A Variance Component Model for Studying Consumer Behaviour

The study of consumer behaviour is a broad field that covers several scientific areas ranging from economics and marketing to psychology and anthropology. Our view in this field is focussed on numbers of product purchases that consumers make. We will try to describe the purchasing process by use of a stochastic model. Classical theory on consumer behaviour assumes a Poisson process for product purchasing at the individual level. Within the class of Poisson based models the Negative Binomial Distribution (NBD) model, introduced by Ehrenberg in 1959 is one of the most popular models (see Ehrenberg, 1988). The NBD model mixes the Poisson process with a Gamma distribution to allow heterogeneity between individuals. The model can be used as a practical tool to evaluate product sales (see Morrison and Schmidtlein, 1988). However, Poisson based models assume that the time between events (purchases) have an exponential distribution. Since the coefficient of variation for exponential distributed interpurchase times is equal to one, the Poisson assumption implies that the process is assumed to have a certain (high) rate of regularity. The assumed rate of regularity may not be reflected by the empirical data. In its turn this may lead to undesired properties of the parameter estimates (see Hoogendoorn and Sikkel, 1999).