Actuarial Modelling of Claim Counts Risk Classification, Credibility ...

24 Actuarial Modelling of Claim Counts
Continuous Mixtures
Multiplying the number of categories in (1.25) often leads to a dramatic increase in the
number of parameters (the q j s and the j s). For large , it is therefore preferable to switch
to a continuous mixture, where the sum in (1.25) is replaced with an integral with respect to
some simple parametric continuous probability density function.
Specifically, if we allow to be continuous with probability density function g·, the
finite mixture model suggested above is replaced by the probability mass function
∫
pk = pkgd
which is often referred to as a mixture distribution. When g· is modelled without parametric
assumptions, the probability mass function p· is a semiparametric mixture model. Often in
actuarial science, g· is taken from some parametric family, so that the resulting probability
mass function is also parametric.
Mixed Poisson Model for the Number of Claims
The Poisson distribution often poorly fits observations made in a portfolio of policyholders.
This is in fact due to the heterogeneity that is present in the portfolio: driving abilities vary
from individual to individual. Therefore it is natural to multiply the mean frequency of
the Poisson distribution by a positive random effect . The frequency will vary within the
portfolio according to the nonobservable random variable . Obviously we will choose
such that E = 1 because we want to obtain, on average, the frequency of the portfolio.
Conditional on , we then have
PrN = k = = pk = exp− k k= 0 1 (1.26)
k!
where p· is the Poisson probability mass function, with mean . The interpretation we
give to this model is that not all policyholders in the portfolio have an identical frequency
. Some of them have a higher frequency ( with ≥ 1), others have a lower frequency
( with ≤ 1). Thus we use a random effect to model this empirical observation.
The annual number of accidents caused by a randomly selected policyholder of the portfolio
is then distributed according to a mixed Poisson law. In this case, the probability that a
randomly selected policyholder reports k claims to the company is obtained by averaging the
conditional probabilities (1.26) with respect to . In general, is not discrete nor continuous
but of mixed type. The probability mass function associated with mixed Poisson models is
defined as
PrN = k = E [ pk ] =
∫
0
exp− k dF
k! (1.27)
where F denotes the distribution function of , assumed to fulfill F 0 = 0. The mixing
distribution described by F represents the heterogeneity of the portfolio of interest; dF
is often called the structure function. It is worth mentioning that the mixed Poisson model
(1.27) is an accident-proneness model: it assumes that a policyholder’s mean claim frequency
does not change over time but allows some insured persons to have higher mean claim
frequencies than others. We will say that N is mixed Poisson distributed with parameter
and risk level , denoted as N ∼ oi when it has probability mass function (1.27).