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February 2020 Weak convergence of quantile and expectile processes under general assumptions
Tobias Zwingmann, Hajo Holzmann
Bernoulli 26(1): 323-351 (February 2020). DOI: 10.3150/19-BEJ1127

Abstract

We show weak convergence of quantile and expectile processes to Gaussian limit processes in the space of bounded functions endowed with an appropriate semimetric which is based on the concepts of epi- and hypo- convergence as introduced in A. Bücher, J. Segers and S. Volgushev (2014), ‘When Uniform Weak Convergence Fails: Empirical Processes for Dependence Functions and Residuals via Epi- and Hypographs’, Annals of Statistics 42. We impose assumptions for which it is known that weak convergence with respect to the supremum norm generally fails to hold. For quantiles, we consider stationary observations, where the marginal distribution function is assumed to be strictly increasing and continuous except for finitely many points and to admit strictly positive – possibly infinite – left- and right-sided derivatives. For expectiles, we focus on independent and identically distributed (i.i.d.) observations. Only a finite second moment and continuity at the boundary points but no further smoothness properties of the distribution function are required. We also show consistency of the bootstrap for this mode of convergence in the i.i.d. case for quantiles and expectiles.

Citation

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Tobias Zwingmann. Hajo Holzmann. "Weak convergence of quantile and expectile processes under general assumptions." Bernoulli 26 (1) 323 - 351, February 2020. https://doi.org/10.3150/19-BEJ1127

Information

Received: 1 August 2017; Revised: 1 April 2019; Published: February 2020
First available in Project Euclid: 26 November 2019

zbMATH: 07140501
MathSciNet: MR4036036
Digital Object Identifier: 10.3150/19-BEJ1127

Rights: Copyright © 2020 Bernoulli Society for Mathematical Statistics and Probability

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Vol.26 • No. 1 • February 2020
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