The Survival Curve with Decreasing Density

Abstract

The nonparametric maximum likelihood estimate (NPMLE) of a distribution function G, based on a random sample from G, under the assumption that G has a monotone decreasing density g, is the least concave majorant of the empirical distribution function (Barlow, Bartholomew, Bremner, and Brunk 1972). This may lead one to believe that the least concave majorant of the Kaplan-Meier (1958) estimate is the NPMLE of G in the “arbitrary right censorship model”—that is, when the data include right censored observations and no assumption on the censoring mechanism is made. This, however, is not correct, as was pointed out by McNichols and Padgett (1982), and the problem of fully characterizing the solution to this problem has been left open, In this article we provide such a characterization in two steps. First, we show that when the largest observation is censored the NPMLE of G is not a proper distribution (the nonparametric likelihood function has a supremum but not a maximum). This suggests that as a numeric too...