Improving the estimation of Kendall's tau when censoring affects only one of the variables

This paper considers the estimation of Kendall's tau for bivariate data (X,Y)(X,Y) when only Y   is subject to right-censoring. Although ττ is estimable under weak regularity conditions, the estimators proposed by Brown et al. [1974. Nonparametric tests of independence for censored data, with applications to heart transplant studies. Reliability and Biometry, 327–354], Weier and Basu [1980. An investigation of Kendall's ττ modified for censored data with applications. J. Statist. Plann. Inference 4, 381–390] and Oakes [1982. A concordance test for independence in the presence of censoring. Biometrics 38, 451–455], which are standard in this context, fail to be consistent when τ≠0τ≠0 because they only use information from the marginal distributions. An exception is the renormalized estimator of Oakes [2006. On consistency of Kendall's tau under censoring. Technical Report, Department of Biostatistics and Computational Biology, University of Rochester, Rochester, NY], whose consistency has been established for all possible values of ττ, but only in the context of the gamma frailty model. Wang and Wells [2000. Estimation of Kendall's tau under censoring. Statist. Sinica 10, 1199–1215] were the first to propose an estimator which accounts for joint information. Four more are developed here: the first three extend the methods of Brown et al. [1974. Nonparametric tests of independence for censored data, with applications to heart transplant studies. Reliability and Biometry, 327–354], Weier and Basu [1980, An investigation of Kendall's ττ modified for censored data with applications. J. Statist. Plann. Inference 4, 381–390] and Oakes [1982, A concordance test for independence in the presence of censoring. Biometrics 38, 451–455] to account for information provided by X  , while the fourth estimator inverts an estimation of Pr(Yi⩽y|Xi=xi,Yi>ci)Pr(Yi⩽y|Xi=xi,Yi>ci) to get an imputation of the value of YiYi censored at Ci=ciCi=ci. Following Lim [2006. Permutation procedures with censored data. Comput. Statist. Data Anal. 50, 332–345], a nonparametric estimator is also considered which averages the τ^i obtained from a large number of possible configurations of the observed data (X1,Z1),…,(Xn,Zn)(X1,Z1),…,(Xn,Zn), where Zi=min(Yi,Ci)Zi=min(Yi,Ci). Simulations are presented which compare these various estimators of Kendall's tau. An illustration involving the well-known Stanford heart transplant data is also presented.