Phase-type distributions for product return data with two-layer censoring

Product return data, such as warranty claims, are usually subject to two layers of right censoring. The first layer, called warranty censoring, applies to the product lifetime due to a fixed warranty limit. The second layer, called end-of-study censoring, applies to the sum of the sales lag and the lifetime due to the end-of-study date for the data collection. The two-layer censoring in the product return data renders traditional nonparametric methods for right-censored data inapplicable. This study develops a generic method for the two-layer censored data using acyclic phase-type distributions (APHDs) in the canonical form. The APHD estimators can be regarded as nonparametric sieve estimators since the family of APHDs is dense in the field of all positive-valued distributions. Based on the property that the class of APHDs is closed under convolution, a dedicated expectation-maximisation algorithm is proposed to compute the maximum likelihood estimators. Comprehensive simulations are conducted to evaluate the performance of the proposed method and compare it with the inverse probability of censoring weighted approach, which is applicable in the absence of warranty censoring. Two real examples from production-delivery supply chains are analysed by the proposed method to provide guidance on warranty reserve management for the manufacturers.