## The Annals of Statistics

- Ann. Statist.
- Volume 18, Number 3 (1990), 1400-1415.

### Kernel and Nearest-Neighbor Estimation of a Conditional Quantile

P. K. Bhattacharya and Ashis K. Gangopadhyay

#### Abstract

Let $(X_1, Z_1), (X_2, Z_2), \ldots, (X_n, Z_n)$ be iid as $(X, Z), Z$ taking values in $R^1$, and for $0 < p < 1$, let $\xi_p(x)$ denote the conditional $p$-quantile of $Z$ given $X = x,$ i.e., $P(Z \leq \xi_p(x)\mid X = x) = p$. In this paper, kernel and nearest-neighbor estimators of $\xi_p(x)$ are proposed. In order to study the asymptotics of these estimates, Bahadur-type representations of the sample conditional quantiles are obtained. These representations are used to examine the important issue of choosing the smoothing parameter by a local approach (for a fixed $x$) based on weak convergence of these estimators with varying $k$ in the $k$-nearest-neighbor method and with varying $h$ in the kernel method with bandwidth $h$. These weak convergence results lead to asymptotic linear models which motivate certain estimators.

#### Article information

**Source**

Ann. Statist. Volume 18, Number 3 (1990), 1400-1415.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aos/1176347757

**Digital Object Identifier**

doi:10.1214/aos/1176347757

**Mathematical Reviews number (MathSciNet)**

MR1062716

**Zentralblatt MATH identifier**

0706.62040

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62G05: Estimation

Secondary: 62J02: General nonlinear regression 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions 60F17: Functional limit theorems; invariance principles

**Keywords**

Conditional quantile kernel estimator nearest-neighbor estimator Bahadur representation weak convergence Browian motion asymptotic linear model order statistics induced order statistics

#### Citation

Bhattacharya, P. K.; Gangopadhyay, Ashis K. Kernel and Nearest-Neighbor Estimation of a Conditional Quantile. Ann. Statist. 18 (1990), no. 3, 1400--1415. doi:10.1214/aos/1176347757. http://projecteuclid.org/euclid.aos/1176347757.