Bayesian Data Combination Approach for Repeated Durations under Unobserved Missing Indicators: Application to Interpurchase-Timing in Marketing

In this study, we focus on intermittent missingness in repeated duration analysis, which is common in applied studies but has not rigorously been considered in statistics. Under intermittent missingness, whether any missing events exist between two observed events is unknown. In other words, the missing indicators are never observed. Thus, if there exist any missing events between two observed events, we observe only the cumulated duration between two or more events. We propose a quasi-Bayes estimation method that utilizes population-level information to identify unobserved intermittent missingness. The proposed model consists of the following: (1) latent variable model, (2) latent missing indicator model which separates true and composite duration, (3) mixtures of duration models and (4) moment restriction from population-level information to deal with nonignorable intermittent missingness. We use a new estimation procedure that combines objective functions of likelihood and GMM simultaneously with latent variables, which we call Bayesian data combination. We apply the proposed model to analyze interpurchase-duration in database marketing using purchase-history data in Japan, which capture purchase incidences and purchase stores.