Prediction from Randomly Right Censored Data

Let X be a random vector taking values in Rd, let Y be a bounded random variable, and let C be a right censoring random variable operating on Y. It is assumed that C is independent of (X, Y), the distribution function of C is continuous, and the support of the distribution of Y is a proper subset of the support of the distribution of C. Given a sample {Xi, min{Yi, Ci}, I[Yi⩽Ci]} and a vector of covariates X, we want to construct an estimate of Y such that the mean squared error is minimized. Without censoring, i.e., for C=∞ almost surely, it is well known that the mean squared error of suitably defined kernel, partitioning, nearest neighbor, least squares, and smoothing spline estimates converges for every distribution of (X, Y) to the optimal value almost surely, if the sample size tends to infinity. In this paper, we modify the above estimates and show that in the random right censoring model described above the same is true for the modified estimates.