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

Mixed Poisson Models for Claim Numbers 31
The random variable N just defined has a compound Poisson distribution. The probability
generating function of a compound distribution is given by
N z = Ez M 1+···+M K
=
=
+∑
k=0
+∑
k=0
PrK = kEz M 1+···+M k
PrK = k ( M z ) k
= K
(
M z ) (1.38)
Note that formula (1.38) is true in general for compound distributions. Replacing K and
M with their expressions gives the probability generating function of N
(
)
N z = exp − 1 − M z
( ) 1 − −/ ln1−
=
1 − z
It can be checked that the probability generating function N corresponds to the probability
generating function (1.37) of a Negative Binomial distribution with d = 1, a =−/ ln1−
and =−/1 − ln1 − .
1.4.6 Poisson-Inverse Gaussian Distribution
There is no reason to restrict ourselves to the Gamma distribution for , except perhaps
mathematical convenience. In fact, any distribution with support in the half positive real line
is a candidate to model the stochastic behaviour of . Here, we discuss the Inverse Gaussian
distribution.
Inverse Gaussian Distribution
The Inverse Gaussian distribution is an ideal candidate for modelling positive, right-skewed
data. Recall that a random variable X is distributed according to the Inverse Gaussian
distribution, which will be henceforth denoted as X ∼ au , if its probability density
function is given by
fx =
(
√ exp − 1 )
2x
3 2x x − 2 x>0 (1.39)
If X ∼ au then the mean is EX = and the variance is VX = . The moment
generating function is given by
∫ +
(
Mt = √ exp − 1
)
0 2x
3 2x x − 2 + tx dx