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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TL;DR: In this article, the authors prove lower bounds for the probability that a multi-parameter Gaussian process has very small values and apply them to a certain class of fractional Brownian sheets.
Abstract: We prove some general lower bounds for the probability that a multi-parameter Gaussian process has very small values. These results, when applied to a certain class of fractional Brownian sheets, yield the exact rate for their so-called small ball probability. We show by example how to use such results to compute the Hausdorff dimension of some exceptional sets determined by maximal increments.
43 citations
TL;DR: In this paper, nonparametric notions of multivariate scatter measures and more scattered measures based on statistical depth functions are investigated, with emphasis on those based on the halfspace depth.
42 citations
Cites background from "The integral of a symmetric unimoda..."
...By Theorem 1 of Anderson (1955) (also see Mudholker(1966) and Anderson (1996)), we have thatP (X + Y 2 Hx) = ZH(x)0 (P (X 2 Hx + t) + P (X 2 Hx + t)) dFY (t) ZH(x)0 (P (X 2 Hx + at) + P (X 2 Hx + at)) dFY (t)= ZRd P (X 2 Hx at) dFY (t)= P (X + aY 2 Hx):Hence > P (X + aY 2 Hx), contradicting the…...
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...By Theorem 1 of Anderson (1955), thelatter function of t is an increasing function of t1....
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TL;DR: In this paper, a sharp bound on the small ball probability of a Gaussian process with mean zero and stationary increments is given, where σ2(h) = EX2(H) is a Gaussian process with concave concave surface.
Abstract: Let {X(t), 0≤t≤1} be a Gaussian process with mean zero and stationary increments. Let σ2(h) =EX
2(h) be nondecreasing and concave on (0,1). A sharp bound on the small ball probability ofX(·) is given in this paper.
42 citations
Cites background from "The integral of a symmetric unimoda..."
...Px=xI{YeA} >>-Pr=x2{YeA} This is the Anderson ( 2 ) inequality (cf....
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TL;DR: In this article, the maximum local times for the Brox and Sinai processes were studied and a somewhat peculiar asymptotic behaviour was observed. But this was only in discrete and continuous time settings.
42 citations
TL;DR: In this paper, omnibus specification tests of parametric dynamic quantile models are proposed, where a possible continuum of quantiles is simultaneously specified under fairly weak conditions on the serial dependence in the underlying data-generating process.
42 citations
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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