In this paper we investigate the problem of estimating the regression function in models with correlated observations. The data is obtained from several experimental units each of them forms a time series. We propose a new estimator based on the inverse of the autocovariance matrix of the observations, assumed known and invertible. Using the properties of the Reproducing Kernel Hilbert spaces, we give the asymptotic expressions of its bias and its variance. In addition, we give a theoretical comparison, by calculating the IMSE, between this new estimator and the classical one proposed by Gasser and Muller. Finally, we conduct a simulation study to investigate the performance of the proposed estimator and to compare it to the Gasser and Muller's estimator in a finite sample set.

中文翻译：

---

具有相关观测值的非参数回归问题中的再生核希尔伯特空间方法

在本文中，我们研究了在具有相关观测值的模型中估计回归函数的问题。数据是从几个实验单元获得的，每个单元形成一个时间序列。我们提出了一个基于观测值自协方差矩阵逆的新估计器，假设已知且可逆。利用再生核希尔伯特空间的性质，我们给出了其偏差和方差的渐近表达式。此外，我们通过计算 IMSE，对这个新估计量与 Gasser 和 Muller 提出的经典估计量进行了理论比较。最后，我们进行了模拟研究，以研究所提出的估计器的性能，并将其与有限样本集中的 Gasser 和 Muller 估计器进行比较。