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Robust optimality of Gaussian noise stability

Elchanan Mossel, Joe Neeman (2015)

Journal of the European Mathematical Society

We prove that under the Gaussian measure, half-spaces are uniquely the most noise stable sets. We also prove a quantitative version of uniqueness, showing that a set which is almost optimally noise stable must be close to a half-space. This extends a theorem of Borell, who proved the same result but without uniqueness, and it also answers a question of Ledoux, who asked whether it was possible to prove Borell’s theorem using a direct semigroup argument. Our quantitative uniqueness result has various...

Rozdělení t a mnohorozměrná geometrie

Vítězslav Línek (2019)

Pokroky matematiky, fyziky a astronomie

V článku odvozujeme hustotu t rozdělení s využitím n -rozměrné geometrie. Oproti obvyklejším metodám k tomu nepotřebujeme předpoklad normality, postačující je nezávislost mnohorozměrného rozdělení na směru. Kromě základů diferenciálního počtu použijeme k odvození jen vzorec pro povrch n -rozměrné koule. Tento přístup byl inspirován metodami R. A. Fishera.

Sample d -copula of order m

José M. González-Barrios, María M. Hernández-Cedillo (2013)

Kybernetika

In this paper we analyze the construction of d -copulas including the ideas of Cuculescu and Theodorescu [5], Fredricks et al. [15], Mikusiński and Taylor [25] and Trutschnig and Fernández-Sánchez [33]. Some of these methods use iterative procedures to construct copulas with fractal supports. The main part of this paper is given in Section 3, where we introduce the sample d -copula of order m with m ≥ 2 , the central idea is to use the above methodologies to construct a new copula based on a sample. The...

Semicopulæ

Fabrizio Durante, Carlo Sempi (2005)

Kybernetika

We define the notion of semicopula, a concept that has already appeared in the statistical literature and study the properties of semicopulas and the connexion of this notion with those of copula, quasi-copula, t -norm.

Semicopulas: characterizations and applicability

Fabrizio Durante, José Quesada-Molina, Carlo Sempi (2006)

Kybernetika

We characterize some bivariate semicopulas and, among them, the semicopulas satisfying a Lipschitz condition. In particular, the characterization of harmonic semicopulas allows us to introduce a new concept of depedence between two random variables. The notion of multivariate semicopula is given and two applications in the theory of fuzzy measures and stochastic processes are given.

Sets of determination for parabolic functions on a half-space

Jarmila Ranošová (1994)

Commentationes Mathematicae Universitatis Carolinae

We characterize all subsets M of ℝ n × ℝ + such that sup X ∈ ℝ n × ℝ + u ( X ) = sup X ∈ M u ( X ) for every bounded parabolic function u on ℝ n × ℝ + . The closely related problem of representing functions as sums of Weierstrass kernels corresponding to points of M is also considered. The results provide a parabolic counterpart to results for classical harmonic functions in a ball, see References. As a by-product the question of representability of probability continuous distributions as sums of multiples of normal distributions is investigated.

Seven Proofs for the Subadditivity of Expected Shortfall

Paul Embrechts, Ruodu Wang (2015)

Dependence Modeling

Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof.With different proofs, different important properties of ES are...

Sharp bounds for expectations of spacings from decreasing density and failure rate families

Katarzyna Danielak, Tomasz Rychlik (2004)

Applicationes Mathematicae

We apply the method of projecting functions onto convex cones in Hilbert spaces to derive sharp upper bounds for the expectations of spacings from i.i.d. samples coming from restricted families of distributions. Two families are considered: distributions with decreasing density and with decreasing failure rate. We also characterize the distributions for which the bounds are attained.

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