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

32 Actuarial Modelling of Claim Counts
( ) ∫ +
(
= exp √
exp − 1 (
x 2 1 − 2t + 2)) dx
0 2x
3 2x
Making the change of variable = x √ 1 − 2t yields
⎛
⎞
( ) ∫ +
1
Mt = exp √
⎝−
0
2 √
2 √ 2 + 2 ⎠ d
1−2t
1−2t 3 exp
( ( √ ) )
= exp 1 − 1 − 2t (1.40)
For the last three decades, the Inverse Gaussian distribution has gained attention in
describing and analyzing right-skewed data. The main appeal of Inverse Gaussian models
lies in the fact that they can accommodate a variety of shapes, from highly skewed to
almost Normal. Moreover, they share many elegant and convenient properties with Gaussian
models. In applied probability, the Inverse Gaussian distribution arises as the distribution of
the first passage time to an absorbing barrier located at a unit distance from the origin in a
Wiener process.
Poisson-Inverse Gaussian Distribution
Let us now complete (1.26)–(1.27) with ∼ au1, that is,
(
1
f = √ exp − 1 )
2
3 2 − 12 >0 (1.41)
The probability mass function is given by
PrN = k =
∫
0
exp−d dk
k!
(
1
√ exp − 1 )
2
3 2 − 12 d (1.42)
The probability mass function can be expressed using modified Bessel functions of the
second kind. Bessel functions have some useful properties that can be used to compute the
Poisson-Inverse Gaussian probabilities and to find the maximum likelihood estimators, for
instance.
Moments and Probability Generating Function
Considering (1.28) and (1.29), we have
EN = and VN = + 2
It can be shown that / √ V = 3 in (1.31) for the Poisson-Inverse Gaussian distribution.
Therefore the skewness of a Poisson-Inverse Gaussian distribution exceeds the skewness of
the Negative Binomial distribution having the same mean and the same variance.
Setting = 1 and = , the probability generating function of N can be obtained from
(1.33) together with (1.40), which gives
( 1
(
N z = exp 1 − √ 1 − 2z − 1) )