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

Kai Schärer, Wolfgang Trutschnig

TL;DR

The paper asks whether conditional distributions can be estimated via Sklar's theorem by separately estimating the conditional copula and the univariate margins. It proposes empirical checkerboard and Bernstein copula approximations, proves uniform conditional convergence under mild regularity (continuous Markov kernel for the copula), and shows that plug-in estimators yield strongly consistent conditional distributions. By aggregating these estimators, the authors derive consistent nonparametric mean, quantile, and expectile regression estimators, with additional results for conditional variance and other functionals. Simulations illustrate when these methods excel or fail (e.g., with copulas lacking a continuous Markov kernel) and compare performance with kernel-based methods, while a real-data example highlights robustness in tails and computational advantages. Overall, the work provides a coherent, copula-based nonparametric regression framework with strong consistency guarantees and practical computational benefits.

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.

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

TL;DR

The paper asks whether conditional distributions can be estimated via Sklar's theorem by separately estimating the conditional copula and the univariate margins. It proposes empirical checkerboard and Bernstein copula approximations, proves uniform conditional convergence under mild regularity (continuous Markov kernel for the copula), and shows that plug-in estimators yield strongly consistent conditional distributions. By aggregating these estimators, the authors derive consistent nonparametric mean, quantile, and expectile regression estimators, with additional results for conditional variance and other functionals. Simulations illustrate when these methods excel or fail (e.g., with copulas lacking a continuous Markov kernel) and compare performance with kernel-based methods, while a real-data example highlights robustness in tails and computational advantages. Overall, the work provides a coherent, copula-based nonparametric regression framework with strong consistency guarantees and practical computational benefits.

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.
Paper Structure (14 sections, 13 theorems, 86 equations, 12 figures)

This paper contains 14 sections, 13 theorems, 86 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Checkerboard Approximations
  4. Bernstein Approximations
  5. Uniform Conditional Convergence of the Empirical Checkerboard Approximations
  6. Uniform Conditional Convergence of the empirical Bernstein approximations
  7. Consequences to Nonparametric Regression
  8. Mean regression
  9. Quantile regression
  10. Expectile Regression
  11. Simulation Study
  12. Copula setting
  13. General regression setting
  14. Real Data Example: Insurance Loss vs. Claim-Handling Costs

Key Result

Lemma 2.2

Suppose that $u \in \mathbb{I}$ and $N \in \mathbb{N}$, consider $t\geq 0$ and set where $\lfloor\cdot\rfloor$ and $\lceil\cdot\rceil$ denote the floor and ceiling functions, respectively. Then the inequality holds.

Figures (12)

  • Figure 1: Convergence of the checkerboard approximation for the AMH copula (a) and the Clayton copula (b).
  • Figure 2: Sample of size $n = 10.000$ from the distribution considered in Example \ref{['ex:gamma_standard']}. CBE for the regression function (blue), NWE (orange) and true regression function (red).
  • Figure 3: (Approximations of the) Mean absolute errors of the CBE (blue) and the NWE (orange) for the distribution considered in Example \ref{['ex:gamma_standard']}; the mean/max absolute error was calculated by evaluating at $m = 2.000$ randomly generated points.
  • Figure 4: Sample of size $n = 10.000$ from the distribution considered in Example \ref{['ex:gamma_sin']}. CBE for the regression function (blue), NWE (orange) and true regression function (red).
  • Figure 5: (Approximations of the) Mean/max absolute errors of the CBE (blue) and the NWE (orange) for the distribution considered in Example \ref{['ex:gamma_sin']}; the mean/max absolute error was calculated by evaluating at $m = 2.000$ randomly generated points. The second panel clearly points at faster convergence of the CBE for the mean absolute error.
  • ...and 7 more figures

Theorems & Definitions (35)

  • Definition 2.1
  • Lemma 2.2
  • Proof 1
  • Theorem 3.1
  • Lemma 3.2
  • Proof 2
  • Lemma 3.3
  • Proof 3
  • Proof 4: Proof of Theorem \ref{['thm:uniform_convergence']}
  • Remark 3.4
  • ...and 25 more