Nonparametric Density Estimation for Spatial Data with Wavelets

TL;DR

This paper develops wavelet-based nonparametric density estimators for high-dimensional spatial data on lattices, establishing their consistency and convergence rates within Besov spaces, supported by numerical simulations.

Contribution

It introduces new wavelet estimators for spatial data, providing theoretical guarantees and convergence rates in Besov spaces, with validation through simulations.

Findings

Establishes consistency criteria for wavelet density estimators.

Derives convergence rates in $L^{p'}$ norms for the estimators.

Validates theoretical results with numerical experiments.

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 $L^{p'}$ for $p'\in [1,\infty)$. For this reason, we study density functions which are elements of a $d$-dimensional Besov space $B^s_{p,q}(\mathbb{R}^d)$. We also verify the analytic correctness of our results in numerical simulations.