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June 2012 Nearly root-$n$ approximation for regression quantile processes
Stephen Portnoy
Ann. Statist. 40(3): 1714-1736 (June 2012). DOI: 10.1214/12-AOS1021

Abstract

Traditionally, assessing the accuracy of inference based on regression quantiles has relied on the Bahadur representation. This provides an error of order $n^{-1/4}$ in normal approximations, and suggests that inference based on regression quantiles may not be as reliable as that based on other (smoother) approaches, whose errors are generally of order $n^{-1/2}$ (or better in special symmetric cases). Fortunately, extensive simulations and empirical applications show that inference for regression quantiles shares the smaller error rates of other procedures. In fact, the “Hungarian” construction of Komlós, Major and Tusnády [Z. Wahrsch. Verw. Gebiete 32 (1975) 111–131, Z. Wahrsch. Verw. Gebiete 34 (1976) 33–58] provides an alternative expansion for the one-sample quantile process with nearly the root-$n$ error rate (specifically, to within a factor of $\log n$). Such an expansion is developed here to provide a theoretical foundation for more accurate approximations for inference in regression quantile models. One specific application of independent interest is a result establishing that for conditional inference, the error rate for coverage probabilities using the Hall and Sheather [J. R. Stat. Soc. Ser. B Stat. Methodol. 50 (1988) 381–391] method of sparsity estimation matches their one-sample rate.

Citation

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Stephen Portnoy. "Nearly root-$n$ approximation for regression quantile processes." Ann. Statist. 40 (3) 1714 - 1736, June 2012. https://doi.org/10.1214/12-AOS1021

Information

Published: June 2012
First available in Project Euclid: 2 October 2012

zbMATH: 1284.62291
MathSciNet: MR3015041
Digital Object Identifier: 10.1214/12-AOS1021

Subjects:
Primary: 62E20 , 62J99
Secondary: 60F17

Keywords: Asymptotic approximation , Hungarian construction , regression quantiles

Rights: Copyright © 2012 Institute of Mathematical Statistics

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Vol.40 • No. 3 • June 2012
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