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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 studied the small deviation problem for a class of symmetric Levy processes, namely, subordinated Levy processes and gave precise estimates (up to a constant multiple in the logarithmic scale) under some mild general assumption.

33 citations

Journal ArticleDOI
TL;DR: In this article, the effect of ambiguous strategies in multidimensional models of policy formation has been studied and some new results on the effects of ambiguities on the general social choice framework have been proved.
Abstract: This paper summarizes some old results and proves some new results on the effects of ambiguous strategies in multidimensional models of policy formation. The effect of lotteries on social choice has been studied in the general social choice framework by Fishburn (1973: 251) and specifically in the context of the spatial model by Shepsle (1970a, 1970b, 1972), Zeckhauser (1969) and McKelvey and Richelson (1974). Here we continue the later approach and consider ambiguity in the context of the spatial model. Thus, the alternative space is R n, as in the usual spatial model, but instead of assuming candidates adopt fixed positions in Rn, it is assumed they may adopt a lottery, or probability density over the space. After introduction of basic notation, Section 2 defines reasonable classes of probability measures from which candidates can select their ambiguous platforms, defines individual utility functions over these ambiguous strategies, and summarizes some known results about ambiguous strategies. The classes of allowable probability measures are generated by assuming that if two measures are in the class, then certain mixtures of these measures are also in the class. Two types of mixtures are defined: probability mixtures and spatial mixtures. By looking at individual preferences for these mixtures, it is seen that there are important differences between the two. If two candidates initially adopt two distinct ambiguous strategies, it is never advantageous for one candidate to converge towards his opponent using a strategy which is probabilistic mixture of their initial position. It may frequently be advantageous to converge towards an opponent using a spatial mixture, however. Completeness of families of distributions under probability mixtures leads to the results of Zeckhauser and Fishburn that nondegenerate lotteries can never be in equilibrium. Completeness under spatial mixtures leads to the theorems on convergence of McKelvey and Richelson. Section 3 studies, in greater detail, the nature of individual preferences over ambiguous strategies. Assuming that individuals evaluate lotteries in

33 citations

Journal ArticleDOI
TL;DR: In this paper, a new class of functions called ''alpha$-quasi-concave'' is introduced and it is proved that the convolution of two unimodal functions each having some additional concavity properties belongs to this class.
Abstract: In the note a new class of functions called $\alpha$-quasi-concave is introduced and it is proved that the convolution of two unimodal functions each having some additional concavity properties belongs to this class. Other results concerning the convolution of unimodal functions are also studied and the extensions of some of them are also given in the paper.

33 citations

Journal ArticleDOI
TL;DR: It is shown that, under the assumption of a Wiener-like probability distribution on the class of singular functions, an adaptive algorithm can locate a singular point accurately with only a small probability of failure.
Abstract: We study from a probabilistic viewpoint the problem of locating singularities of functions using function evaluations. We show that, under the assumption of a Wiener-like probability distribution on the class of singular functions, an adaptive algorithm can locate a singular point accurately with only a small probability of failure. As an application, we show that an integration algorithm that adaptively locates a singular point is probabilistically superior to nonadaptive algorithms.

32 citations


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

  • ...are independent Gaussian random vectors, each with zero mean, Anderson's inequality (see [1]) implies that Probz({|X,(/; A)| < T(h): V; = /,....

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Book ChapterDOI
TL;DR: In this paper, the normalized Lebesgue measure on S n −1 is defined as the probability measure of a harmonic function whose radial limits are equal to μ-almost everywhere to f.
Abstract: Let μ be the normalized Lebesgue measure on S n −1. For x = (x 1,…,x n ) with ||x||2 < 1 we denote by μ x the probability measure on S n −1 given by \( \frac{{1 - {{\left\| x \right\|}^2}}} {{{{\left\| {y - x} \right\|}^n}}}d\mu \left( y \right) \). We recall that if f is an integrable function on S n -1 then u(x) = \( u(x) = {\int_{{S^{n - 1}}} {f(y)d\mu }^x}(y) \) is a harmonic function whose radial limits are equal μ-almost everywhere to f.

32 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)....

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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....

    [...]

  • ...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....

    [...]