Adaptive wavelet estimation of a function from an m-dependent process with possibly unbounded m

ABSTRACT

The estimation of a multivariate function from a stationary m-dependent process is investigated, with a special focus on the case where m is large or unbounded. We develop an adaptive estimator based on wavelet methods. Under flexible assumptions on the nonparametric model, we prove the good performances of our estimator by determining sharp rates of convergence under two kinds of errors: the pointwise mean squared error and the mean integrated squared error. We illustrate our theoretical result by considering the multivariate density estimation problem, the derivatives density estimation problem, the density estimation problem in a GARCH-type model and the multivariate regression function estimation problem. The performance of proposed estimator has been shown by a numerical study for a simulated and real data sets.

KEYWORDS:

• Density estimation
• m-dependence
• Nonparametric regression
• Rates of convergence
• Wavelet methods.

AMS 2000 SUBJECT CLASSIFICATIONS:

• 62G07
• 62G20

Acknowledgements

We thank the reviewers for providing comments that helped us to improve the presentation of our work. Hassan Doosti and Lewi Stone gratefully acknowledge the support of the Australian Research Council grant DP150102472.