Estimation of conditional cumulative distribution function from current status data

TL;DR

This paper introduces a new method for estimating the conditional distribution function of a positive variable based on current status data, using model selection and minimax optimality, with demonstrated effectiveness through simulations.

Contribution

It develops a model selection approach for estimating the conditional CDF from current status data, achieving minimax rates over Besov spaces.

Findings

Estimator converges at optimal rates.

Method performs well in simulations.

Parameters affecting accuracy are identified.

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: \delta = 1_{Y \leq T}. 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(R^2) called models are built as tensor products of classical approximation spaces of L^2(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.