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.

翻译：针对一类具有随机回归函数的非参数回归模型，本文提出了相合加权最小二乘估计量。该实值随机函数具有$k$个自变量，且以概率1连续。我们获得了新估计量在固定设计和随机设计下对不可观测随机回归函数的一致概率收敛速度的显式上界。与前人的结果相比，该收敛速度上界对$k$维设计点的相关结构不敏感。作为应用，我们研究了稠密数据条件下含加性噪声随机场均值函数与协方差函数的估计问题。通过仿真算例验证了理论结果，表明新估计量在某些情形下比Nadaraya—Watson估计量更精确。文中还包含对2012—2021年日本地震真实数据处理的实例。