Local polynomial trend regression for spatial data on $\mathbb{R}^d$

TL;DR

This paper develops a comprehensive asymptotic theory for local polynomial regression applied to irregularly spaced spatial data in multiple dimensions, including confidence interval construction and applications to hypothesis testing.

Contribution

It introduces a general asymptotic framework for LP regression on spatial data with irregular sampling, covering various dependence structures and establishing normality and convergence results.

Findings

Asymptotic normality of LP estimators for spatial data.

Methods for confidence interval construction for spatial regression.

Application to two-sample testing for mean functions.

Abstract

This paper develops a general asymptotic theory of local polynomial (LP) regression for spatial data observed at irregularly spaced locations in a sampling region $R_n \subset \mathbb{R}^d$. We adopt a stochastic sampling design that can generate irregularly spaced sampling sites in a flexible manner including both pure increasing and mixed increasing domain frameworks. We first introduce a nonparametric regression model for spatial data defined on $\mathbb{R}^d$ and then establish the asymptotic normality of LP estimators with general order $p \geq 1$. We also propose methods for constructing confidence intervals and establishing uniform convergence rates of LP estimators. Our dependence structure conditions on the underlying processes cover a wide class of random fields such as L\'evy-driven continuous autoregressive moving average random fields. As an application of our main results, we discuss a two-sample testing problem for mean functions and their partial derivatives.