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March, 1992 Nonparametric Function Estimation Involving Time Series
Young K. Truong, Charles J. Stone
Ann. Statist. 20(1): 77-97 (March, 1992). DOI: 10.1214/aos/1176348513


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 $\theta(\mathbf{X}_0) = \operatorname{med}(Y_0\mid\mathbf{X}_0)$.


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Young K. Truong. Charles J. Stone. "Nonparametric Function Estimation Involving Time Series." Ann. Statist. 20 (1) 77 - 97, March, 1992.


Published: March, 1992
First available in Project Euclid: 12 April 2007

zbMATH: 0764.62038
MathSciNet: MR1150335
Digital Object Identifier: 10.1214/aos/1176348513

Primary: 62G05
Secondary: 62E20

Keywords: conditional mean function , conditional median function , local aveage , local median , rate of convergence , stationarity

Rights: Copyright © 1992 Institute of Mathematical Statistics

Vol.20 • No. 1 • March, 1992
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