Nonparametric regression for locally stationary random fields under stochastic sampling design

In this paper, we introduce nonparametric regression framework for locally stationary random fields $\{X_{s, A_{n}}: s \in R_{n} \}$ in $\mathbb{R}^{p}$ which is observed at finite number of locations in a sampling region $R_{n} \subset \mathbb{R}^{d}$ with volume $O(A_{n}^{d})$. We develop asymptotic theory for the estimation problem on the mean function of the model. In particular, we first derive uniform convergence rate of general kernel estimators on compact sets and then derive asymptotic normality of kernel estimators for mean functions. Moreover, we consider additive models to avoid the curse of dimensionality that comes from the dependence of the convergence rate of general estimators on the number of covariates. We also drive a uniform convergence rate and joint asymptotic normality of kernel estimators for additive functions. Additionally, we introduce a notion of \textit{approximately $m_{n}$-dependent} locally stationary random field ($m_{n} \to \infty$ as $n \to \infty$) to discuss and give examples of locally stationary random fields that satisfy our regularity conditions. We find that approximately $m_{n}$-dependent locally stationary random fields include a wide class of locally stationary version of L\'evy-driven moving average random fields.