A Systematic Relationship between Minimum Bias and Generalized Linear Models

The minimum bias method is a natural tool to use in parameterizing classification ratemaking plans. Such plans build rates for a large, heterogeneous group of insureds using arithmetic operations to combine a small set of parameters in many different ways. Since the arithmetic structure of a class plan is usually not wholly appropriate, rates for some individual classification cells may be biased. Classification ratemaking therefore requires measures of bias, and minimum bias is a natural objective to use when determining rates. This paper introduces a family of linear bias measures and shows how classification rates with minimum (zero) linear bias for each class are the same as those obtained by solving a related generalized linear model using maximum likelihood. The examples considered include the standard additive and multiplicative models used by Insurance Services Office (ISO) for private passenger auto ratemaking and general liability ratemaking, see ISO [10] and Graves and Castillo [7], respectively. Knowing how to associate a generalized linear model to a linear bias function is useful for several reasons. It makes the underlying statistical assumptions explicit so the user can judge their appropriateness for a given application. It provides an alternative method to solve for the model parameters, which is computationally more efficient than using the minimum bias iterative method. In fact not all linear bias functions allow an iterative solution; in these cases, solving a generalized linear model using maximum likelihood provides an effective way to determine model parameters. Finally, it opens up the possibility of using statistical techniques for parameter estimates, analysis of residuals and model fit, significance of effects, and comparison of different models.