Technical Note—Why Does the NBD Model Work? Robustness in Representing Product Purchases, Brand Purchases and Imperfectly Recorded Purchases

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Abstract
One of the most managerially useful constructs that emerge from the stochastic modelling of brand choice is that of conditional expectations. In this paper the conditional expectations are derived for a generalization of the NBD model, called the beta binomial/negative binomial distribution (BB/NBD) model, first described by Jeuland, Bass and Wright. The model, developed to jointly represent the product class purchase and brand selection processes, is also particularly appropriate for analyzing panel data when not every purchase occasion is recorded. The conditional expectations for the NBD model increase linearly with the number of previous purchases made by an individual. Therefore the degree of nonlinearity of the conditional expectations in our more complicated model allows us to assess the robustness of NBD results when in fact the basic assumptions are not true. This paper gives the simple conditions when the NBD will exactly or approximately represent the BB/NBD; and describes the qualitative deviations between these two models when they are not identical.product purchases, brand choice, conditional expectations, NBD model

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