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

Risk Classification 107
PrN i1 = k i1 N iTi = k iTi
( )
∫ + Ti
(
∏
1
= PrN it = k it i = √ exp − 1 )
0
2
3 2 − 12 d
t=1
Again, numerical methods are needed to obtain the solutions and the SAS R /STAT procedure
NLMIXED can be used to maximize the log-likelihood.
The fit of the Poisson-Inverse Gaussian model is described in Table 2.14. We get
̂ = 05575. The variance-covariance matrix of the estimated regression coefficients and
dispersion parameter ̂ is
⎛
⎞
0004216 −0000289 −0003723 −0003709 −0000534 −0000567 0000115
−0000289 0000830 −0000002 −0000004 0000000 0000009 −0000002
−0003723 −0000002 0004118 0003755 −0000097 −0000058 −0000027
̂̂
=
−0003709 −0000004 0003755 0004193 −0000069 −0000052 0000011
⎜ −0000534 0000000 −0000097 −0000069 0001152 0000616 0000030 ⎟
⎝ −0000567 0000009 −0000058 −0000052 0000616 0001169 −0000010 ⎠
0000115 −0000002 −0000027 0000011 0000030 −0000010 0002492
The log-likelihood is equal to −19 6437 and Type 3 analysis gives
Source DF Chi-square Pr>Chi-sq
Gender 1 54 00201
Age ∗ Power 2 1488