In this study, we develop an asymptotic theory of nonparametric regression for locally stationary random fields (LSRFs) $\{{\bf X}_{{\bf s}, A_{n}}: {\bf s} \in R_{n} \}$ in $\mathbb{R}^{p}$ observed at irregularly spaced locations in $R_{n} =[0,A_{n}]^{d} \subset \mathbb{R}^{d}$. 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 $m_{n}$-dependent RFs to provide examples of LSRFs. We find that these RFs include a wide class of L\'evy-driven moving average RFs.
翻译：在此研究中,我们为本地固定随机字段(LSRFs) $ ⁇ bf X ⁇ bf s}, A ⁇ n ⁇ :\bf s} $$\mathb{R ⁇ p} 美元观察到的不定期间距地点的美元的非参数回归性理论($ ⁇ n} =[0,A ⁇ n} ⁇ d} \ subset\mathb{R ⁇ d}。我们首先得出普通内核估测器的统一趋同率,然后得出模型平均功能的估测器的均匀性常态。此外,我们考虑添加模型,以避免因估算器的趋同率对同变量数的依赖性而导致的诅咒。随后,我们得出添加功能估测器的统一趋同率和联合的均匀性常态性。我们还引入了大约 $ ⁇ n} $依赖的普通估测器,以提供LSRFs平均功能的典型。我们发现这些累动式模型包括平均RFs。
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