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August 2014 When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs
Axel Bücher, Johan Segers, Stanislav Volgushev
Ann. Statist. 42(4): 1598-1634 (August 2014). DOI: 10.1214/14-AOS1237


In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those spaces fails to hold. Examples include empirical copula and tail dependence processes and residual empirical processes in linear regression models in case the underlying distributions lack a certain degree of smoothness. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epi- and hypo-convergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies $L^{p}$ convergence. For the examples mentioned above, weak convergence with respect to the new metric is established in situations where it does not occur with respect to the supremum distance. The results are applied to obtain asymptotic properties of resampling procedures and goodness-of-fit tests.


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Axel Bücher. Johan Segers. Stanislav Volgushev. "When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs." Ann. Statist. 42 (4) 1598 - 1634, August 2014.


Published: August 2014
First available in Project Euclid: 7 August 2014

zbMATH: 1323.60038
MathSciNet: MR3262462
Digital Object Identifier: 10.1214/14-AOS1237

Primary: 60F05 , 62G30
Secondary: 62G32 , 62M09

Keywords: bootstrap , copula , epigraph , hypograph , Linear regression , local alternative , power curve , residual empirical process , stable tail dependence function , weak convergence

Rights: Copyright © 2014 Institute of Mathematical Statistics


Vol.42 • No. 4 • August 2014
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