Locally Efficient Estimation With Bivariate Right-Censored Data

Abstract

Estimation of the survival curve for independently right-censored bivariate failure time data is a problem that has been studied extensively over the past 20 years. In this article we propose a new class of estimators for the bivariate survivor function based on locally efficient (LE) estimation. The LE estimator takes bivariate estimators Fn and Gn of the distributions of the time variables (T1, T2) and the censoring variables (C1, C2), and maps them to the resulting estimator ŜLE. If Fn and Gn are appropriate consistent estimators of F and G, then ŜLE will be nonparametrically efficient (thus the term “locally efficient”). However, if either Fn or Gn (but not both) is not a consistent estimator of F or G, then ŜLE will still be consistent and asymptotically normally distributed. We propose a locally efficient estimator that uses a consistent, nonparametric estimator for G and allows the user to supply lower-dimensional (semiparametric or parametric) working model for F. Because the estimator that we choose for G is consistent, the resulting LE estimator will always be consistent and asymptotically normal, and our simulation studies have indicated that using a lower-dimensional model for F gives excellent small-sample performance. In addition, our algorithm for calculation of the efficient influence curve at true distributions for F and G computes the efficiency bound for the model that can be used to calculate relative efficiencies for any bivariate estimator. In this article we introduce the LE estimator for bivariate right-censored data, present an asymptotic result, present the results of simulation studies, and perform a brief data analysis illustrating the use of the LE estimator.