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Partial heteroscedastic deconvolution estimation in nonparametric regression

Baba Thiam

TL;DR

This work tackles nonparametric regression when covariates are measured with heteroscedastic error by extending partial deconvolution kernel methods. It proposes a kernel-based estimator $\widehat{r}_n(x,t)$ that incorporates heteroscedastic measurement errors through $L_{U_j}$ and a joint-density estimator $\widehat{f}_n(x,t)$, achieving optimal convergence rates under mild regularity and bandwidth conditions. Theoretical results establish pointwise consistency and minimax-optimal rates over a smoothness class, while simulations demonstrate robustness to error type (supersmooth vs. ordinary smooth) and superiority over naive error-ignoring approaches. The methodology enables reliable nonparametric regression in practical settings with heterogeneous measurement processes, with clear asymptotic guarantees and a framework adaptable to unknown error structures in future work.

Abstract

In this paper, we consider a partial deconvolution kernel estimator for nonparametric regression when some covariates are measured with error while others are observed without error. We focus on a general and realistic setting in which the measurement errors are heteroscedastic. We propose a kernel-based estimator of the regression function in this framework and show that it achieves the optimal convergence rate under suitable regularity conditions. The finite-sample performance of the proposed estimator is illustrated through simulation studies.

Partial heteroscedastic deconvolution estimation in nonparametric regression

TL;DR

This work tackles nonparametric regression when covariates are measured with heteroscedastic error by extending partial deconvolution kernel methods. It proposes a kernel-based estimator that incorporates heteroscedastic measurement errors through and a joint-density estimator , achieving optimal convergence rates under mild regularity and bandwidth conditions. Theoretical results establish pointwise consistency and minimax-optimal rates over a smoothness class, while simulations demonstrate robustness to error type (supersmooth vs. ordinary smooth) and superiority over naive error-ignoring approaches. The methodology enables reliable nonparametric regression in practical settings with heterogeneous measurement processes, with clear asymptotic guarantees and a framework adaptable to unknown error structures in future work.

Abstract

In this paper, we consider a partial deconvolution kernel estimator for nonparametric regression when some covariates are measured with error while others are observed without error. We focus on a general and realistic setting in which the measurement errors are heteroscedastic. We propose a kernel-based estimator of the regression function in this framework and show that it achieves the optimal convergence rate under suitable regularity conditions. The finite-sample performance of the proposed estimator is illustrated through simulation studies.
Paper Structure (4 sections, 4 theorems, 44 equations, 3 figures, 2 tables)

This paper contains 4 sections, 4 theorems, 44 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Assumptions and main results
  3. Numerical studies
  4. Proofs

Key Result

Theorem 1

Assume that Assumptions exist_ass--var_ass holds.

Figures (3)

  • Figure 1: Estimation of cross-sections at $x=1$ (on the left) and at $t=0$ (on the right) from model \ref{['modele1']} under the Normal contaminated error. The dashed line is the estimated curve $\widehat{r}_n$ and the solid line is the true curve. The sample size is $n=500$.
  • Figure 2: Estimation of cross-sections at $x=1$ (on the left) and at $t=0$ (on the right) from model \ref{['modele1']} under the Laplace contaminated error. The dashed line is the estimated curve $\widehat{r}_n$ and the solid line is the true curve. The sample size is $n=500$.
  • Figure 3: Estimation of cross-sections at $x=1$ (on the left) and at $t=1$ (on the right) from model \ref{['modele2']} under the Normal contaminated error. The dashed line is the estimated curve $\widehat{r}_n$, the dotdashed line is $\widetilde{r}_n$ defined in \ref{['rntilde']} and the solid line is the true curve.The sample size is $n=500$.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2