ(𝑦𝑗 ))
𝑗−1

𝑖=1

𝑗

𝑝(𝑦𝑗 ) = 𝑆(𝑦𝑗−1 ) − 𝑆(𝑦𝑗 ) = ∏ (1 − ℎ(𝑦𝑗 )) − ∏ (1 − ℎ(𝑦𝑗 ))
𝑗−1

𝑖=1

𝑖=1

𝑗−1

= ∏ (1 − ℎ(𝑦𝑗 )) (1 − (1 − ℎ(𝑦𝑗 ))) = ℎ(𝑦𝑗 ) ∏ (1 − ℎ(𝑦𝑗 ))
𝑖=1

𝑖=1

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Let 𝜆𝑗 = ℎ(𝑦𝑗 ) and using maximum likelihood to estimate 𝜆𝑗
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Withdrawals at the time of deaths do count in the risk set
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Tail correction
- Def: Extrapolation of the survival function past the last time in the study
...
Otherwise, there are two extreme
options:
o Efron’s tail correction: assume nobody survives past 𝑦𝑚𝑎𝑥 , 𝑆𝑛 (𝑦) ≥ 0 for 𝑦 ≥ 𝑦𝑚𝑎𝑥
o Klein and Moeschberger’s tail correction: assume nobody dies past 𝑦𝑚𝑎𝑥 until some
plausible upper limit 𝛾 for survival time,
𝑆 (𝑦 ),
𝑦𝑚𝑎𝑥 ≤ 𝑦 < 𝛾
𝑆𝑛 (𝑦) = { 𝑛 𝑘
0,
𝑦≥𝛾
- Brown, Hollander, and Korwar’s exponential tail correction:
𝑦

𝑆𝑛 (𝑦) = 𝑆𝑛 (𝑦𝑘 )𝑦𝑚𝑎𝑥 , 𝑦 ≥ 𝑦𝑚𝑎𝑥



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