Likelihood -based estimation of stationary semi -Markov processes under window censoring.

Abstract

This thesis considers processes that jump among a finite set of states, with a random amount of time spent in between. Examples of these occur in Engineering, Economics and the Social Sciences. In Semi-Markov processes these transitions follow a Markov Chain and the sojourn times are governed by distributions <italic>Fij</italic>(<italic>t</italic>) that depend only on the connecting states. Suppose that the process started far in the past and that we are able to observe <italic>n</italic> copies of it for a finite amount of time. We view these data as <italic>n</italic> i.i.d. double censored sample paths from a stationary semi-Markov process. In this dissertation we study the estimation of such process from window censored data, under either Maximum Likelihood or Penalized Maximum Likelihood. Our work is first done parametrically by assuming either exponential or Weibull distributions for the sojourn times. Next, we propose a non-parametric treatment by modeling the log-hazard function through linear splines. This allows for processes with non-monotonic failure risks. We prove consistency of these estimators.