Biometrika (1997), 84,4, pp. 863-880 Printed in Great Britain Nonparametric estimators of the bivariate survival function under simplified censoring conditions BY WEIJING WANG Institute of Statistics, Academia Sinica, Taiwan e-mail: wjwang@stat.sinica.edu.tw AND MARTIN T. WELLS Department of Social Statistics & Statistics Center, Cornell University, New York 14853, U.S.A. e-mail: mtwl@cornell.edu SUMMARY New bivariate survival function estimators are proposed in the case where the depen- dence relationship between the censoring variables are modelled. Specific examples include the cases when censoring variables are univariate, mutually independent or specified by a marginal model. Large sample properties of the proposed estimators are discussed. The finite sample performance of the proposed estimators compared with other fully non- parametric estimators is studied via simulations. A real data example is given. Some key words: Archimedean copula; Bivariate failure time data; Independent censoring Marginal modelling; Univariate censoring. 1. INTRODUCTION Unlike the univariate Kaplan & Meier (1958) estimator, which has the usual optimal properties, estimators of the bivariate survival function proposed in literature have some unsatisfactory features and are in general quite complex (Gill, 1992). Roughly speaking, the bivariate censoring complicates the analysis. This paper considers situations when the relationship between the censoring variables can be simplified so that estimation of the joint survival function is more direct. Let (X,, Y i )(i=l,...,n)ben independent and identically distributed pairs of bivariate failure times with a common joint survival function F(x, y) = pr(X ^ x, Y^-y) and let (CK, C 2i ) (i=l,...,n) be n independent and identically distributed pairs of censoring variables with a common joint survival function G(x, y) = pr(C t ^ x, C 2 ^ y)- Let F ; (.) and G,(.) (i = 1, 2) denote the marginal survival functions of X, Y, C x and C 2 , respectively. If we assume right censoring, the observed variables become {X h %) = {{X t A C u ), (Y t A C 2i )}, {5f, 5ft = {I(X i < C u ), I(Y t < C 2i )} (i = 1,..., n), where A denotes the minimum and /(.) denotes the indicator function. Denote by H(x, y) = pi(X ^ x, Y^y) the joint survival function of observables (X h %) (i = 1,..., n). It is usually assumed that (X h Y t ) are independent of (C u , C 2i ) for all i in order to ensure identifiability (Pruitt, 1993) of the survival function. Several nonparametric estimators of F(x, y) have been proposed such as those by Hanley & Parnes (1983), Tsai, Leurgans & at National Chiao Tung University Library on December 7, 2011 http://biomet.oxfordjournals.org/ Downloaded from