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.

翻译：本文针对采样区域$R_n \subset \mathbb{R}^d$内不规则分布的空间数据，发展了局部多项式（LP）回归的一般渐近理论。我们采用一种随机采样设计，该设计能以灵活方式生成不规则分布的采样点，涵盖纯递增框架和混合递增框架。首先引入定义在$\mathbb{R}^d$上空间数据的非参数回归模型，然后建立一般阶数$p \geq 1$下LP估计量的渐近正态性。同时提出构建置信区间的方法，并证明LP估计量的一致收敛速率。本文对底层过程的相依结构条件涵盖了一类广泛的随机场，如Lévy驱动的连续自回归移动平均随机场。作为主要结果的应用，我们探讨了均值函数及其偏导数的双样本检验问题。