Adaptive pointwise estimation of conditional density function

TL;DR

This paper introduces adaptive kernel and projection-based estimators for the conditional density function, achieving minimax optimality and effectively incorporating the influence of the marginal density at a fixed point.

Contribution

It develops two adaptive estimation procedures for the conditional density, proven to be optimal and capable of quantifying the impact of the marginal density on convergence rates.

Findings

Establishes minimax optimality of the proposed estimators.

Derives oracle inequalities for the adaptive procedures.

Demonstrates good practical performance through simulations.

Abstract

In this paper we consider the problem of estimating $f$, the conditional density of $Y$ given $X$, by using an independent sample distributed as $(X,Y)$ in the multivariate setting. We consider the estimation of $f(x,.)$ where $x$ is a fixed point. We define two different procedures of estimation, the first one using kernel rules, the second one inspired from projection methods. Both adapted estimators are tuned by using the Goldenshluger and Lepski methodology. After deriving lower bounds, we show that these procedures satisfy oracle inequalities and are optimal from the minimax point of view on anisotropic H{\"o}lder balls. Furthermore, our results allow us to measure precisely the influence of $\mathrm{f}\_X(x)$ on rates of convergence, where $\mathrm{f}\_X$ is the density of $X$. Finally, some simulations illustrate the good behavior of our tuned estimates in practice.