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

46 Actuarial Modelling of Claim Counts
Negative Binomial against Poisson-Inverse Gaussian Vuong test statistic equal to -0.1086,
with p-value 91.36 %.
Poisson-LogNormal against Negative Binomial
with p-value 96.54 %.
Vuong test statistic equal to −00435,
Poisson-LogNormal against Poisson-Inverse Gaussian Vuong test statistic equal to
−03254, with p-value 74.48 %.
We cannot discriminate between the three competing models given the data, and they all
fit the model equally well.
Remark 1.3 (Chi-Square Goodness-of-Fit Tests) In many papers appearing in the
actuarial literature devoted to the analysis of claim numbers, as well as in most empirical
studies, Chi-square goodness-of-fit tests are performed to select the optimal model. However,
this approach neglects the exposures-to-risk (acting as if all the policies were in the portfolio
for the whole year). We do not rely on Chi-square goodness-of-fit tests here since they do
not allow for unequal risk exposures. Note that the vast majority of papers appearing in the
actuarial literature disregard risk exposures (and proceed as if all the risk exposures were
equal to 1).
1.7 Further Reading and Bibliographic Notes
1.7.1 Mixed Poisson Distributions
Mixed Poisson distributions are often used to model insurance claim numbers. The statistical
analysis of counting random variables is described in much detail in Johnson ET AL. (1992).
An excellent introduction to statistical inference is provided by Franklin (2005). In the
actuarial literature, Klugman ET AL. (2004) provide a good account of statistical inference
applied to insurance data sets, and in particular the analysis of counting random variables.
Generating functions are described in Kendall & Stuart (1977) and Feller (1971).
The axiomatic approach for which the (mixed) Poisson distribution is the counting
distribution for a (mixed) Poisson process is presented in Grandell (1997). Mixture models
are discussed in Lindsay (1995). See also Titterington ET AL. (1985). Let us mention the
work by Karlis (2005), who applied the EM algorithm for maximum likelihood estimation
in mixed Poisson models.
1.7.2 Survey of Empirical Studies Devoted to Claim Frequencies
Kestemont & Paris (1985), using mixtures of Poisson processes, defined a large class of
probability distributions and developed an efficient method for estimating their parameters.
For the six data sets in Gossiaux & Lemaire (1981), they proposed a law depending on
three parameters and they always obtained extremely good fits. As particular cases of the
laws introduced in Kestemont & Paris (1985), we find the ordinary Poisson distribution,
the Poisson-Inverse Gaussian distribution, and the Negative Binomial distribution.
Tremblay (1992) used the Poisson-Inverse Gaussian distribution. Willmot (1987)
compared the Poisson-Inverse Gaussian distribution to the Negative Binomial one and
concluded that the fits are superior with the Poisson-Inverse Gaussian in all the six cases