Estimation, Evaluation, and Selection of Actuarial Models

14 CHAPTER 2. MODEL ESTIMATION
i d i x i u i i d i x i u i
1 0 - 0.1 16 0 4.8 -
2 0 - 0.5 17 0 - 4.8
3 0 - 0.8 18 0 - 4.8
4 0 0.8 - 19—30 0 - 5.0
5 0 - 1.8 31 0.3 - 5.0
6 0 - 1.8 32 0.7 - 5.0
7 0 - 2.1 33 1.0 4.1 -
8 0 - 2.5 34 1.8 3.1 -
9 0 - 2.8 35 2.1 - 3.9
10 0 2.9 - 36 2.9 - 5.0
11 0 2.9 - 37 2.9 - 4.8
12 0 - 3.9 38 3.2 4.0 -
13 0 4.0 - 39 3.4 - 5.0
14 0 - 4.0 40 3.9 - 5.0
15 0 - 4.1
j y j s j r j
1 0.8 1 32 − 0 − 2=30or 0+32− 0 − 2=30
2 2.9 2 35 − 1 − 8=26or 30 + 3 − 1 − 6=26
3 3.1 1 37 − 3 − 8=26or 26 + 2 − 2 − 0=26
4 4.0 2 40 − 4 − 10 = 26 or 26 + 3 − 1 − 2=26
5 4.1 1 40 − 6 − 11 = 23 or 26 + 0 − 2 − 1=23
6 4.8 1 40 − 7 − 12 = 21 or 23 + 0 − 1 − 1=21
¤
Exercise 3 Repeat the above example, treating “surrender” as “death.” The easiest way to do this
is to reverse the x and u labels and then use the above formula. In this case death produces censoring
because those who die are lost to observation and thus their surrender time is never observed. Treat
those who lasted the entire fiveyearsassurrendersatthattime.
Despitealltheworkwehavedonetothispoint,wehaveyettoproduceanestimatorofthe
survival function. The one most commonly used is called the Kaplan-Meier Product-Limit
Estimator. Begin with S(0) = 1. Because no one died prior to y 1 , the survival function remains
at 1 until this value. Thinking conditionally, just before y 1 , there were r 1 people available to die, of
which s 1 did so. Thus, the probability of surviving past y 1 is (r 1 − s 1 )/r 1 .Thisbecomesthevalue
of S(y 1 ) and the survival function remains at that value until y 2 . Again, thinking conditionally,
the new survival value at y 2 is S(y 1 )(r 2 − s 2 )/r 2 . The general formula is
⎧
1, 0 ≤ t