Covariance function of a bivariate distribution function estimator for left truncated and right censored data

In left truncation and right censoring models one observes i.i.d. samples from the triplet (T, Z, δ) only if T ≤ Z, where Z = min(Y, C) and δ is one if Z = Y and zero otherwise. Here, Y is the variable of interest, T is the truncating variable and C is the censoring variable. Recently, Gürler and Gijbels (1996) proposed a nonparametric estimator for the bivariate distribution function when one of the components is subject to left truncation and right censoring. An asymptotic representation of this estimator as a mean of i.i.d. random variables with a negligible remainder term has been developed. This result establishes the convergence to a two time parameter Gaussian process. The covariance structure of the limiting process is quite complicated however, and is derived in this paper. We also consider the special case of censoring only. In this case the general expression for the variance function reduces to a simpler formula.