Comment on “Modeling Purchasing Behavior with Sudden ‘Death’: A Flexible Customer Lifetime Model”

In their 2012 paper, Bemmaor and Glady introduced the gamma/Gompertz/negative binomial distribution model for customer base analysis. Their model uses exponentially distributed interpurchase times and a Gompertz distributed customer lifetime, where the latter distribution is nonmemoryless. This comment corrects an error in their expression for the conditional expected number of individual future purchases $\mathbb{E}(x_i^{*})$ in a forecasting interval of length $T^{*}$. Contrary to their approach, the correct derivation of the conditional expectation must be based on the conditional survival and density functions of the lifetime distribution. Using the wrong formula leads managers to overestimate the expected future customer purchases. Further, the comment corrects the erroneous expressions for the conditional variance $\mathit{Var}(x_i^{*})$ and the conditional mean residual lifetime $\mathbb{E}(\tau -T_i)$.

This paper was accepted by Raphael Thomadsen, marketing.

Supplemental Material: The e-companion is available at https://doi.org/10.1287/mnsc.2022.4422.