Is Each NPMLE of a Continuous Bivariate Distribution Function with Singly Right-Censored Data Really Inconsistent?

Abstract

We consider non-parametric estimation of a continuous cdf of a random vector (X 1, X 2). With bivariate RC data, it is stated in van der Laan (1996, p. 59810, Ann. Statist.), Quale et al. (2006, JASA) etc. that “it is well known that the NPMLE for continuous data is inconsistent (Tsai et al. (1986)).” The claim is based on a result in Tsai et al. (1986, p.1352, Ann. Statist.) that if X 1 is right censored but not X 2, then common ways for defining one NPMLE lead to inconsistency. If X 1 is right censored and X 2 is type I right-censored (which includes the case in Tsai et al.), we present a consistent NPMLE. The result corrects a common misinterpretation of Tsai's example (Tsai et al., 1986, Ann. Statist.).