Large sample theory of empirical distributions in a window censoring model for renewal processes.

Abstract

In reliability or medical studies, we may only observe each ongoing renewal process for a certain period of time, and therefore, the observations are "window censored". In general, the nonparametric maximum likelihood estimator (hereafter the NPMLE) does not exist for this problem. Vardi devised the RT algorithm for finding the NPMLE restricted to a compact support (M-restricted MLE). Here a slightly different approach is proposed and the algorithm is modified to seek the NPMLE in a larger space. It is shown that the modified algorithm converges monotonically in likelihood and all its limit points are fixed points. It is also shown that for equal observation periods, the NPMLE is unique and is given by the algorithm. There are four types of observations involved in this problem: uncensored, right censored, left censored and doubly censored. The four types of observations are dependent and follow different distributions. To investigate the asymptotic properties of the estimators, first we studied the properties of the empirical processes induced by them. Some laws of large numbers are proved and some distributional properties are derived. Uniform consistency of the estimators for fixed observation periods is proved through the uniqueness of the solution for the score equation, by showing certain contraction properties of the equation. A simulation study shows that some problems with bias may occur. Specifically, the tail probability is typically inflated by the NPMLE. To correct this problem, we propose "Step estimators", which are based on the RT algorithm, but with a prescribed initial estimator to start. The uniform consistency and distributional theory are fully investigated for these estimators. Unlike the typical censored data case, the weak convergences of these estimators do not hold in uniform metric. A weight function has to be introduced.