Robustness of the negative binomial distribution : a simulation study

Abstract

Negative Binomial Distribution (NBD) has been widely employed in marketing to model purchase incidence. In this paper, we undertake a simulation study to examine the sensitivity of NBD when the number of purchases in a given period is Poisson or Erlang 2 but the underlying heterogeneity is different from gamma. In particular, we allow the heterogeneity to follow three other flexible and well-known distributions: inverse Gaussian, lognormal and Weibull. All these distributions have shapes that are similar to the gamma. Moreover, we evaluate the performance of three alternative models namely, the Condensed Negative Binomial (CNBD), the Generalized Negative Binomial (GNBD) and the Generalized Poisson (GPD).

Our simulation results indicate that NBD is robust to the violation of gamma heterogeneity when the interpurchase time is exponential and the coefficient of variation of the heterogeneity distribution is either low or moderate. CNBD exhibits similar robustness when the interpurchase time is Erlang 2. However, when the coefficient of variation exceeds one, all models exhibit extreme sensitivity to variations in the heterogeneity distribution for both interpurchase times. Interestingly, NBD and CNBD do not appear to be good substitutes for each other. GNBD and GPD models are possible alternatives to the NBD model.

Keywords:

• Marketing
• simulation
• purchase incidence models
• NBD models
• probability distributions