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Asymptotic confidence bands for the histogram regression estimator

Natalie Neumeyer, Jan Rabe, Mathias Trabs

Abstract

Asymptotic uniform confidence bands are constructed for a multivariate nonparametric regression model with heteroscedastic noise, employing histogram estimators under flexible partition conditions. The construction is especially applicable to unsmooth regression functions of Hölder regularity less than one. While the radius of the confidence bands could be approximated via the Gumbel distribution, our construction does not depend on an extreme value distribution, but instead can be explicitly calculated for the chosen partition.

Asymptotic confidence bands for the histogram regression estimator

Abstract

Asymptotic uniform confidence bands are constructed for a multivariate nonparametric regression model with heteroscedastic noise, employing histogram estimators under flexible partition conditions. The construction is especially applicable to unsmooth regression functions of Hölder regularity less than one. While the radius of the confidence bands could be approximated via the Gumbel distribution, our construction does not depend on an extreme value distribution, but instead can be explicitly calculated for the chosen partition.

Paper Structure

This paper contains 11 sections, 8 theorems, 100 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Histogram estimator and confidence bands
  3. Proof strategy
  4. Numerical illustration
  5. Conclusion
  6. Proofs
  7. Proof of Theorem \ref{['thm:CB histo']} and its corollaries
  8. Proof of Theorem \ref{['thm:sup m tilde approx']}
  9. Moment bounds for binomial distributions
  10. Extensions
  11. Approximation of suprema of empirical processes

Key Result

Theorem 2.2

Let the partition satisfy Assumption ass:part. Let $\alpha\in(0,1],C_H>0$ and assume that Assume that $\sup_{x\in \mathcal{X}}\mathbb{E}[|\varepsilon|^3|X=x]<\infty$ and $\mathbb{E}\left[\vert\varepsilon\vert^{\nu}\right] <\infty$ for some $4\leq \nu\in\mathbb{N}$ such that For independent standard normally distributed random variables $(Z_{j})_{j\in\mathbb{N}}$ and some $\beta\in(0,1)$ let $c_{

Figures (2)

  • Figure 1: Absolute error $\widetilde{c}_\Delta(\beta)- c_\Delta(\beta)$ and relative error $(\widetilde{c}_\Delta(\beta)- c_\Delta(\beta))/c_\Delta(\beta)$ for $\beta=0.05$ in dependence of $\Delta$.
  • Figure 2: Regression function (top left) together with realizations of the Nadaraya-Watson estimator with automatic bandwidth choice (top right) and the two histogram estimators (bottom row) with two different partitions based on $n=800$ observations with noise level $\sigma=1$.

Theorems & Definitions (18)

  • Theorem 2.2
  • Example 2.3
  • Corollary 2.4
  • Corollary 2.5
  • Corollary 2.6
  • Theorem 3.1
  • proof : Proof of Theorem \ref{['thm:CB histo']}
  • proof : Proof of Corollary \ref{['limit-distribution']}
  • proof : Proof of Corollary \ref{['cor-rate']}
  • proof : Proof of Corollary \ref{['cor-estimated-px']}
  • ...and 8 more