## The Annals of Statistics

### Nonparametric Estimates of Regression Quantiles and Their Local Bahadur Representation

Probal Chaudhuri

#### 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

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