A General Approximation to the Distribution of Count Data with Applications to Inventory Modeling

Abstract

We derive a general approximation to the distribution of count data based on the first two moments of the underlying interarrival distribution. The result is a variant of the Birnbaum-Saunders (BISA) distribution. This distribution behaves like the lognormal in several respects; however, we show that the BISA can fit both simulated and empirical data better than the lognormal and that the BISA possesses additive properties that the lognormal does not. This results in computational advantages for operational models that involve summing random variables. Moreover, although the BISA can be fit to count data (as we demonstrate empirically), it can also be fit directly to transaction-level interarrival data. This provides a simple, practical way to sidestep distributional fitting problems that arise from count data that is censored by inventory stockouts. In numerical experiments involving dynamic inventory models, we compare the BISA distribution to other commonly used distributions and show how it leads to better managerial decisions.

Keywords: Birnbaum-Saunders, inverse Gaussian, gamma, confluent hypergeometric functions, news vendor model