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Journal ArticleDOI

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.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the authors investigate four orthant stochastic orderings between random vectors X and Y that have finitely discrete probability distributions in R k. They have applications to multiattribute decision under risk, dependency of random vectors, and the statistical comparison of two k-variate samples.

19 citations

Book ChapterDOI
01 Jan 1978

19 citations

Journal ArticleDOI
TL;DR: In this paper, the authors introduce several methods to test the null hypothesis that the mean structure of a time series of curves does not change, i.e., a change point is not detected with a large probability.
Abstract: Motivated by the problem of the detection of a change point in the mean structure of yield curves, we introduce several methods to test the null hypothesis that the mean structure of a time series of curves does not change. The mean structure does not refer merely to the level of the curves, but also to their range and other aspects of their shape, most prominently concavity. The performance of the tests depends on whether possible break points in the error structure, which refers to the random variability in the aspects of the curves listed above, are taken into account or not. If they are not taken into account, an existing change point in the mean structure may fail to be detected with a large probability. The paper contains a complete asymptotic theory, a simulation study and illustrative data examples, as well as details of the numerical implementation of the testing procedures. This article is protected by copyright. All rights reserved

19 citations


Cites methods from "The integral of a symmetric unimoda..."

  • ...It follows from Anderson (1955) that P { ∫ 1 0 ‖Ḡ0(x)‖2dx ≥ t } ≥ P { ∫ 1 0 ‖Ḡ(x)‖2dx ≥ t } for all t, and therefore the level of rejection is at least α when aN = N−1/2....

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  • ...We thank Professor Davar Khoshnevisan for directing us to the paper of Anderson (1955). This research has been partially supported by NSF Grants DMS–1462067 and DMS–1305858....

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  • ...We thank Professor Davar Khoshnevisan for directing us to the paper of Anderson (1955)....

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Journal ArticleDOI
TL;DR: In this paper, the authors prove that a centered real-valued operator-scaling Gaussian random field with stationary increments satisfies a form of strong local nondeterminism and establish its exact uniform and local moduli of continuity.
Abstract: Let $X=\{X(t),t\in\mathrm{R}^{N}\}$ be a centered real-valued operator-scaling Gaussian random field with stationary increments, introduced by Bierme, Meerschaert and Scheffler (Stochastic Process. Appl. 117 (2007) 312–332). We prove that $X$ satisfies a form of strong local nondeterminism and establish its exact uniform and local moduli of continuity. The main results are expressed in terms of the quasi-metric $\tau_{E}$ associated with the scaling exponent of $X$. Examples are provided to illustrate the subtle changes of the regularity properties.

18 citations

01 Jan 2012
TL;DR: In this paper, the authors present a survey of small deviation probabilistic analysis of Gaussian measures with respect to the inequalities commonly used in small deviation theory and apply it to the law of iterated logarithm and quantization problems.
Abstract: of small deviation prob-abilitiesWe tried also to mention the articles that contain small deviation resultsimplicitly, eg those on entropy numbers of operators related to stochasticprocessesSome general theoretical works on Gaussian measures are also includedwhenever they treat inequalities widely used in small deviation theoryAlso included are the relevant surveys of small deviation theoryThe articles on applications of small deviation theory (eg to the lawsof iterated logarithm or to quantization problems) are NOT included unlessthey contain original small deviation results This could well be a subject ofseparate bibliographic search

18 citations

References
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Book
01 Jan 1953

10,512 citations

Journal ArticleDOI
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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BookDOI
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

Journal ArticleDOI
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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