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

• Activity Number: 29
• Type: Contributed
• Date/Time: Sunday, August 2, 2009 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Qiqing Yu*+ and Chingfu Sen and Jinlong Huang and Chinsan Lee
• Companies: Binghamton University and National SunYat-sen University and National Sun Yat-sen University and Shu-Te University of Technology
• Keywords: nonparametric MLE ; bivariate survival analysis ; uniform strong consistency ; right censoring

Abstract

We investigate the claim in the literature that each nonparametric MLE (NPMLE) of a continuous cdf F of the random vector (X1,X2) under right censoring is inconsistent (see, e.g., van der Laan (1996, Annals) and Quale et al. (2006, JASA)). Tsai et al. (1986, Annals) present an example that if X1 is right censored but X2 is continuous and exact, then the NPMLE is not uniquely defined and the common ways for defining an NPMLE lead to inconsistency. Since then it has been interpreted as a counter example to the consistency of the NPMLE of a continuous F with general bivariate right-censored data. In this paper, assuming X1 is right censored and X2 is continuous and subject to type I right censoring, we present a consistent NPMLE and an inconsistent NPMLE. Our results show that the aforementioned claim is not justified.