Nonparametric Density Estimation for Spatial Data with Wavelets

Abstract

Nonparametric density estimators are studied for $d$-dimensional, strongly spatial mixing data which is defined on a general $N$-dimensional lattice structure. We consider linear and nonlinear hard thresholded wavelet estimators which are derived from a $d$-dimensional multiresolution analysis. We give sufficient criteria for the consistency of these estimators and derive rates of convergence in $L^{p^{\prime }}$ for $p^{\prime }\in [1,\infty )$. For this reason, we study density functions which are elements of a $d$-dimensional Besov space $B_{p,q}^s({\mathbb{R}}^d)$. We also verify the analytic correctness of our results in numerical simulations.