In the present paper, we consider the nonparametric regression model with random design based on \((\mathbf{X}_\mathrm{t},\mathbf{Y}_\mathrm{t})_{\mathrm{t}\ge 0}\) a \(\mathbb {R}^{d}\times \mathbb {R}^{q}\)-valued strictly stationary and ergodic continuous time process, where the regression function is given by \(m(\mathbf{x},\psi ) = \mathbb {E}(\psi (\mathbf{Y}) \mid \mathbf{X} = \mathbf{x}))\), for a measurable function \(\psi : \mathbb {R}^{q} \rightarrow \mathbb {R}\). We focus on the estimation of the location \({\varvec{\Theta }}\) (mode) of a unique maximum of \(m(\cdot , \psi )\) by the location \( \widehat{{\varvec{\Theta }}}_\mathrm{T}\) of a maximum of the Nadaraya–Watson kernel estimator \(\widehat{m}_\mathrm{T}(\cdot , \psi )\) for the curve \(m(\cdot , \psi )\). Within this context, we obtain the consistency with rate and the asymptotic normality results for \( \widehat{{\varvec{\Theta }}}_\mathrm{T}\) under mild local smoothness assumptions on \(m(\cdot , \psi )\) and the design density \(f(\cdot )\) of \(\mathbf{X}\). Beyond ergodicity, any other assumption is imposed on the data. This paper extends the scope of some previous results established under the mixing condition. The usefulness of our results will be illustrated in the construction of confidence regions.

在本文中，我们考虑基于\((\mathbf{X}_\mathrm{t},\mathbf{Y}_\mathrm{t})_{\mathrm{t}的随机设计的非参数回归模型\ge 0}\)一个\(\mathbb {R}^{d}\times \mathbb {R}^{q}\) -值严格平稳和遍历的连续时间过程，其中回归函数由\( m(\mathbf{x},\psi ) = \mathbb {E}(\psi (\mathbf{Y}) \mid \mathbf{X} = \mathbf{x}))\)，对于可测函数\ (\psi : \mathbb {R}^{q} \rightarrow \mathbb {R}\)。我们专注于通过位置\ ( \ widehat { {\ varvec{\Theta }}}_\mathrm{T}\)曲线\(m(\cdot , \psi )\)的最大 Nadaraya–Watson 核估计量\(\widehat{m}_\mathrm{T}(\cdot , \psi )\) 。在这种情况下，我们在\(m( \ cdot , \psi )\)和\(\mathbf{X}\)的设计密度\(f(\cdot )\ ) 。除了遍历性之外，还对数据施加了任何其他假设。本文扩展了在混合条件下建立的一些先前结果的范围。我们的结果的有用性将在置信区域的构建中得到说明。