The Annals of Statistics

Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation

Probal Chaudhuri

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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.

Article information

Ann. Statist., Volume 19, Number 2 (1991), 760-777.

First available in Project Euclid: 12 April 2007

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Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62G35: Robustness 62G20: Asymptotic properties 62E20: Asymptotic distribution theory

Regression quantiles Bahadur representation optimal nonparametric rates of convergence


Chaudhuri, Probal. Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation. Ann. Statist. 19 (1991), no. 2, 760--777. doi:10.1214/aos/1176348119.

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