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Adaptive thresholding for wavelet-based nonparametric heteroskedastic variance estimation on the sphere

Claudio Durastanti, Radomyra Shevchenko

TL;DR

A needlet-based estimator is proposed, combining multiresolution analysis with hard thresholding, which exploits the spatial and spectral localization of needlets to adapt to unknown smoothness and is shown to attain minimax-optimal convergence rates over Besov spaces.

Abstract

This paper investigates the nonparametric estimation of a heteroskedastic variance function on the sphere in a regression framework, assuming the variance belongs to a Besov regularity class. A needlet-based estimator is proposed, combining multiresolution analysis with hard thresholding. The method exploits the spatial and spectral localization of needlets to adapt to unknown smoothness and is shown to attain minimax-optimal convergence rates over Besov spaces.

Adaptive thresholding for wavelet-based nonparametric heteroskedastic variance estimation on the sphere

TL;DR

A needlet-based estimator is proposed, combining multiresolution analysis with hard thresholding, which exploits the spatial and spectral localization of needlets to adapt to unknown smoothness and is shown to attain minimax-optimal convergence rates over Besov spaces.

Abstract

This paper investigates the nonparametric estimation of a heteroskedastic variance function on the sphere in a regression framework, assuming the variance belongs to a Besov regularity class. A needlet-based estimator is proposed, combining multiresolution analysis with hard thresholding. The method exploits the spatial and spectral localization of needlets to adapt to unknown smoothness and is shown to attain minimax-optimal convergence rates over Besov spaces.
Paper Structure (58 sections, 6 theorems, 299 equations, 1 table)

This paper contains 58 sections, 6 theorems, 299 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background and preliminary results
  3. Notation.
  4. Needlet frames on the sphere and their properties
  5. Besov spaces on the sphere
  6. Besov balls.
  7. Norm embeddings and coefficient inequalities.
  8. Stability under multiplication.
  9. Jackson-type inequalities.
  10. Mixed-norm Jackson inequalities.
  11. Heteroskedastic spherical regression and estimation
  12. The model
  13. Needlet representation.
  14. Heteroskedasticity.
  15. Needlet-based estimation procedure
  16. ...and 43 more sections

Key Result

Proposition 3.1

Let $\widehat{h}_{j,k}$ be defined by eq:vest. Then, under model eq:model, assuming $\mathbb{E}[\varepsilon_i^2]=1$, and denoting $\mathbb{E}[\varepsilon_i^m]=\gamma_m$, $m=3,4$, it holds that and with

Theorems & Definitions (22)

  • Remark 1: Choice of the threshold rule
  • Remark 2: Thresholding, regimes and optimal scale
  • Proposition 3.1: Expectation and variance of $\widehat{h}_{j,k}$
  • Proposition 3.2: Deviation and moment control for the estimator $\widehat{h}_{j,k}$
  • Remark 3: Data-driven construction of $\widehat{(g^2)}_{j,k}$
  • Remark 4: Cut-off frequency for $\widehat{(g^2)}$
  • Lemma 3.3: Asymptotic unbiasedness of $\widehat{(g^2)}$
  • Lemma 3.4
  • Remark 5: Alternative estimation approach: cross-fitted product estimator
  • Proposition 4.1: Upper bound for $\widehat{h} - h$
  • ...and 12 more