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：本文发展了采样区域 $R_n \subset \mathbb{R}^d$ 内不规则分布空间数据的局部多项式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，涵盖纯递增和混合递增域框架。首先引入定义在 $\mathbb{R}^d$ 上空间数据的非参数回归模型，随后建立一般阶数 $p \geq 1$ 的 LP 估计量的渐近正态性。同时提出置信区间构建方法以及 LP 估计量的一致收敛速率。对底层过程的依赖结构条件覆盖了广泛随机场类别，例如 Lévy 驱动的连续自回归滑动平均随机场。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。