The Annals of Statistics

Kernel and Nearest-Neighbor Estimation of a Conditional Quantile

P. K. Bhattacharya and Ashis K. Gangopadhyay

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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

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

First available in Project Euclid: 12 April 2007

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Primary: 62G05: Estimation
Secondary: 62J02: General nonlinear regression 62G20: Asymptotic properties 62G30: Order statistics; empirical distribution functions 60F17: Functional limit theorems; invariance principles

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


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.

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