Exploring the equivalence of two common mixture models for duration data

The beta-geometric (BG) distribution and the Pareto distribution of the second kind (P(II)) are two basic models for duration-time data that share some underlying characteristics (i.e., continuous mixtures of memoryless distributions), but differ in two important respects: first, the BG is the natural model to use when the event of interest occurs in discrete time, while the P(II) is the right choice for a continuous-time setting. Second, the underlying mixing distributions (the beta and gamma for the BG and P(II), respectively, are very different—and often believed to be non-comparable with each other. Despite these and other key differences, the two models are strikingly similar in terms of their fit and predictive performance as well as their parameter estimates. We explore this equivalence, both empirically and analytically, and discuss the implications from both a substantive and methodological standpoint