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

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.

💡 Research Summary

This paper addresses the fundamental question of whether conditional distributions can be estimated by exploiting Sklar’s theorem, i.e., by separately estimating the copula that captures the dependence structure and the marginal distributions. The authors focus on the bivariate case and propose a fully non‑parametric procedure that builds on two types of approximations for the copula: empirical checkerboard approximations and empirical Bernstein approximations. Both approximations are constructed from the empirical copula of the sample, but they differ in smoothness and computational properties. The checkerboard approximation partitions the unit square into an N×N grid and assigns to each cell the empirical mass of observations falling into that cell, yielding an absolutely continuous copula whose density is piecewise constant. The Bernstein approximation uses Bernstein polynomials to obtain a smooth copula that can be expressed as a weighted sum of the original copula evaluated at the grid points.

A key theoretical contribution is the proof that, under the mild condition that the true copula admits a continuous Markov kernel (i.e., its regular conditional distribution given one coordinate is a continuous function of that coordinate), both the checkerboard and Bernstein approximations converge uniformly in the conditional sense to the true copula when the grid resolution N grows appropriately with the sample size n. This notion of uniform conditional convergence is stronger than the weak conditional convergence previously established in the literature: it requires the supremum over the unit square of the absolute difference between the conditional distribution functions of the approximations and the true copula to vanish almost surely.

Having established uniform conditional convergence for the copula, the authors plug in the empirical marginal distribution functions (the usual empirical CDFs of X and Y) to obtain an estimator of the full conditional distribution K̂_n(x,·) of Y given X=x. They prove strong consistency of this estimator in the sense that, for almost every x, the estimated conditional distribution converges uniformly to the true conditional distribution as n→∞.

The paper then shows how to aggregate the estimated conditional distributions to obtain non‑parametric estimators for three important regression functionals: the conditional mean m(x)=E