The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities
01 Feb 1955-Vol. 6, Iss: 2, pp 170-176
About: The article was published on 1955-02-01 and is currently open access. It has received 552 citations till now. The article focuses on the topics: Convex set & Subderivative.
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01 Jan 2013
TL;DR: In this paper, a stochastic tail order is introduced to compare right tails of distributions and related closure properties, which is then used to compare the dependence structure of multivariate extreme value distributions in terms of upper tail behaviors of their underlying samples.
Abstract: A stochastic tail order is introduced to compare right tails of distributions and related closure properties are established. The stochastic tail order is then used to compare the dependence structure of multivariate extreme value distributions in terms of upper tail behaviors of their underlying samples.
13 citations
Additional excerpts
...Anderson in [1], Fefferman, Jodeit and Perlman in [6] show that if μ1 = μ2, R1 d = R2, and ξΣ1ξ ≤ ξΣ2ξ, ∀ ξ ∈ R, then E(ψ(X)) ≤ E(ψ(Y )) for all symmetric and convex functions ψ : R 7→ R, such that the expectations exist....
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TL;DR: A survey of known proofs of the antitonicity of the inverse matrix function for positive definite matrices w.r.t. the Lowner partial ordering is given in this paper.
Abstract: A survey is given of known proofs of the antitonicity of the inverse matrix function for positive definite matrices w.r.t. the Lowner partial ordering, and of the corresponding result for the Moore-Penrose inverse of nonnegative definite matrices [the theorem of Milliken and Akdeniz (1977)]. A short new proof of the latter result is obtained by employing an extremal representation of a nonnegative definite quadratic form. Another proof of this result involving Schur complements is also given, and is seen to be extendable to the case of symmetric (not necessarily nonnegative definite) matrices. A geometrical interpretation of Milliken and Akdeniz's theorem is presented. As an application, the relationship between the concepts of greater (maximum) concentration and smaller (minimum) dispersion is considered for a pair (class) of vector-valued statistics with possibly degenerate distributions.
13 citations
TL;DR: In this article, it was shown that the first eigenvalue λ 1 satisfies the asymptotic relation ln P {λ 1 ⩽t}∼−2 d ω d j (d−2)/2 d ·t −d/2 when t→0+, where ωd and j(d− 2)/2 are respectively the Lebesgue measure of the unit ball in R d and the first zero of the Bessel function J(d −2)/ 2.
Abstract: Denote by ϕ(t)=∑ n⩾1 e −λ n t , t>0 , the spectral function related to the Dirichlet Laplacian for the typical cell C of a standard Poisson–Voronoi tessellation in R d , d⩾2 . We show that the expectation E ϕ(t) , t>0, is a functional of the convex hull of a standard d-dimensional Brownian bridge. This enables us to study the asymptotic behaviour of E ϕ(t) , when t→0+,+∞. In particular, we prove that the law of the first eigenvalue λ1 of C satisfies the asymptotic relation ln P {λ 1 ⩽t}∼−2 d ω d j (d−2)/2 d ·t −d/2 when t→0+, where ωd and j(d−2)/2 are respectively the Lebesgue measure of the unit ball in R d and the first zero of the Bessel function J(d−2)/2.
12 citations
Cites methods from "The integral of a symmetric unimoda..."
...Moreover, it is obvious from the definition of the Voronoi cell C(0) that for every c ∈ [1, 2), C(0) ⊂ [ ∪x∈ΦB ( 1 c · x, ( 1 c − 1 2 ) · L )]c ....
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...For every fixed ε > 0 and c ∈ [1, 2), we obtain ∫ +∞...
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...since the set Au, u > 0, is convex and symmetric, we may apply Anderson’s lemma [1] which gives P {W ∈ Au + x} ≤ P {W ∈ Au} ....
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TL;DR: In this article, the properties of Hotelling's T 2 test under model misspecification in the model for a multivariate experiment are studied under misspecified location and scale.
Abstract: Properties of Hotelling's (1931) T 2 are studied under model misspecification in the model for a multivariate experiment. Stochastic bounds on T 2 and further properties of the T 2 test are studied under misspecified location and scale. The bounds are evaluated numerically in selected cases.
12 citations
Posted Content•
TL;DR: In this article, the extremal point process of critical points is studied, i.e., the point process associated to all critical values in the vicinity of the ground-state.
