Efficiency of the sieved-NPMLE in CAR-Censored Data Models

Abstract

Suppose that a random variable X of interest is grouped or censored or missing so that one only observes a coarsening of X ie a random set containing X with probability It is assumed that the coarsening mechanism has the coarsening at random property Suppose furthermore that the coarsening either equals X itself or that is a set with positive X-probability We modify the NPMLE of the distribution of X by demanding that its support is the set of observed data points We provide a general theorem giving sucient conditions for eciency of this NPMLE or eciency of the NPMLE after a small data reduction We apply the theorem to a number of examples