论文标题
$ \ mathbb {r}^d $的空间数据的本地多项式趋势回归
Local polynomial trend regression for spatial data on $\mathbb{R}^d$
论文作者
论文摘要
本文开发了一种局部多项式（LP）回归的一般渐近理论，用于在采样区域中不规则间隔的位置观察到的空间数据$ r_n \ subset \ subset \ mathbb {r}^d $。我们采用随机抽样设计，该设计可以以灵活的方式生成不规则间隔的采样位点，包括纯增加和混合增加的域框架。我们首先引入了一个非参数回归模型，用于在$ \ mathbb {r}^d $上定义的空间数据，然后建立LP估计器的渐近正态性，并使用通用订单$ p \ geq 1 $。我们还提出了构建置信区间和建立LP估计量统一收敛速率的方法。我们对基础过程的依赖性结构条件涵盖了一系列随机场，例如莱维驱动的连续自动回归移动平均随机场。作为我们主要结果的应用，我们讨论了平均功能及其部分导数的两样本测试问题。
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 Lévy-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.
