Finite Mixtures of Generalized Linear Regression Models

Abstract

Finite mixture models have now been used for more than hundred years (Newcomb (1886), Pearson (1894)). They are a very popular statistical modeling technique given that they constitute a flexible and-easily extensible model class for (1) approximating general distribution functions in a semi-parametric way and (2) accounting for unobserved heterogeneity. The number of applications has tremendously increased in the last decades as model estimation in a frequentist as well as a Bayesian framework has become feasible with the nowadays easily available computing power. The simplest finite mixture models are finite mixtures of distributions which are used for model-based clustering. In this case the model is given by a convex combination of a finite number of different distributions where each of the distributions is referred to as component. More complicated mixtures have been developed by inserting different kinds of models for each component. An obvious extension is to estimate a generalized linear model (McCullagh and Nelder (1989)) for each component. Finite mixtures of GLMs allow to relax the assumption that the regression coefficients and dispersion parameters are the same for all observations. In contrast to mixed effects models, where it is assumed that the distribution of the parameters over the observations is known, finite mixture models do not require to specify this distribution a-priori but allow to approximate it in a data-driven way. In a regression setting unobserved heterogeneity for example occurs if important covariates have been omitted in the data collection and hence their influence is not accounted for in the data analysis. In addition in some areas of application the modeling aim is to find groups of observations with similar regression coefficients. In market segmentation (Wedel and Kamakura (2001)) one kind of application among others of finite mixtures of GLMs aims for example at determining groups of consumers with similar price elasticities in order to develop an optimal pricing policy for a market segment.