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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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Book ChapterDOI
01 Jan 1965
TL;DR: In this article, a variety of interesting inequalities centering on the theme of matrices are discussed, and the basic concept of positive number can be extended to matrices in many different and significant ways.
Abstract: As soon as we leave the well-traversed fields of real and complex numbers for the broader and relatively unexplored domains of hyper-complex numbers, we open the way for the introduction of many different types of ordering relationships. In this chapter, we shall discuss a variety of interesting inequalities centering about the theme of matrices. As we shall see, the basic concept of positive number can be extended to matrices in many different and significant ways.

2 citations

Journal ArticleDOI
TL;DR: In this paper , the adaptive minimal grid neighbors algorithm (AMN) was proposed to compute the desired zero set of the Bargmann transform of a signal contaminated with complex white noise, or equivalently the computation of the zeros of its short-time Fourier transform with Gaussian window.
Abstract: Abstract We study the computation of the zero set of the Bargmann transform of a signal contaminated with complex white noise, or, equivalently, the computation of the zeros of its short-time Fourier transform with Gaussian window. We introduce the adaptive minimal grid neighbors algorithm (AMN), a variant of a method that has recently appeared in the signal processing literature, and prove that with high probability it computes the desired zero set. More precisely, given samples of the Bargmann transform of a signal on a finite grid with spacing $$\delta $$ δ , AMN is shown to compute the desired zero set up to a factor of $$\delta $$ δ in the Wasserstein error metric, with failure probability $$O(\delta ^4 \log ^2(1/\delta ))$$ O ( δ 4 log 2 ( 1 / δ ) ) . We also provide numerical tests and comparison with other algorithms.

2 citations

Journal ArticleDOI
TL;DR: In this paper, when a realization of the homogeneous Gaussian field belongs to the class Hpr(G)0, questions of approximation of functions are discussed, and the authors consider the class of functions that can be approximated by a Gaussian approximation.
Abstract: This paper is the continuation of [1], [2]. In it questions of approximation of functions are discussed, when a realization of the homogeneous Gaussian field belongs to the class Hpr(G)0.

2 citations

Journal ArticleDOI
TL;DR: In this paper, the optimality of the estimator θ(Qα ) when L(θ ) is the set of all normalized linear functions of θ and all elements in L( θ ) are of equal importance is investigated.
Abstract: We investigate the optimality of the estimator θ(Qα ) when L(θ ) is the set of all normalized linear functions of θ and all elements in L(θ ) are of “equal importance.”

2 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....

    [...]

  • ...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)....

    [...]

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....

    [...]

  • ...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....

    [...]