Wavelet Estimation of an Unknown Function Observed with Correlated Noise

Abstract

In many practical applications of nonparametric regression, it is desirable to allow for the possibility that the noise is correlated. In this article, we focus on wavelet-based nonparametric function estimation and propose two distinct methods for estimating the correlation structure of the noise, one based in the time domain and the other based in the wavelet domain. Once the correlation structure has been estimated, there are various methods that may be used for reconstructing the unknown signal; we focus here on the empirical Bayes block shrinkage method proposed by Wang and Wood (2006). A simulation study is described. Our numerical results indicate that the proposed methods do a good job of reconstructing the signal even when the noise is highly correlated.

Keywords:

• Bayes block shrinkage
• Correlation structure
• Durbin–Levinson algorithm
• Innovations algorithm
• Pseudo likelihood estimation

Mathematics Subject Classification:

• 62G08
• 62M10

Acknowledgment

The authors would like to thank the anonymous reviewer for the positive comments and constructive suggestions.