Article

Title: Minimax Estimation of the Conditional Cumulative Distribution Function

Author(s): Elodie Brunel, Fabienne Comte and Claire Lacour
Issue: Volume 72 Series A Part 2 Year 2010
Pages: 293 -- 330
Abstract
Consider an i.i.d. sample $(X_i; Y_i)$, $i= 1, \ldots, n$ of observations and denote by $F(y|x)$ the conditional cumulative distribution function of $Y_i$ given $X_i = x$. We provide a data driven nonparametric strategy to estimate $F$. We prove that, in term of the integrated mean square risk on a compact set, our estimator performs a squared-bias variance compromise. We deduce from this an upper bound for the rate of convergence of the estimator, in a context of anisotropic function classes. A lower bound for this rate is also proved, which implies the optimality of our estimator. Then our procedure can be adapted to positive censored random variables $Y_i$'s, i.e. when only $Z_i=\inf (Y_i, C_i)$ and $\delta_i={\mathbf 1}_{\{Y_i \le C_i\}}$ are observed, for an i.i.d. censoring sequence $(C_i)_{1 \le i \le n}$ independent of $(X_i, Y_i)_{1 \le i \le n}$. Simulation experiments and a real data example illustrate the method.
AMS (2000) subject classification. Primary 62N02, 62G07.
Keywords and phrases: Adaptive estimation, censored data, cumulative conditional distribution function, nonparametric methods.