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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01 Jan 1980
TL;DR: In this paper, the authors studied convexity properties of multivariate distribution functions, properties of certain probabilities of convex sets and related multivariate inequalities, leading to the study of the problem of maximization and optimization.
Abstract: Problems of maximization and optimization appear frequently in statistics. Such problems lead to the study of convexity properties of multivariate distribution functions, properties of certain probabilities of convex sets and related multivariate inequalities.
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TL;DR: In this paper, the residuals of the fit are considered in the goodness-of-fit process for VARMA (p,q) models and a new goodness of fit process based on a transformed correlationmatrix sequence is proposed.
Abstract: As an extension of the univariate technique in Ubierna and Velilla (2007), we present a goodness-of-fit process for VARMA (p,q) models in which the residuals of the fit are considered. We also formulatean explicit form of the asymptotic covariance function, as well as a suitable representation of the limitprocess. More importantly, we propose a new goodness-of-fit process based on a transformed correlationmatrix sequence. The new goodness-of-fit process is proved to converge weakly to the Brownian bridge.Several simulations, comparisons, and examples are presented. These results illustrate the scope of bothour theoretical findings and contributions. Our method is shown to be sensitive to detect lack of fit.Thus, it can be considered as a useful tool tool for identifying a proper time series model.
Cites background or methods from "The integral of a symmetric unimoda..."
...We extend to a multivariate setting the univariate goodness-of-fit process studied by Durlauf (1991) and Anderson (1993). We study weak convergence of this new method, whose application does not depend on the choice of a particular lag, and that uses a distribution that is free of unknown parameters....
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...This indicates that the consequences of an inequality in Anderson (1955) can be very severe in practice....
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TL;DR: In this article, the degree, iterates of the H-index, and the coreness measures of centrality were studied for random neighborhood graphs and it was shown that in the large-sample limit and under some standard condition on the connectivity radius, the degree converges to the likelihood depth.
Abstract: Given a sample of points in a Euclidean space, we can define a notion of depth by forming a neighborhood graph and applying a notion of centrality. In the present paper, we focus on the degree, iterates of the H-index, and the coreness, which are all well-known measures of centrality. We study their behaviors when applied to a sample of points drawn i.i.d. from an underlying density and with a connectivity radius properly chosen. Equivalently, we study these notions of centrality in the context of random neighborhood graphs. We show that, in the large-sample limit and under some standard condition on the connectivity radius, the degree converges to the likelihood depth (unsurprisingly), while iterates of the H-index and the coreness converge to new notions of depth.
TL;DR: In this article , the authors present a set of rules for the operation of a large-scale network of data centers, such as a network of sensors and communication networks, which they call "operatorname{\mathbf E}w(mathbf X-\mathbf v)/\operatourname{\Mathbf E}){to[0,\infty)
Abstract: Получены точные верхние и нижние грани для отношения $\operatorname{\mathbf E}w(\mathbf X-\mathbf v)/\operatorname{\mathbf E}w(\mathbf X)$ для центрированного гауссовского случайного вектора $\mathbf X$ в $\mathbf R^n$, а также оценки скорости изменения $\operatorname{\mathbf E}w(\mathbf X-t\mathbf v)$ по отношению к $t$, где $w\colon\mathbf R^n\to[0,\infty)$ - произвольная одновершинная функция и $\mathbf v$ - произвольный вектор в $\mathbf R^n$. В качестве следствия таких результатов даны точные верхние и нижние грани для функции мощности статистических критериев для математического ожидания многомерного нормального распределения.
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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