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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, it was shown that the probability that the Brownian sheet has a supremum at most at most ε(1/επσon) is of order ϵ(επ ϵ −2 ϵ/(log(1 /επsilon))^3 ).
Abstract: We show that the logarithm of the probability that the Brownian sheet has a supremum at most $\epsilon$ over $\lbrack 0, 1\rbrack^2$ is of order $\epsilon^{-2}(\log(1/\epsilon))^3$.

89 citations

Journal ArticleDOI
TL;DR: It is shown that an approximation of arbitrary distributions P on d-dimensional space by distributions with log-concave density exists if and only if P has finite first moments and is not supported by some hyperplane, and that this approximation depends continuously on P with respect to Mallows distance D 1.
Abstract: We study the approximation of arbitrary distributions $P$ on $d$-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback--Leibler-type functional. We show that such an approximation exists if and only if $P$ has finite first moments and is not supported by some hyperplane. Furthermore we show that this approximation depends continuously on $P$ with respect to Mallows distance $D_1(\cdot,\cdot)$. This result implies consistency of the maximum likelihood estimator of a log-concave density under fairly general conditions. It also allows us to prove existence and consistency of estimators in regression models with a response $Y=\mu(X)+\epsilon$, where $X$ and $\epsilon$ are independent, $\mu(\cdot)$ belongs to a certain class of regression functions while $\epsilon$ is a random error with log-concave density and mean zero.

87 citations


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

  • ...This follows from a more general inequality which is somewhat reminiscent of Anderson’s lemma [Anderson (1955)]: THEOREM 3.5....

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  • ...Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 170-176. Ayer, M., Brunk, H. D., Ewing, G. M., Reid, W. T. and Silverman, E. (1955). An empirical distribution function for sampling with incomplete information....

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  • ...Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities....

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  • ...Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 170-176. Ayer, M., Brunk, H. D., Ewing, G. M., Reid, W. T. and Silverman, E. (1955). An empirical distribution function for sampling with incomplete information. Ann. Math. Statist. 26 641-647. Bagnoli, M. and Bergstrom, T. (2005). Log-concave probability and its applications....

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  • ...Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 170-176. Ayer, M., Brunk, H. D., Ewing, G. M., Reid, W. T. and Silverman, E. (1955). An empirical distribution function for sampling with incomplete information. Ann. Math. Statist. 26 641-647. Bagnoli, M. and Bergstrom, T. (2005). Log-concave probability and its applications. Econometric Theory 26 445-469. Balabdaoui, F., Rufibach, K. and Wellner, J. A. (2009). Limit distribution theory for maximum likelihood estimation of a log-concave density....

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Journal ArticleDOI
TL;DR: In this paper, the authors study the approximation of arbitrary distributions P on d-dimensional space by distributions with log-concave density and show that such an approximation exists if and only if P has finite first moments and is not supported by some hyperplane.
Abstract: We study the approximation of arbitrary distributions P on d-dimensional space by distributions with log-concave density. Approximation means minimizing a Kullback―Leibler-type functional. We show that such an approximation exists if and only if P has finite first moments and is not supported by some hyperplane. Furthermore we show that this approximation depends continuously on P with respect to Mallows distance D 1 (·, ·). This result implies consistency of the maximum likelihood estimator of a log-concave density under fairly general conditions. It also allows us to prove existence and consistency of estimators in regression models with a response Y = μ(X) + e, where X and e are independent, μ(·) belongs to a certain class of regression functions while e is a random error with log-concave density and mean zero.

87 citations

Journal ArticleDOI
TL;DR: In this paper, a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body was proved for the case of convex bodies, where the curvature measure is defined by a convex convex manifold.
Abstract: We prove a tight subspace concentration inequality for the dual curvature measures of a symmetric convex body. © 2018 International Press of Boston, Inc. All Rights Reserved.

86 citations


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

  • ...It was shown by Anderson [1] that the integral of an even unimodal function over translates of a symmetric convex region is maximal if the center of symmetry is moved to the origin....

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  • ...If q ∈ [1, 2), then an even finite Borel measure μ on S1 is a qth dual curvature measure if and only if μ(S1 ∩ L) μ(S1) < 1 q for every one-dimensional subspace L of R2....

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  • ...It was shown by Anderson [1] that the integral of...

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Journal ArticleDOI
TL;DR: In this paper, the authors consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior, in particular when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter H 2.
Abstract: We consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter H 2 ( 1 , 1 ). Our contribution in this work is twofold. First, when the driving vector fields satisfy Hormander's c elebrated "Lie bracket condition", we derive explicit quantitative bounds on the i nverse of the Malliavin ma- trix. En route to this, we provide a novel "deterministic" version of Norris's lemma for differential equations driven by rough paths. This result, with the added assumption that the linearized equation has moments, will then yield that the transition laws have a smooth density with respect to Lebesgue measure. Our second main result states that under Hormander's condi tion, the solutions to rough differential equations driven by fractional Brownian motion with H 2 ( 1 , 1 ) enjoy a suitable version of the strong Feller property. Under a standard controllability condition, this implies that they admit a unique stationary solution that is physical in the sense that it does not "look into the future".

81 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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