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.).

Keywords:

• Bivariate right censoring
• Bivariate survival analysis
• Consistency
• Generalized MLE
• Nonparametric MLE
• Simulation

Mathematics Subject Classification:

• Primary 62G05
• Secondary 62G20

Acknowledgments

Professor Yu is partially supported by NSF Grant DMS-0803456. Professor Lee is partially supported by Grant NSC 97–2118-M-366–001 from the National Science Council of Taiwan. The author thanks the Editor and a referee for helpful suggestions.