Estimation, Evaluation, and Selection of Actuarial Models

Chapter 3
Sampling properties of estimators
3.1 Introduction
Regardless of how a model is estimated, it is extremely unlikely that the estimated model will
exactlymatchthetruedistribution.Ideally,wewouldliketobeabletomeasuretheerrorwewill
be making when using the estimated model. But this is clearly impossible! If we knew the amount
of error we had made, we could adjust our estimate by that amount and then have no error at
all. The best we can do is discover how much error is inherent in repeated use of the procedure, as
opposed to how much error we made with our current estimate. Therefore, this Chapter is about
the quality of the ensemble of answers produced from the procedure, not about the quality of a
particular answer.
When constructing models, there are a number of types of error. Several will not be covered
in this Note. Among them are model error (choosing the wrong model) and sampling frame error
(trying to draw inferences about a population that differs from the one sampled). An example of
model error is selecting a Pareto distribution when the true distribution is Weibull. An example of
sampling frame error is sampling claims from policies sold by independent agents in order to price
policies sold over the internet.
The type of error we can measure is that due to using a sample from the population to make
inferences about the entire population. Errors occur when the items sampled do not represent the
population. As noted earlier, we cannot know if the particular items sampled today do or do not
represent the population. We can, however, estimate the extent to which estimators are affected
by the possibility of a non-representative sample.
The approach taken in this Chapter is to consider all the samples that might be taken from
the population. Each such sample leads to an estimated quantity (for example, a probability, a
parameter value, or a moment). We do not expect the estimated quantities to always match the
true value. For a sensible estimation procedure, we do expect that for some samples, the quantity
will match the true value, for many it will be close, and for only a few will it be quite different.
If we can construct a measure of how well the set of potential estimates matches the true value,
we have a handle on the quality of our estimation procedure. The approach outlined here is often
called the “classical” or “frequentist” approach to estimation.
Finally, we need a word about the difference between “estimate” and “estimator.” The former
refers to the specific value obtained when applying an estimation procedure to a set of numbers.
The latter refers to a rule or formula that produces the estimate. An estimate is a number or
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