NBD-Dirichlet Model

The NBD-Dirichlet model is the canonical stochastic model of repeat buying and brand choice in stationary, competitive consumer-goods markets. Introduced by Gerald Goodhardt, Andrew Ehrenberg and Christopher Chatfield in their 1984 Journal of the Royal Statistical Society paper "The Dirichlet," it integrates two processes: how often households buy in a product category, modeled by the negative binomial distribution (NBD), and how those purchases are split across competing brands, modeled by a multinomial-Dirichlet process. From just a few parameters, the model reproduces a remarkably wide set of empirical regularities, including each brand's penetration (how many people buy it), its buyers' purchase frequency, repeat-purchase rates, the share of category requirements each brand earns, and the duplication of purchase between brands. The model encodes Ehrenberg's classic 'laws' of buying behavior, most famously double jeopardy, whereby small brands suffer twice over by having both fewer buyers and slightly less loyal buyers. It assumes a stationary, non-partitioned market with brand choice that looks like sampling 'as if from an urn,' and it serves as a benchmark of what normal, no-loyalty-segmentation buying looks like, against which deviations such as genuine partitioning or excess loyalty can be detected.