Universal kernel-type estimation of random fields

TL;DR

This paper introduces universal kernel-type estimators for nonparametric regression of random fields, providing explicit convergence bounds that are insensitive to the design's correlation structure, with applications to estimating mean and covariance functions.

Contribution

It proposes new weighted least squares estimators for random fields with explicit convergence bounds unaffected by design correlation, improving estimation accuracy over existing methods.

Findings

Estimators achieve uniform convergence with explicit bounds.

Simulation shows improved accuracy over Nadaraya--Watson estimators.

Application to earthquake data demonstrates practical utility.

Abstract

Consistent weighted least square estimators are proposed for a wide class of nonparametric regression models with random regression function, where this real-valued random function of $k$ arguments is assumed to be continuous with probability 1. We obtain explicit upper bounds for the rate of uniform convergence in probability of the new estimators to the unobservable random regression function for both fixed or random designs. In contrast to the predecessors' results, the bounds for the convergence are insensitive to the correlation structure of the $k$-variate design points. As an application, we study the problem of estimating the mean and covariance functions of random fields with additive noise under dense data conditions. The theoretical results of the study are illustrated by simulation examples which show that the new estimators are more accurate in some cases than the Nadaraya--Watson ones. An example of processing real data on earthquakes in Japan in 2012--2021 is included.