Wavelet-based estimation of regression function with strong mixing errors under fixed design

ABSTRACT

We consider wavelet-based non linear estimators, which are constructed by using the thresholding of the empirical wavelet coefficients, for the mean regression functions with strong mixing errors and investigate their asymptotic rates of convergence. We show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes B^s_{p, q}. The theory is illustrated with some numerical examples.

A new ingredient in our development is a Bernstein-type exponential inequality, for a sequence of random variables with certain mixing structure and are not necessarily bounded or sub-Gaussian. This moderate deviation inequality may be of independent interest.

KEYWORDS:

• Minimax estimation
• Mixing errors
• Non linear wavelet-based estimator
• Rates of convergence.

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 62G05
• Secondary: 62G08, 62G20

Acknowledgments

The authors are grateful to two referees for their careful reading of an earlier version of the manuscript and for their extremely helpful suggestions, in particular, for pointing out to us the paper of Chesneau (2013). These have led to significant improvement of the article.

Funding

Research partially supported by NSF grant DMS-1309856.

Notes

1 We thank the anonymous referee for bringing this interesting article to our attention. See Remark 3.2 below for a comparison of our result to those in Chesneau, (2013).