Electronic Journal of Statistics

Quantile processes for semi and nonparametric regression

Shih-Kang Chao, Stanislav Volgushev, and Guang Cheng

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A collection of quantile curves provides a complete picture of conditional distributions. A properly centered and scaled version of the estimated curves at various quantile levels gives rise to the so-called quantile regression process (QRP). In this paper, we establish weak convergence of QRP in a general series approximation framework, which includes linear models with increasing dimension, nonparametric models and partial linear models. An interesting consequence is obtained in the last class of models, where parametric and non-parametric estimators are shown to be asymptotically independent. Applications of our general process convergence results include the construction of non-crossing quantile curves and the estimation of conditional distribution functions. As a result of independent interest, we obtain a series of Bahadur representations with exponential bounds for tail probabilities of all remainder terms. Bounds of this kind are potentially useful in analyzing statistical inference procedures under the divide-and-conquer setup.

Article information

Electron. J. Statist., Volume 11, Number 2 (2017), 3272-3331.

Received: March 2017
First available in Project Euclid: 2 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F12: Asymptotic properties of estimators 62G20: Asymptotic properties 62G08: Nonparametric regression

Bahadur representation quantile regression process semi/nonparametric model series estimation

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Chao, Shih-Kang; Volgushev, Stanislav; Cheng, Guang. Quantile processes for semi and nonparametric regression. Electron. J. Statist. 11 (2017), no. 2, 3272--3331. doi:10.1214/17-EJS1313. https://projecteuclid.org/euclid.ejs/1506931550

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