Orthogonal Series Estimates on Strong Spatial Mixing Data

TL;DR

This paper introduces a nonparametric regression estimator for high-dimensional spatial data using Haar wavelets, demonstrating its consistency, adaptability to local smoothness, and convergence rates through theoretical analysis and simulations.

Contribution

It develops a novel orthonormal wavelet-based estimator for spatially mixing data, providing theoretical guarantees and demonstrating adaptability to local function smoothness.

Findings

Estimator is consistent under strong spatial mixing.

It adapts to local smoothness of the regression function.

Simulation results support theoretical convergence rates.

Abstract

We study a nonparametric regression model for sample data which is defined on an $N$-dimensional lattice structure and which is assumed to be strong spatial mixing: we use design adapted multidimensional Haar wavelets which form an orthonormal system w.r.t. the empirical measure of the sample data. For such orthonormal systems, we consider a nonparametric hard thresholding estimator. We give sufficient criteria for the consistency of this estimator and we derive rates of convergence. The theorems reveal that our estimator is able to adapt to the local smoothness of the underlying regression function and the design distribution. We illustrate our results with simulated examples.