Introduction and Background
The negative binomial distribution (NBD) has been widely used in marketing to model purchase frequency counts, particularly in the packaged goods context. It is known as one of the true marketing laws (Ehrenberg, 1996; Sharp, 2010), and was first applied to marketing science by Ehrenberg (1959) to model brand purchasing behavior. Ehrenberg made two assumptions: (a) Purchases of a given consumer in successive time periods follow a Poisson distribution. This implies that the variance of purchases within individual consumers is "as if" random over time (i.e., Poisson process). (b) The mean rates of purchasing of different consumers in the long run differ and their distribution is a Gamma distribution. Following these assumptions, the frequency of consumers making 0, 1, 2, 3, . . ., x purchases in a given time period can be modeled by the NBD. Ehrenberg (1959) shows the earliest published example of the NBD model fit to purchasing data in consumer packaged goods, where the theoretical values closely match the actual data. This means that brand buying behavior follows a predictable pattern, and this pattern can provide a useful baseline for managers to evaluate their marketing activities as well as how to grow their brands (Sharp, 2010).