Abstract: Recently, sharp results concerning the critical points of the Hamiltonian of the $p$-spin spherical spin glass model have been obtained by means of moments computations. In particular, these moments computations allow for the evaluation of the leading term of the ground-state, i.e., of the global minimum. In this paper, we study the extremal point process of critical points - that is, the point process associated to all critical values in the vicinity of the ground-state. We show that the latter converges in distribution to a Poisson point process of exponential intensity. In particular, we identify the correct centering of the ground-state and prove the convergence in distribution of the centered minimum to a (minus) Gumbel variable. These results are identical to what one obtains for a sequence of i.i.d variables, correctly normalized; namely, we show that the model is in the universality class of REM.
12 citations
References
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TL;DR: In this article, a general method for calculating the limiting distributions of these criteria is developed by reducing them to corresponding problems in stochastic processes, which in turn lead to more or less classical eigenvalue and boundary value problems for special classes of differential equations.
Abstract: The statistical problem treated is that of testing the hypothesis that $n$ independent, identically distributed random variables have a specified continuous distribution function $F(x)$. If $F_n(x)$ is the empirical cumulative distribution function and $\psi(t)$ is some nonnegative weight function $(0 \leqq t \leqq 1)$, we consider $n^{\frac{1}{2}} \sup_{-\infty
3,082 citations
"The integral of a symmetric unimoda..." refers background in this paper
...In Theorem 1 the equality in (1) holds for k<l if and only if, for every u, (E+y)r\Ku=Er\Ku-\-y....
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...It will be noticed that we obtain strict inequality in (1) if and only if for at least one u, H(u)>H*(u) (because H(u) is continuous on the left)....
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1,660 citations
01 Jan 1934
TL;DR: In this article, Minkowski et al. den engen Zusammenhang dieser Begriffbildungen und Satze mit der Frage nach der bestimmung konvexer Flachen durch ihre GAusssche Krtim mung aufgedeckt und tiefliegende diesbeztigliche Satze bewiesen.
Abstract: Konvexe Figuren haben von jeher in der Geometrie eine bedeutende Rolle gespielt. Die durch ihre KonvexiUitseigenschaft allein charakteri sierten Gebilde hat aber erst BRUNN zum Gegenstand umfassender geometrischer Untersuchungen gemacht. In zwei Arbeiten "Ovale und EifHichen" und "Kurven ohne Wendepunkte" aus den Jahren 1887 und 1889 (vgl. Literaturverzeichnis BRUNN [1J, [2J) hat er neben zahl reichen Satzen der verschiedensten Art tiber konvexe Bereiche und Korper einen Satz tiber die Flacheninhalte von parallelen ebenen Schnitten eines konvexen K6rpers bewiesen, der sich in der Folge als fundamental herausgestellt hat. Die Bedeutung dieses Satzes hervor gehoben zu haben, ist das Verdienst von MINKOWSKI. In mehreren Arbeiten, insbesondere in "Volumeri. und Oberflache" (1903) und in der groBztigig angelegten, unvollendet geblieben n Arbeit "Zur Theorie der konvexen K6rper" (Literaturverzeichnis [3], [4J) hat er durch Ein fUhrung von grundlegenden Begriffen wie Stutzfunktion, gemischtes VolulIl, en usw. die dem Problemkreis angemessenen formalen Hilfsmittel geschaffen und vor allem den Weg zu vielseitigen Anwendungen, speziell auf das isoperimetrische (isepiphane) und andere Extremalprobleme fUr konvexe Bereiche und K6rper er6ffnet. Weiterhin hat MINKOWSKI den engen Zusammenhang dieser Begriffsbildungen und Satze mit der Frage nach der Bestimmung konvexer Flachen durch ihre GAusssche Krtim mung aufgedeckt und tiefliegende diesbeztigliche Satze bewiesen.
927 citations
TL;DR: In this paper, the authors extended the Cramer-Smirnov and von Mises test to the parametric case, a suggestion of Cramer [1], see also [2].
Abstract: The "goodness of fit" problem, consisting of comparing the empirical and hypothetical cumulative distribution functions (cdf's), is treated here for the case when an auxiliary parameter is to be estimated. This extends the Cramer-Smirnov and von Mises test to the parametric case, a suggestion of Cramer [1], see also [2]. The characteristic function of the limiting distribution of the test function is found by consideration of a Guassian stochastic process.
140 citations
"The integral of a symmetric unimoda..." refers background in this paper
...f ud[H*(u) - H(u)} = b[H*(b) - H(b)] - a[H*(a) - H(a)} (3) " + f [(H(u) - H*(u)]du....
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...J a Since/(x) has a finite integral over E, bH(b)—>0 as b—>oo and hence also bH*(b)—>0 as b—*<x>; therefore the first term on the right in (3) can be made arbitrarily small in absolute value....
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