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

28 Actuarial Modelling of Claim Counts
From (1.33), we see that the knowledge of the mixed Poisson distribution oi
is equivalent to the knowledge of F . The mixed Poisson distributions are thus identifiable,
that is, having N 1 ∼ oi 1 and N 2 ∼ oi 2 then N 1 and N 2 are identically
distributed if, and only if, 1 and 2 are identically distributed.
1.4.5 Negative Binomial Distribution
Gamma Distribution
Recall that a random variable X is distributed according to the two-parameter Gamma
distribution, which will henceforth be denoted as X ∼ am , if its probability density
function is given by
fx = x−1 exp−x
x>0 (1.34)
Note that when = 1, the Gamma distribution reduces to the Negative Exponential one
(which is denoted as X ∼ xp) with probability density function
fx = exp−x
x > 0
The distribution function F of X can be expressed in terms of the incomplete Gamma
function. Specifically, if X ∼ am , then Fx = x.
Probability Mass Function
The Negative Binomial distribution is a widely used alternative to the Poisson distribution for
handling count data when the variance is appreciably greater than the mean (this condition,
known as overdispersion, is frequently met in practice, as discussed above).
There are several models that lead to the Negative Binomial distribution. A classic example
arises from the theory of accident proneness which was developed after Greenwood &
Yule (1920). This theory assumes that the number of accidents suffered by an individual
is Poisson distributed, but that the Poisson mean (interpreted as the individual’s accident
proneness) varies between individuals in the population under study. If the Poisson mean
is assumed to be Gamma distributed, then the Negative Binomial is the resultant overall
distribution of accidents per individual.
Specifically, completing (1.26)–(1.27) with ∼ ama a, that is, with probability
density function
f = 1
a aa a−1 exp−a > 0 (1.35)
yields the Negative Binomial probability mass function
( )
a + k − 1 ···a a a ( ) d k
PrN = k =
k! a + d a + d
( )
a + k a a ( ) d k
= k= 0 1 2
ak! a + d a + d