Estimation, Evaluation, and Selection of Actuarial Models

44 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS
3.3 Variance and confidence intervals
3.3.1 Introduction
The examples in the previous Section were relatively simple. Problems of actuarial interest are
often more complex. Complexity may be introduced via censoring or truncation or by having a
goal that is not as easy to work with as a population mean. In this Section, accuracy measures for
the estimators introduced in Chapter 2 will be introduced. These accuracy measures will then be
used to provide interval estimates for the quantity of interest.
3.3.2 Methods for empirical distributions
Three cases will be covered. The first case is complete, individual data. That will be followed by
an analysis of the grouped data case. The final case involves the effects of censoring or truncation.
In Chapter 2, most of the time, the goal was estimation of the survival function. That case will
be the only one covered in this Subsection. Estimation of related quantities will be covered in the
following Subsection.
When all of the information is available, analyzing the empirical estimate of the survival function
is straightforward.
Example 3.16 Demonstrate that the empirical estimator of the survival function is unbiased and
consistent.
Recall that the empirical estimate of S(x) is S n (x) =Y/n where Y is the number of observations
in the sample that are greater than x. ThenY must have a binomial distribution with parameters
n and S(x). Then,
E[S n (x)] = E(Y/n)=nS(x)/n = S(x),
demonstrating that the estimator is unbiased. The variance is
Var[S n (x)] = Var(Y/n)=
S(x)[1 − S(x)]
n
which has a limit of zero, thus verifying consistency. ¤
In order to make use of the result, the best we can do for the variance is estimate it. It is
unlikely we know the value of S(x), because that is the quantity we are trying to estimate. The
estimated variance is given by
dVar[S n (x)] = S n(x)[1 − S n (x)]
.
n
The same results hold for empirically estimated probabilities. Let p =Pr(a