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

34 Actuarial Modelling of Claim Counts
LogNormal Distribution
Recall that a random variable X is Normally distributed with mean and variance 2 ,
denoted as X ∼ or 2 , if its distribution function is
where
( x −
)
Fx =
x = 1 √
2
∫ x
−
exp−y 2 /2 dy (1.44)
Now, a random variable X is LogNormally distributed with parameters and (notation
X ∼ or 2 )iflnX is Normally distributed with mean and variance 2 , that is, if
its probability density function is given by
fx =
(
1
√ exp − 1
)
2x 2 ln x − 2 2
If X ∼ or 2 , then its mean is
)
EX = exp
( + 2
2
and its variance
VX = exp ( 2 + 2)( exp 2 − 1 )
x>0
Poisson-LogNormal Distribution
Taking =− 2 /2 (to ensure that E = 1), the probability density function of ∼
or− 2 /2 2 is
(
1
f =
√ 2 exp − ln + )
2 /2 2
>0 (1.45)
2 2
The probability mass function of the Poisson-LogNormal distribution is given by
PrN = k = 1
√ d k
2 k!
∫
Coming back to (1.28) and (1.29), we easily see that
0
exp−d k−1 exp
(− ln + )
2 /2 2
d
2 2
EN = and VN = + 2( exp 2 − 1 )
It can be shown that / √ V = 2 + exp 2 in (1.31) for the Poisson-LogNormal
distribution. Therefore the skewness of a Poisson-LogNormal distribution exceeds the
skewness of the Poisson-Inverse Gaussian distribution having the same mean and the same
variance.