Non-parametric regression for spatially dependent data with wavelets

ABSTRACT

We study non-parametric regression estimates for random fields. The data satisfies certain strong mixing conditions and is defined on the regular N-dimensional lattice structure. We show consistency and obtain rates of convergence. The rates are optimal modulo a logarithmic factor in some cases. As an application, we estimate the regression function with multidimensional wavelets which are not necessarily isotropic. We simulate random fields on planar graphs with the concept of concliques (cf. [Kaiser MS, Lahiri SN, Nordman DJ. Goodness of fit tests for a class of markov random field models. Ann Statist. 2012;40:104–130]) in numerical examples of the estimation procedure.

KEYWORDS:

• Multidimensional wavelets
• non-parametric regression
• random fields
• rates of convergence
• strong spatial mixing conditions

MSC 2010:

• Primary: 62G08
• 62H11
• 65T60
• Secondary: 65C40
• 60G60

Acknowledgments

The author is very grateful to two referees and an associate editor, their comments helped to improve the manuscript.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was supported by the Deutsche Forschungsgemeinschaft (DFG), Grant Number KR-4977/1 and by the Fraunhofer ITWM, 67663 Kaiserslautern, Germany.