Summary: 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 \(T:\delta=\mathbb I_{\{Y\leq T\}}\) with \(\mathbb I\) 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 \((X,T,\delta)\).
A collection of finite-dimensional linear subsets of \(L^2(\mathbb R^2)\) called models are built as tensor products of classical approximation spaces of \(L^2(\mathbb R)\). 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.