Bayesian estimation of a lifetime distribution under double truncation caused by time-restricted data collection

Abstract

We study a type of double truncation where units are observed if and only if their death event occurs within a specific timespan. The resulting missing data mechanism is nonignorable and thus has to be reconsidered. Based on the density function of observed lifetimes and the random sample size, we derive a likelihood model that enables simultaneous estimation of the lifetime distribution and the parameters governing the birth process. In particular, knowledge of the population size is not required. We show that the model is identifiable under certain conditions by using results on exponential families. Bayesian estimators and corresponding standard errors for all involved parameters become available by using MCMC simulation. We describe how the simulation can be performed efficiently while maintaining sufficiently good mixing behaviour of the resulting chains. Both finite-sample and asymptotic properties of the investigated estimators are examined through a simulation study. The proposed method is applied to estimate the lifetime distribution of German companies.