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，建立了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，能够灵活生成不规则分布的采样点，既涵盖纯递增框架也涵盖混合递增框架。首先引入定义在 $\mathbb{R}^d$ 上的空间数据非参数回归模型，随后建立一般阶数 $p \geq 1$ 的局部多项式估计量的渐近正态性。同时提出构建置信区间的方法并建立局部多项式估计量的一致收敛速率。对底层过程依赖结构条件的设定覆盖了包括 Lévy 驱动连续自回归滑动平均随机场在内的广泛随机场类别。作为主要结果的应用，我们讨论了均值函数及其偏导数的双样本检验问题。