Estimation, Evaluation, and Selection of Actuarial Models

Chapter 2
Model estimation
2.1 Introduction
In this Chapter it is assumed that the type of model is known, but not the full description of the
model. In the Part 3 Note, models were divided into two types–data-dependent and parametric.
The definitions are repeated below.
Definition 2.1 A data-dependent distribution is at least as complex as the data or knowledge
that produced it and the number of “parameters” increases as the number of data points or amount
of knowledge increases.
Definition 2.2 A parametric distribution is a set of distribution functions, each member of
which is determined by specifying one or more values called parameters. Thenumberofparameters
is fixed and finite.
For this Note, only two data-dependent distributions will be considered. They depend on the
data in similar ways. The simplest definitions for the two types considered appear below.
Definition 2.3 The empirical distribution is obtained by assigning probability 1/n to each data
point.
Definition 2.4 A kernel smoothed distribution is obtained by replacing each data point with a
continuous random variable and then assigning probability 1/n to each such random variable. The
random variables used must be identical except for a location or scale change that is related to its
associated data point.
Note that the empirical distribution is a special type of kernel smoothed distribution in which
the random variable assigns probability one to the data point. The following Sections will introduce
an alternative to the empirical distribution that is similar in spirit, but produces different numbers.
They will also show how the definition can be modified to account for data that have been altered
through censoring and truncation (to be defined later). With regard to kernel smoothing, there are
several distributions that could be used, a few of which are introduced in this Chapter.
A large number of parametric distributions have been encountered in previous readings. The
issue for this Chapter is to learn a variety of techniques for using data to estimate the values of the
distribution’s parameters.
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