A Model of the Joint Distribution of Purchase Quantity and Timing

Prediction of purchase timing and quantity decisions of a household is an important element for success of any retailer. This is especially so for an online retailer, as the traditional brick-and-mortar retailer would be more concerned with total sales. A number of statistical models have been developed in the marketing literature to aid traditional retailers in predicting sales and analyzing the impact of various marketing activities on sales. However, there are two important differences between traditional retail outlets and the increasingly important online retail/delivery companies, differences that prevent these firms from using models developed for the traditional retailers: 1) the profits of the online retailer/delivery company depend on purchase frequency and on purchase quantity, while the profits of traditional retailers are simply tied to total sales, and 2) customers in the tails of the frequency distribution are more important to the delivery company than to the retail outlet. Both of these differences are due to the fact that the delivery companies incur a delivery cost for each sale, while customers themselves travel to retail outlets when buying from traditional retailers. These differences in costs translate directly into needs that a model must address. For a model intended to be useful to online retailers the dependent variable should be a bivariate distribution of frequency and quantity, and frequency distribution must accurately represent consumers in the tails. In this article we develop such a model and apply it to predicting the consumer?s joint decision of when to shop and how much to spend at the store. Our approach is to model the marginal distribution of purchase timing and the distribution of purchase quantity conditional on purchase timing. We propose a hierarchical Bayes model that disentangles the weekly and daily components of the purchase timing. The daily component has a dependence on the weekly component thereby accounting for strong observed periodicity in the data. For the purchase times, we use the Conway-Maxwell-Poisson distribution, which we find useful to fit data in the tail regions (extremely frequent and infrequent purchasers).