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

Mixed Poisson Models for Claim Numbers 23
units consists of several sub-populations within each of which a relatively simpler model
applies. The source of heterogeneity could be gender, age, geographical area, etc.
Discrete Mixtures
In order to define a mixture model mathematically, suppose the distribution of N can be
represented by a probability mass function of the form
PrN = k = pk = q 1 p 1 k 1 +···+q p k (1.24)
where = q T T T , q T = q 1 q , T = 1 . The model is usually referred
to as a discrete (or finite) mixture model. Here j is a (vector) parameter characterizing the
probability mass function p j · j and the q j s are mixing weights.
Example 1.1 A particular example of finite mixture is the zero-inflated distribution. It has
been observed empirically that counting distributions often show excess of zeros against the
Poisson distribution. In order to accommodate this feature, a combination of the original
distribution p k k= 0 1(be it Poisson or not) together with the degenerate distribution
with all probability concentrated at the origin, gives a finite mixture with
PrN = 0 = + 1 − p 0
PrN = k = 1 − p k k= 1 2
A mixture of this kind is usually referred to as zero-inflated, zero-modified or as a distribution
with added zeros.
Model (1.24) allows each component probability mass function to belong to a different
parametric family. In most applications, a common parametric family is assumed and thus
the mixture model takes the following form
pk = q 1 pk 1 +···+q pk (1.25)
which we assume to hold in the sequel. The mixing weight q can be regarded as a discrete
probability function over , describing the variation in the choice of across the population
of interest.
This class of mixture models includes mixtures of Poisson distributions. Such a mixture
is adequate to model count data (number of claims reported to an insurance company,
number of accidents caused by an insured driver, etc.) where the components of the
mixture are Poisson distributions with mean j . In that respect, (1.25) means that there
are categories of policyholders, with annual expected claim frequencies 1 2 ,
respectively. The proportion of the portfolio in the different categories is q 1 q 2 q ,
respectively. Considering a given policyholder, the actuary does not know to which category
he belongs, but the probability that he comes from category j is q j . The probability mass
function of the number of claims reported by this insured driver is thus a weighted average
of the probability mass functions associated with the k categories.