Adaptive estimation of the conditional cumulative distribution function from current status data

Abstract

Consider a positive random variable of interest Y depending on a covariate X, and a random observation time T independent of Y given X. Assume that the only knowledge available about Y is its current status at time T: δ=I{Y≤T} with I the indicator function. This paper presents a procedure to estimate the conditional cumulative distribution function F of Y given X from an independent identically distributed sample of (X,T,δ).A collection of finite-dimensional linear subsets of L2(R2) called models are built as tensor products of classical approximation spaces of L2(R). Then a collection of estimators of F is constructed by minimization of a regression-type contrast on each model and a data driven procedure allows to choose an estimator among the collection. We show that the selected estimator converges as fast as the best estimator in the collection up to a multiplicative constant and is minimax over anisotropic Besov balls. Finally simulation results illustrate the performance of the estimation and underline parameters that impact the estimation accuracy.