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.

针对具有随机回归函数的一类广泛的非参数回归模型，提出了一致的加权最小二乘估计量，其中k 个参数的实值随机函数被假设为以概率 1 连续。我们获得了一致收敛率的明确上限对于固定或随机设计，新估计量对不可观察的随机回归函数的概率。与前人的结果相比，收敛的界限对k的相关结构不敏感- 变化的设计点。作为一个应用，我们研究了在密集数据条件下估计具有加性噪声的随机场的均值和协方差函数的问题。研究的理论结果通过模拟例子进行了说明，这些例子表明新的估计器在某些情况下比 Nadaraya-Watson 估计器更准确。其中包括处理 2012 年至 2021 年日本地震真实数据的示例。