Predicting Future Random Events Based on Past Performance

There are many situations where one is interested in predicting the expected number of events in period 2 given that x events occurred in period 1. For example, insurance companies must decide whether or not to cancel the insurance of drivers who had 3 or more accidents during the previous year. In analyzing marketing research data an analyst may wish to predict the number of future purchases to be made by those customers who made x purchases in the previous 3 months. The owner of a baseball team may wish to estimate the number of home runs a batter will hit this year given he hit (say) 40 home runs last year. When the events (e.g., accidents, purchase occasions, home runs) can be assumed to occur randomly over time a very general result due to Robbins (Robbins, H. 1977. Prediction and estimation for the compound Poisson distribution. Proc. National Acad Sci. USA 74 2670--2671.) is available. This approach has certain advantages and disadvantages with respect to the negative binomial model that is commonly used to analyze purchasing and accident data. The Robbins result is particularly appropriate for predicting what the zero class in period 1 will do in period 2. The main purpose of this paper is to present the Robbins result in a form that can be readily understood by applied researchers in a variety of disciplines. Data on consumer purchases, motor vehicle violations and accidents are analyzed and applications to other areas are discussed.