Some Asymptotic Properties of Kernel Regression Estimators of the Mode for Stationnary and Ergodic Continuous Time Processes

S. Bouzebda and S. Sidi

In the present paper, we consider the nonparametric regression model with random design based on \((\mathbf{X}_{\rm t},\mathbf{Y}_{\rm t})_{\rm t \geq 0}\) an $\mathbb{R}^{d}\times\mathbb{R}^{q}$-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 $m(\mathbf{x},\psi) = \mathbb{E}(\psi(\mathbf{ Y}) \mid \mathbf{ X} = \mathbf{ x}))$ $\mathbf{\Theta}$ (mode) of a unique maximum of $m(\cdot, \psi)$ by the location \(\widehat{\mathbf{\Theta}}_{\rm T}\) of a maximum of the Nadaraya-Watson kernel estimator \(\widehat{m}_{\rm T}(\cdot, \psi)\) for the curve \(m(\cdot, \psi)\). Within this context, we obtain the consistency with rate and the asymptotic normality results for \(\widehat{\mathbf{ \Theta}}_{\rm T}\) 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.

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