Nonparametric Regression with Censored Covariates

The paper discusses weak convergence results for an estimate of the conditional survival function F(t | z) = Pr(T> t | Z = z) where T is a multivariate response variable and Z is a vector of covariates. It is assumed that both T and Z are subject to right censoring. The estimate is obtained by kernel smoothing the empirical analogue of a product integral representation of multivariate survival functions. Under regularity conditions we show that a standardized version of the regression estimate converges weakly to a mean zero Gaussian process and give the form of the asymptotic covariance in the case of univariate response variables. As a by product we also discuss asymptotic normality results for density estimates based on the smoothed product integral estimate.