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Tobias Zwingmann

Publications and source records attributed to Tobias Zwingmann.

3 recordsLinked to original sources
arXiv2017

Weak convergence of quantile and expectile processes under general assumptions

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 \citet{buecher2014}. We impose assumptions for which it is known that weak convergence with respect to the supremum norm or the Skorodhod metric generally fails to hold. For expectiles, we only require a distribution with finite second moment but no further smoothness properties of distribution function, for quantiles, the distribution is assumed to be absolutely continuous with a version of its Lebesgue density which is strictly positive and has left- and right-sided limits. We also show consistency of the bootstrap for this mode of convergence.

math.ST↗
arXiv2016

Asymptotics for the expected shortfall

We derive the joint asymptotic distribution of empirical quantiles and expected shortfalls under general conditions on the distribution of the underlying observations. In particular, we do not assume that the distribution function is differentiable at the quantile with strictly positive derivative. Hence the rate of convergence and the asymptotic distribution for the quantile can be non-standard, but our results show that the expected shortfall remains asymptotically normal with a $\sqrt{n}$-rate, and we even give the joint distribution in such non-standard cases. In the derivation we use the bivariate scoring functions for quantile and expected shortfall as recently introduced by Fissler and Ziegel (2016). The main technical issue is to deal with the distinct rates for quantile and expected shortfall when applying the argmax-continuity theorem. We also consider spectral risk measures with finitely-supported spectral measures, and illustrate our results in a simulation study.

math.ST↗