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

Risk Classification 73
with the Poisson likelihood is based on a false distributional assumption. The Poisson
maximum likelihood estimator ̂ remains nevertheless consistent for the true parameter 0 ,
i.e. ̂ → proba 0 as the sample size n →+. This explains why the Poisson regression
model is so useful: it continues to give reliable estimations for the annual expected claim
frequency even if the true model is not Poisson, provided the sample size is large enough.
However, the variances of the ̂ j s are mis-estimated. Inference must then be based on the
robust information matrix estimate of the variance-covariance matrix of ̂ that is based on the
empirical estimate of the observed information. Specifically, because of the misspecification,
the asymptotic variance-covariance matrix of ̂ is now given by
where
i=1
In practice, it will be estimated by
̂
= −1
n∑
n∑
= ˜x i˜x T i d i exp T 0˜x i and = ˜x i˜x T i VN ix i
i=1
̂̂
= ̂ −1̂̂ (2.9)
where
n∑
̂ = ˜x i˜x T̂ n∑
i i and ̂ = ˜x i˜x T i ̂ i − k i 2
i=1
with ̂ i = d i exp̂ T˜x i . Let us point out that ̂ − remains approximately Normally
distributed with mean 0 and covariance matrix ̂,
provided the sample size is large enough.
i=1
2.3.15 Numerical Illustration
Within SAS R , the GENMOD procedure can be used to fit Poisson regression models. This
procedure supports the Normal, Binomial, Poisson, Gamma, Inverse Gaussian, Negative
Binomial and Multinomial distributions, in the framework of generalized linear models. A
typical use of the GENMOD procedure is to perform Poisson regression with a log link
function. This type of model is usually called a loglinear model.
The logarithm of the exposure-to-risk is used as an offset, that is, a regression variable
with a constant coefficient of 1 for each observation. A log linear relationship between the
mean and the explanatory factors is specified by the log link function. The log link function
ensures that the mean number of insurance claims predicted from the fitted model is positive.
The results obtained from the Poisson regression for Portfolio A presented in Section 2.2
are shown in Table 2.1. Table 2.1 is similar to the ‘Analysis of Parameter Estimates’ table
produced by the GENMOD procedure. Such a table summarizes the results of the iterative
parameter estimation process. For each parameter in the model, the GENMOD procedure
displays columns with the parameter name, the degrees of freedom associated with the
parameter, the estimated parameter value, the standard error of the parameter estimate, the