Nonparametric regression for locally stationary random fields under stochastic sampling design

Abstract

In this study, we develop an asymptotic theory of nonparametric regression for locally stationary random fields (LSRFs) $\{{\mathit{X}}_{\mathit{s},{\mathit{A}}_{\mathit{n}}}:\mathit{s}\in {\mathit{R}}_{\mathit{n}}\}$ in ${\mathbb{R}}^{\mathit{p}}$ observed at irregularly spaced locations in ${\mathit{R}}_{\mathit{n}}={[0,{\mathit{A}}_{\mathit{n}}]}^{\mathit{d}}\subset {\mathbb{R}}^{\mathit{d}}$. We first derive the uniform convergence rate of general kernel estimators, followed by the asymptotic normality of an estimator for the mean function of the model. Moreover, we consider additive models to avoid the curse of dimensionality arising from the dependence of the convergence rate of estimators on the number of covariates. Subsequently, we derive the uniform convergence rate and joint asymptotic normality of the estimators for additive functions. We also introduce approximately ${\mathit{m}}_{\mathit{n}}$-dependent RFs to provide examples of LSRFs. We find that these RFs include a wide class of Lévy-driven moving average RFs.