Estimation, Evaluation, and Selection of Actuarial Models

Chapter 5
Models with covariates
5.1 Introduction
It may be that the distribution of the random variable of interest depends on certain characteristics
of the underlying situation. For example, the distribution of time to death may be related to the
individual’s age, gender, smoking status, blood pressure, height and weight. Or, consider the
number of automobile accidents a vehicle has in a year. The distribution of this variable might be
related to the number of miles it is to be driven, where it is driven, and various characteristics of
the primary driver such as age, gender, marital status, and driving history.
Example 5.1 Suppose we believe that the distribution of the number of accidents a driver has in a
year is related to the driver’s age and gender. Provide three approaches to modeling this situation.
Of course there is no limit to the number of models that could be considered. Three that might
be used are given below.
1. Construct a model for each combination of gender and age. Collect data separately for
each combination and construct each model separately. Either parametric or data-dependent
models could be selected.
2. Construct a single, fully parametric model for this situation. As an example, the number of
accidents for a given driver could be assumed to have the Poisson distribution with parameter
λ. Thevalueofλ is then assumed to depend on the age x and the gender (g =1for males,
g =0for females) in some way such as
λ =(α 0 + α 1 x + α 2 x 2 )β g .
3. Begin with a model for the density, distribution, or hazard rate function that is similar to a
data-dependent model. Then use the age and gender to modify this function. For example,
select a survival function S 0 (n) and then the survival function for a particular driver might
be
S(n|x, g) =[S 0 (n)] (α 0+α 1 x+α 2 x 2 )β g .
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