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Simultaneous confidence bands for extremal quantile regression with splines

  • Takuma Yoshida


This study investigates simultaneous confidence bands for extremal quantile regressions using the spline method. We construct the spline estimator for intermediate order quantiles using a conventional quantile regression framework, and we obtain the extreme order quantile estimator by extrapolating the spline estimator for intermediate order quantiles. We establish the asymptotic normality of the spline and extrapolated estimators for intermediate and extreme order quantiles. By applying the volume of tube formula to the above two estimators, we construct simultaneous conditional quantile confidence bands for intermediate and extreme order quantiles. To confirm the performance of the proposed confidence bands, we use a Monte Carlo simulation and an example with real data.


Extrapolation Extremal quantile regression Simultaneous confidence band Volume of tube formula 

AMS 2000 Subject Classifications

62G08 62G15 62G32 


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The author gratefully acknowledge the valuable input of the Editor, Associate Editor and the anonymous two referees that improved the presentation of this paper. The research of the author was partially supported by KAKENHI 26730019 and KAKENHI 18K18011.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.KagoshimaJapan

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