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September, 1990 Kernel and Nearest-Neighbor Estimation of a Conditional Quantile
P. K. Bhattacharya, Ashis K. Gangopadhyay
Ann. Statist. 18(3): 1400-1415 (September, 1990). DOI: 10.1214/aos/1176347757


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.


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P. K. Bhattacharya. Ashis K. Gangopadhyay. "Kernel and Nearest-Neighbor Estimation of a Conditional Quantile." Ann. Statist. 18 (3) 1400 - 1415, September, 1990.


Published: September, 1990
First available in Project Euclid: 12 April 2007

zbMATH: 0706.62040
MathSciNet: MR1062716
Digital Object Identifier: 10.1214/aos/1176347757

Primary: 62G05
Secondary: 60F17, 62G20, 62G30, 62J02

Rights: Copyright © 1990 Institute of Mathematical Statistics


Vol.18 • No. 3 • September, 1990
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