Estimation, Evaluation, and Selection of Actuarial Models

52 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS
Exercise 58 Repeat the previous example, using time to surrender as the variable. Interpret 2 q 3
as the probability of surrendering before the five years expire.
The previous example indicated that the usual method of constructing a confidence interval
can lead to an unacceptable result. An alternative approach can be constructed as follows. Let
Y =ln[− ln S n (t)]. Using the delta method (see Theorem 3.33 in the next Subsection), the variance
of Y can be approximated as follows. The function of interest is g(x) = ln(− ln x). Its
derivative is g 0 (x) = 1
be approximated by
− ln x
−1
x
=(x ln x)−1 . According to the delta method, the variance of Y can
{g 0 [E(S n (t))]} 2 Var[S n (t)] = Var[S n(t)]
[S n (t)lnS n (t)] 2
wherewehaveusedthefactthatS n (t) is an unbiased estimator of S(t). Then, an estimated 95%
confidence interval for θ =ln[− ln S(t)] is
q
dVar[S n (t)]
ln[− ln S n (t)] ± 1.96
S n (t)lnS n (t) .
Because S(t) =exp(−e θ ), putting each endpoint through this formula will provide a confidence
interval for S(t). Fortheupperlimitwehave(whereˆv = Var[S d n (t)])
n
exp −e ln[− ln S n(t)]+1.96 √ˆv/[S o n (t)lnS n (t)]
= exp
n[ln o
S n (t)]e 1.96√ˆv/[S n (t)lnS n (t)]
"
= S n (t) U 1.96 √ˆv
#
, U =exp
.
S n (t)lnS n (t)
Similarly, the lower limit is S n (t) 1/U . This interval will always be inside the range zero to one and
is referred to as the log-transformed confidence interval.
Example 3.27 Obtain the log-transformed confidence interval for S(3) as in Example 3.26.
We have
"
1.96 √ #
0.0034671
U =exp
=0.32142.
0.8923 ln(0.8923)
The lower limit of the interval is 0.8923 1/0.32142 =0.70150 and the upper limit is 0.8923 0.32142 =
0.96404. ¤
Exercise 59 Obtain the log-transformed confidence interval for S(3) in Exercise 58.
Similar results are available for the Nelson-Åalen estimator. An intuitive derivation of a variance
estimate proceeds as follows. As in the derivation for the Kaplan-Meier estimator, all results are
obtained assuming the risk set numbers are known, not random. The number of deaths at death
time t i has approximately a Poisson distribution 3 with parameter r i h(t i ) and so its variance is
3 A binomial assumption (as was used for the Kaplan-Meier derivation) could also have been made. Similarly, a
Poisson assumption could have been used for the Kaplan-Meier derivation. The formulas in this Note are the ones
most commonly used.