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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.