Estimation, Evaluation, and Selection of Actuarial Models

38 CHAPTER 3. SAMPLING PROPERTIES OF ESTIMATORS
function while an estimator is a random variable or a random function. Usually, both the words
and the context will make clear which is being referred to.
3.2 Measures of quality
3.2.1 Introduction
There are a number of ways to measure the quality of an estimator. Three of them are discussed
in this Section. Two examples will be used throughout to illustrate them.
Example 3.1 A population contains the values 1, 3, 5, and 9. We want to estimate the population
mean by taking a sample of size two with replacement.
Example 3.2 A population has the exponential distribution with a mean of θ. We want to estimate
the population mean by taking a sample of size three with replacement.
Both examples are clearly artificial in that we know the answers prior to sampling (4.5 and θ).
However, that knowledge will make apparent the error in the procedure we select. For practical
applications, we will need to be able to estimate the error when we do not know the true value of
the quantity being estimated.
3.2.2 Unbiasedness
When constructing an estimator, it would be good if, on average, the errors we make cancel each
other out. More formally, let θ bethequantitywewanttoestimate.Letˆθ be the random variable
that represents the estimator and let E(ˆθ|θ) be the expected value of the estimator ˆθ when θ is the
true parameter value.
Definition 3.3 An estimator, ˆθ, isunbiased if E(ˆθ|θ) =θ for all θ. The bias is biasˆθ(θ) =
E(ˆθ|θ) − θ.
θ.
The bias depends on the estimator being used and may also depend on the particular value of
Example 3.4 For Example 3.1 determine the bias of the sample mean as an estimator of the
population mean.
The population mean is θ =4.5. The sample mean is the average of the two observations.
It is also the estimator we would use employing the empirical approach. In all cases, we assume
that sampling is random. In other words, every sample of size n has the same chance of being
drawn. Such sampling also implies that any member of the population has the same chance of
being observed as any other member. For this example, there are 16 equally likely ways the sample
could have turned out. They are listed below.
1,1 1,3 1,5 1,9 3,1 3,3 3,5 3,9
5,1 5,3 5,5 5,9 9,1 9,3 9,5 9,9
This leads to the following 16 equally likely values for the sample mean.