Generalized Linear Models

ABSTRACT Multiple linear regression is the most widely used statistical technique in practical econometrics. In actuarial statistics, situations occur that do not fit comfortably in that setting. Regression assumes normally distributed disturbances with a constant variance around a mean that is linear in the collateral data. In many actuarial applications, a symmetric normally distributed random variable with a variance that is the same whatever the mean does not adequately describe the situation. For counts, a Poisson distribution is generally a good model, if the assumptions of a Poisson process described in Chapter 4 are valid. For these random variables, the mean and variance are the same, but the datasets encountered in practice generally exhibit a variance greater than the mean. A distribution to describe the claim size should have a thick right-hand tail. The distribution of claims expressed as a multiple of their mean would always be much the same, so rather than a variance not depending of the mean, one would expect the coefficient of variation to be constant. Furthermore, the phenomena to be modeled are rarely additive in the collateral data. A multiplicative model is much more plausible. If other policy characteristics remain the same, moving from downtown to the country would result in a reduction in the average total claims by some fixed percentage of it, not by a fixed amount independent of the original risk. The same holds if the car is replaced by a lighter one.