#### Abstract

Estimation for bivariate right censored data is a problem that has had much study over the past 15 years. In this paper we propose a new class of estimators for the bivariate survivor function based on locally efficient estimation. The locally efficient estimator takes bivariate estimators Fn and Gn of the distributions of the time variables T1,T2 and the censoring variables C1,C2, respectively, and maps them to the resulting estimator. If Fn and Gn are consistent estimators of F and G, respectively, then the resulting estimator 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, respectively, then the estimator will still be consistent and asymptotically normally distributed. We propose a locally efficient estimator which uses a consistent, non-parametric estimator for G and allows the user to supply lower dimensional (semi-parameteric or parametric) model for F. Since the estimator we choose for G will be a consistent estimator of G, the resulting locally efficient 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 yields also the efficiency bound which can be used to calculate relative efficiencies for any bivariate estimator. In this paper we will introduce the locally efficient estimator for bivariate right censored data, present an asymptotic theorem, present the results of simulation studies and perform a brief data analysis illustrating the use of the locally efficient estimator.

#### Disciplines

Statistical Methodology | Statistical Models | Statistical Theory | Survival Analysis

#### Suggested Citation

Quale, Christopher M.; van der Laan, Mark J.; and Robins, James M., "Locally Efficient Estimation with Bivariate Right Censored Data " (October 2002). *U.C. Berkeley Division of Biostatistics Working Paper Series.* Working Paper 119.

http://biostats.bepress.com/ucbbiostat/paper119

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Statistical Methodology Commons, Statistical Models Commons, Statistical Theory Commons, Survival Analysis Commons