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

108 Actuarial Modelling of Claim Counts
PrN i1 = k i1 N iTi = k iTi
( )
∫ + ∏ Ti
= exp− it it k it
1
(
0
k it ! √ 2 exp − ln + 2 /2 2 )
d
2 2
t=1
The integral has no closed-form solution so that it is not possible to derive the log-likelihood
equations. Therefore, numerical procedures are needed to solve the integral and to find
maximum likelihood estimates (the NLMIXED procedure of SAS R /STAT, for instance).
Now,
EN it = it and VN it = it + ( exp 2 − 1 ) it 2
The fit of the Poisson-LogNormal model is described in Table 2.15. We get ̂ 2 = 04581.
The variance-covariance matrix of the estimated regression coefficients and dispersion
parameter ̂ 2 is
̂̂
=
⎛
⎞
0004225 −0000291 −0003729 −0003714 −0000536 −0000569 0000091
−0000291 0000832 −0000001 −0000004 −0000001 0000009 −0000003
−0003729 −0000001 0004125 0003760 −0000097 −0000058 −0000022
−0003714 −0000004 0003760 0004199 −0000069 −0000052 0000006
⎜ −0000536 −0000001 −0000097 −0000069 0001155 0000618 0000022
⎟
⎝ −0000569 0000009 −0000058 −0000052 0000618 0001173 −0000007 ⎠
0000091 −0000003 −0000022 0000006 0000022 −0000007 0001193
The log-likelihood is −19 6423 and Type 3 analysis gives
Source DF Chi-square Pr>Chi-sq
Gender 1 52 00226
Age ∗ Power 2 1488