The negative binomial (NBD) model, as used to characterize repeat-purchasing in numerous markets, assumes stationary individual-level buying rates. In many situations (e.g., for new products), this assumption is rather tenuous -- repeat buying behavior will evolve over time, thereby requiring the use of a nonstationary model to properly capture and forecast the observed sales patterns.
We introduce a model -- the nonstationary exponential-gamma (NSEG) model -- that accomplishes these tasks while retaining the well-known robustness, interpretability, and other desirable properties of the basic NBD framework. Starting with an exponential-gamma process -- the timing equivalent of the NBD -- we introduce a stochastic renewal process, which allows consumers to revise their preferences for the new product after one or more repeat purchases of it. The nature of this renewal process can change over time, allowing for the possibility that preference revisions are common in the early stages of the new product launch, but less likely to occur after a consumer has made several repeat purchases of the product. Over time, the nonstationarity component can disappear completely, allowing the model to become exactly equivalent to the NBD for all subsequent purchases.
Our empirical analysis examines the model's validity on the basis of forecasting accuracy and parameter stability across calibration periods of different lengths. We demonstrate that NSEG performs very well on both dimensions, especially in contrast to the benchmark NBD model.