Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1611.07222

Asymptotics for the expected shortfall

Tobias Zwingmann·Hajo Holzmann
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.STstat.TH

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Tobias Zwingmann, Hajo Holzmann. 2016-11-25. Asymptotics for the expected shortfall. https://arxiv.org/abs/1611.07222

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Functional independent component analysis by choice of norm: a framework for near-perfect classification

We develop a theory for functional independent component analysis in an infinite-dimensional framework using Sobolev spaces that accommodate smoother functions. The notion of penalized kurtosis is introduced motivated by Silverman's method for smoothing principal components. This approach allows for a classical definition of independent components obtained via projection onto the eigenfunctions of a smoothed kurtosis operator mapping a whitened functional random variable. We discuss the theoretical properties of this operator in relation to a generalized Fisher discriminant function and the relationship it entails with the Feldman-Hájek dichotomy for Gaussian measures, both of which are critical to the principles of functional classification. The proposed estimators are a particularly competitive alternative in binary classification of functional data and can eventually achieve the so-called near-perfect classification, which is a genuine phenomenon of high-dimensional data. Our methods are illustrated through simulations, various real datasets, and used to model electroencephalographic biomarkers for the diagnosis of depressive disorder.

math.ST↗
arXiv2026

Trace-Class Results for MCMC Algorithms for Student-$t$ Regression Models

In this paper, we consider MCMC algorithms for Student-$t$ regression models. In three cases, we investigate the efficiency of Markov chains based on the algorithms in terms of whether trace-class results hold or not. First, we consider the case where the parameters follow a matrix-normal-inverse-Wishart distribution and show that the Markov operator associated with a standard data augmentation algorithm is trace-class. Second, we consider the case of an improper prior and univariate outcomes. In this case, the standard Markov operator is not trace-class but the Markov operator associated with a collapsed Gibbs algorithm is trace-class. Third, we consider the case of an improper prior and multivariate outcomes. We obtain a trace-class result for a parameter expanded data augmentation algorithm which is based on a univariate working parameter. Finally, we consider the problem of numerially estimating a convergence rate of the trace-class Markov operator in the second case.

math.ST↗
arXiv2026

The Manifold Hypothesis under Unknown Gaussian Noise:Conditional Certificates and Consistent Dimension Estimation

We study what noisy data can establish about the Manifold Hypothesis under explicit identification and regularity conditions. A population residual certificate combines independent-view localization, Gaussian concentration, membership uncertainty, and population transfer. Existing rectifiability criteria then yield a covered-scale consequence. For a local smooth manifold with positive Hölder density, the actual-ball covariance limit identifies the spectral crossing with geometric dimension. We prove almost-sure eventual recovery under repeated observations. Reusing accurate localization averages improves the sufficient point-sample condition from $Nr^{d+4}\gg\log N$ to $Nr^d\gg\log N$, with replication $kr^2\gg\log N$. A two-mass certificate controls incorrect geometric-dimension emissions under declared class bounds. For single observations with unknown Gaussian noise, affine-support or known coordinate-bound restrictions provide noise intervals and consistent Gaussian correlation-dimension estimators. Ahlfors regularity identifies this exponent with Hausdorff dimension and with the geometric dimension of a homogeneous smooth class. Exact Cantor calculations delineate the limits of integer spectral counts and adjacent-radius slopes. We credit established local PCA, rectifiability, concentration, binomial inference, and deconvolution results before specifying our constructions. Reproducible experiments distinguish point estimation, finite-scale coverage, and certificate emission.

math.ST↗