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June, 1991 Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation
Probal Chaudhuri
Ann. Statist. 19(2): 760-777 (June, 1991). DOI: 10.1214/aos/1176348119


Let $(X, Y)$ be a random vector such that $X$ is $d$-dimensional, $Y$ is real valued and $Y = \theta(X) + \varepsilon$, where $X$ and $\varepsilon$ are independent and the $\alpha$th quantile of $\varepsilon$ is $0$ ($\alpha$ is fixed such that $0 < \alpha < 1$). Assume that $\theta$ is a smooth function with order of smoothness $p > 0$, and set $r = (p - m)/(2p + d)$, where $m$ is a nonnegative integer smaller than $p$. Let $T(\theta)$ denote a derivative of $\theta$ of order $m$. It is proved that there exists a pointwise estimate $\hat{T}_n$ of $T(\theta)$, based on a set of i.i.d. observations $(X_1, Y_1),\cdots,(S_n, Y_n)$, that achieves the optimal nonparametric rate of convergence $n^{-r}$ under appropriate regularity conditions. Further, a local Bahadur type representation is shown to hold for the estimate $\hat{T}_n$ and this is used to obtain some useful asymptotic results.


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Probal Chaudhuri. "Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation." Ann. Statist. 19 (2) 760 - 777, June, 1991.


Published: June, 1991
First available in Project Euclid: 12 April 2007

zbMATH: 0728.62042
MathSciNet: MR1105843
Digital Object Identifier: 10.1214/aos/1176348119

Primary: 62G05
Secondary: 62E20 , 62G20 , 62G35

Keywords: Bahadur representation , optimal nonparametric rates of convergence , regression quantiles

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.19 • No. 2 • June, 1991
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