The Annals of Statistics

Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation

Probal Chaudhuri

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176348119

Digital Object Identifier
doi:10.1214/aos/1176348119

Mathematical Reviews number (MathSciNet)
MR1105843

Zentralblatt MATH identifier
0728.62042

JSTOR
links.jstor.org

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

Keywords
Regression quantiles Bahadur representation optimal nonparametric rates of convergence

Citation

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. https://projecteuclid.org/euclid.aos/1176348119


Export citation