Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic equivalence for nonparametric additive regression

Moritz Jirak, Alexander Meister, Angelika Rohde

Abstract

We prove asymptotic equivalence of nonparametric additive regression and an appropriate Gaussian white noise experiment in which a multidimensional shifted Wiener process is observed, whose dimension equals the number of additive components. The shift depends on the additive components of the regression function and solely the one- and two-dimensional marginal distributions of the covariates via an explicitly specified bounded but non-compact linear operator~$Γ$. The number of additive components $d$ is allowed to increase moderately with respect to the sample size. In the special case of pairwise independent components of the covariates, the white noise model decomposes into $d$ independent univariate processes. Moreover, we study approximation in some semiparametric setting where $Γ$ splits into a multiplication operator and an asymptotically negligible Hilbert-Schmidt operator.

Asymptotic equivalence for nonparametric additive regression

Abstract

We prove asymptotic equivalence of nonparametric additive regression and an appropriate Gaussian white noise experiment in which a multidimensional shifted Wiener process is observed, whose dimension equals the number of additive components. The shift depends on the additive components of the regression function and solely the one- and two-dimensional marginal distributions of the covariates via an explicitly specified bounded but non-compact linear operator~. The number of additive components is allowed to increase moderately with respect to the sample size. In the special case of pairwise independent components of the covariates, the white noise model decomposes into independent univariate processes. Moreover, we study approximation in some semiparametric setting where splits into a multiplication operator and an asymptotically negligible Hilbert-Schmidt operator.
Paper Structure (12 sections, 15 theorems, 106 equations)

This paper contains 12 sections, 15 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Regression model and pilot approximation
  3. Dimension reduction
  4. Localization and removal of the covariates
  5. Localization procedure
  6. Pilot estimators
  7. Removal of the covariates
  8. White noise model
  9. Special cases and extensions
  10. Pairwise independent components of the covariates
  11. Further approximation in semiparametric submodels
  12. Proofs of the side results

Key Result

Lemma 2.1

There exists an orthonormal basis $\psi_{n,j}$, $j=1,\ldots,K_n^*$, of $\Phi_n$ with respect to the inner product of $L_2(p_X)$ such that

Theorems & Definitions (28)

  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • Lemma 4.3
  • Proposition 4.4
  • Proposition 5.1
  • proof
  • ...and 18 more