Mixed Poisson Processes with Dropout for Consumer Studies

We adapt the classical mixed Poisson process models for investigation of consumer behaviour in a situation where after a random time we can no longer identify a customer despite the customer remaining in the panel and continuing to perform buying actions. We derive explicit expressions for the distribution of the number of purchases by a random customer observed at a random subinterval for a given interval. For the estimation of parameters in the gamma–Poisson scheme, we use the estimator minimizing the Hellinger distance between the sampling and model distributions, and demonstrate that this method is almost as efficient as the maximum likelihood being much simpler. The results can be used for modelling internet user behaviour where cookies and other user identifiers naturally expire after a random time.