Estimating Heterogeneity in Consumers' Purchase Rates

Consumers are different in their purchase rates and it is important to determine this heterogeneity. If a consumer's purchases follow a Poisson process (hence exponential interpurchase time), and purchase rates are distributed gamma across consumers, then a simple measure of heterogeneity is the shape parameter r of the gamma distribution. Although we can use either the number of purchases or the interpurchase time data to estimate this heterogeneity, we suggest that number of purchases data are better and easier to use. It is also suggested that while method of moments (MOM) gives good parameter estimates for models using number of purchases data (e.g. NBD), it may be very misleading for models using interpurchase time data (e.g. Pareto). We also recommend caution when using maximum likelihood estimation procedure for the interpurchase time data if multiple observations are available for each consumer. This is to ensure that the model captures heterogeneity across consumers and not across observations.