Estimating Conditional Distributions via Sklar's Theorem and Empirical Checkerboard Approximations, with Consequences to Nonparametric Regression

TL;DR

This paper develops a method to estimate conditional distributions nonparametrically using Sklar's theorem and empirical checkerboard approximations, demonstrating strong consistency and applications to regression functions.

Contribution

It introduces a novel approach combining empirical checkerboard approximations with Sklar's theorem for consistent estimation of conditional distributions.

Findings

Uniform weak convergence of copula estimators under mild assumptions

Strong consistency of conditional distribution estimators

Effective nonparametric regression function estimation

Abstract

We tackle the natural question of whether it is possible to estimate conditional distributions via Sklar's theorem by separately estimating the conditional distributions of the underlying copula and the marginals. Working with so-called empirical checkerboard/Bernstein approximations with suitably chosen resolution/degree, we first show that uniform weak convergence to the true underlying copula can be established under very mild regularity assumptions. Building upon these results and plugging in the univariate empirical marginal distribution functions we then provide an affirmative answer to the afore-mentioned question and prove strong consistency of the resulting estimators for the conditional distributions. Moreover, we show that aggregating our estimators allows to construct consistent nonparametric estimators for the mean, the quantile, and the expectile regression function, and beyond. Some simulations illustrating the performance of the estimators and a real data example complement the established theoretical results.