Nonparametric Function Estimation Involving Time Series

Abstract

Consider a stationary time series $(\mathbf{X}_t, Y_t), t = 0, \pm 1,\ldots,$ with $\mathbf{X}_t$ being $\mathbb{R}^d$-valued and $Y_t$ real-valued. The conditional mean function is given by $\theta(\mathbf{X}_0) = E(Y_0\mid\mathbf{X}_0)$. Under appropriate regularity conditions, a local average estimator of this function based on a finite realization $(\mathbf{X}_1, Y_1),\ldots,(\mathbf{X}_n, Y_n)$ can be chosen to achieve the optimal rate of convergence $n^{-1/(2 + d)}$ both pointwise and in $L_2$ norms restricted to a compact; and it can also be chosen to achieve the optimal rate of convergence $(n^{-1} \log(n))^{1/(2 + d)}$ in $L_\infty$ norm restricted to a compact. Similar results hold for local median estimators of the conditional median function, which is given by $\theta(\mathbf{X}_0) = \operatorname{med}(Y_0\mid\mathbf{X}_0)$.