Statistics Papers

Document Type

Journal Article

Date of this Version

1-2009

Publication Source

Journal of Multivariate Analysis

Volume

100

Issue

1

Start Page

126

Last Page

136

DOI

10.1016/j.jmva.2008.03.007

Abstract

Variance function estimation in multivariate nonparametric regression is considered and the minimax rate of convergence is established in the iid Gaussian case. Our work uses the approach that generalizes the one used in [A. Munk, Bissantz, T. Wagner, G. Freitag, On difference based variance estimation in nonparametric regression when the covariate is high dimensional, J. R. Stat. Soc. B 67 (Part 1) (2005) 19–41] for the constant variance case. As is the case when the number of dimensions d=1, and very much contrary to standard thinking, it is often not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean. Instead it is desirable to use estimators of the mean with minimal bias. Another important conclusion is that the first order difference based estimator that achieves minimax rate of convergence in the one-dimensional case does not do the same in the high dimensional case. Instead, the optimal order of differences depends on the number of dimensions.

Copyright/Permission Statement

© 2009. This manuscript version is made available under the CC-BY-NC-ND 4.0 license

Keywords

minimax estimation, nonparametric regression, variance estimation

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Date Posted: 27 November 2017

This document has been peer reviewed.