  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.
