Local polynomial regression for spatial data on ℝ^d
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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 ⊂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 ℝ^d and then establish the asymptotic normality of LP estimators with general order p ≥ 1. We also propose methods for constructing confidence intervals and establish 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.
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