A Note on Block-Thresholded Wavelet Estimators with Correlated Noise

Abstract

Hall et al. (1999) proposed block-thresholding methods to estimate mean regression functions with independent random errors. They showed that block-thresholded wavelet estimators attain minimax-optimal convergence rates when the mean functions belong to a large class of functions that involve a wide variety of irregularities, including chirp and Doppler functions, and functions with jump discontinuities. In this article, we show that block-thresholded wavelet estimators still attain minimax convergence rates when the mean functions belong to a wide range of Besov classes (where s > 1/p, p ≥ 1 and q ≥ 1) with long-memory Gaussian errors. Therefore, in the presence of long-memory Gaussian errors, wavelet estimators still provide extensive adaptivity.

Keywords:

• Block thresholding
• Long range dependence data
• Minimax estimation
• Nonlinear wavelet-based estimator
• Rates of convergence

Mathematics Subject Classification:

• Primary 62G07
• Secondary 62C20

Acknowledgments

The authors are grateful to the editor and referees for their careful reading of an earlier version of the manuscript and for their helpful suggestions.

The first author's research was supported in part by the NSF grant DMS-0604499; the second author by NSF grant DMS-0706728.