Nonparametric bivariate density estimation for censored lifetimes

Abstract

It is well known that estimation of a bivariate cumulative distribution function of a pair of right censored lifetimes presents challenges unparalleled to the univariate case where a product-limit Kaplan–Meyer’s methodology typically yields optimal estimation, and the literature on optimal estimation of the joint probability density is next to none. The paper, for the first time in the survival analysis literature, develops the theory and methodology of sharp minimax and adaptive nonparametric estimation of the joint density under the mean integrated squared error (MISE) criterion. The theory shows how an underlying joint density, together with the bivariate distribution of censoring variables, affect the estimation, and what and how may or may not be estimated in the presence of censoring. Practical example illustrates the problem.