Estimation, Evaluation, and Selection of Actuarial Models

3.3. VARIANCE AND CONFIDENCE INTERVALS 51
It can be demonstrated that when there is no censoring or truncation, the two formulas will always
produce the same answer. Recall that the development of Greenwood’s formula produced the
variance only at death ages. The convention for non-death ages is to take the sum up to the last
death age that is less than or equal to the age under consideration.
With regard to 2ˆq 3 , arguing as in Example 3.17 produces an estimated (conditional) variance
of
dVar( 2ˆq 3 )= (4/27)(23/27) = 92
27 27 3 .
For Greenwood’s formula, we first must note that we are estimating
2q 3 =
S(3) − S(5)
S(3)
=1− S(5)
S(3) .
As with the empirical estimate, all calculations must be done given the 27 people alive at duration
3. Furthermore, the variance of 2ˆq 3 is the same as the variance of Ŝ(5) using only information from
duration 3 and beyond. Starting from duration 3 there are three death times, 3.1, 4.0, and 4.8 with
r 1 =27, r 2 =26, r 3 =25, s 1 =1, s 2 =1,ands 3 =2. Greenwood’s approximation is
µ 23 2 µ 1
27 27(26) + 1
26(25) + 2
= 92
25(23) 27 3 . ¤
Exercise 57 Repeat the previous example, using time to surrender as the variable.
Example 3.26 Repeat the previous example, this time using all 40 observations in Data Set D2
and the incomplete information due to censoring and truncation.
For this example, the direct empirical approach is not available. That is because it is unclear
what the sample size is (it varies over time as subjects enter and leave due to truncation and
censoring). From Example 2.16, the relevant values within the first 3 years are r 1 =30, r 2 =26,
s 1 =1,ands 2 =2.FromExample2.17,S 40 (3) = 0.8923. Then, Greenwood’s estimate is
(0.8923) 2 µ 1
30(29) + 2
26(24)
=0.0034671.
An approximate 95% confidence interval can be constructed using the normal approximation. It is
0.8923 ± 1.96 √ 0.0034671 = 0.8923 ± 0.1154
which corresponds to the interval (0.7769, 1.0077). For small sample sizes, it is possible to create
confidence intervals that admit values less than 0 or greater than 1.
With regard to 2ˆq 3 , the relevant quantities are (starting at duration 3, but using the subscripts
from the earlier examples for these data) r 3 =26, r 4 =26, r 5 =23, r 6 =21, s 3 =1, s 4 =2, s 5 =1,
and s 6 =1. This gives an estimated variance of
µ 0.7215 2 µ 1
0.8923 26(25) + 2
26(24) + 1
23(22) + 1
=0.0059502.
21(20)
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