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

Mixed Poisson Models for Claim Numbers 45
Poisson the maximum likelihood estimate of the Poisson mean is ̂ = 01462. The 95 %
confidence interval for is (0.1395;0.1532). The log-likelihood of the Poisson model is
−5579.339.
Negative Binomial the maximum likelihood estimate of the mean is ̂ = 01474 and
the dispersion parameter â = 0889. The variance of the random effect is estimated as
̂V = 1/â = 11253. The respective 95 % confidence intervals are (0.1402;0.1551) for
and (0.8144;1.4361) for V. The log-likelihood of the Negative Binomial model is
-5534.36, which is better than the Poisson log-likelihood.
Poisson-Inverse Gaussian the maximum likelihood estimation of the mean is ̂ = 01475,
and the variance of the random effect is estimated to ̂V = ̂ = 11770. The respective
95 % confidence intervals are (0.1402;0.1552) for and (0.8258;1.5282) for V. The
log-likelihood of the Poisson-Inverse Gaussian model is −553428, which is better than the
Poisson log-likelihood and almost equivalent to the Negative Binomial log-likelihood.
Poisson-LogNormal the maximum likelihood estimation of the mean is ̂ = 01476, and
̂ 2 = 07964. The variance of the random effect is estimated to ̂V = 12175. The respective
95 % confidence intervals are (0.1403;0.1553) for and (0.6170;0.9758) for 2 . The loglikelihood
of the Poisson-LogNormal model is −553444, which is better than the Poisson
log-likelihood and almost equivalent to the Negative Binomial and Poisson-Inverse Gaussian
log-likelihoods.
The results have been obtained with the help of the SAS R procedure GENMOD for the
Poisson and Negative Binomial distributions (details will be given in the next chapter) and
by a direct maximization of the log-likelihood using the Newton–Raphson procedure (coded
in the SAS R environment IML) in the Poisson-Inverse Gaussian and Poisson-LogNormal
cases.
It is interesting to note that the values of ̂ are different in the Poisson and mixed Poisson
models. If all the risk exposures were equal then these values would have been the same in
all cases.
Let us now compare the Poisson fit to Portfolio A with each of the mixed Poisson fits.
To this end, we use a likelihood ratio test, with an adjusted Chi-square approximation (since
the Poisson case is at the border of the mixed Poisson family). Comparing the Poisson fit to
any of the three mixed Poisson models leads to a clear rejection of the former one:
Poisson against Negative Binomial
less than 10 −10 .
likelihood ratio test statistic of 89.95, with a p-value
Poisson against Poisson-Inverse Gaussian
p-value less than 10 −10 .
likelihood ratio test statistic of 90.12, with a
Poisson against Poisson-LogNormal likelihood ratio test statistic of 89.80, with a p-value
less than 10 −10 .
The rejection of the Poisson assumption in favour of a mixed Poisson model is interpreted
as a sign that the portfolio is composed of different types of drivers (i.e. the portfolio is
heterogeneous).
Now, comparing the three mixed Poisson models with the Vuong test gives: