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.
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TL;DR: In this paper, the packing dimension of the graph of a Gaussian random field with continuous sample paths over a Borel map was shown to be upper bounded by a very mild assumption.
Abstract: Let $X=\{(X_1(t),\dots,X_d(t)): t\in \mathbb{R}^n\}$ be a Gaussian random field in $\mathbb{R}^d$ such that $X_1,\dots,X_d$ are independent, centered Gaussian random fields with continuous sample paths. Let $f\colon \mathbb{R}^n\to \mathbb{R}^d$ be a Borel map and let $A\subset \mathbb{R}^n$ be an analytic set. The main goal of the paper is to determine the almost sure value of the packing dimension of the image and graph of $X+f$ restricted to $A$ under a very mild assumption. This generalizes a result of Du, Miao, Wu and Xiao, who calculated the packing dimension of $X(A)$ if $X_1,\dots,X_d$ are independent copies of the same Gaussian random field $X_0$. Provided that $X$ is a fractional Brownian motion, our result is new even if $n=d=1$ and $f$ is continuous, and even if $f\equiv 0$ in the case of graphs.
For a fractional Brownian motion $X$ we also obtain the sharp lower bound for the packing dimension of the graph of $X$ over $A$ in terms of the Hurst index of $X$ and the packing dimension of $A$. The analogous result for images was obtained by Talagrand and Xiao.
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TL;DR: In this article, it was shown that the set of alternatives for which a test may improve upon the likelihood ratio test is always asymptotically negligible in a relative volume sense.
Abstract: To assess whether there is some signal in a big database, aggregate tests for the global null hypothesis of no effect are routinely applied in practice before more specialized analysis is carried out. Although a plethora of aggregate tests is available, each test has its strengths but also its blind spots. In a Gaussian sequence model, we study whether it is possible to obtain a test with substantially better consistency properties than the likelihood ratio (i.e., Euclidean norm based) test. We establish an impossibility result, showing that in the high-dimensional framework we consider, the set of alternatives for which a test may improve upon the likelihood ratio test -- that is, its superconsistency points -- is always asymptotically negligible in a relative volume sense.
24 Apr 2023
TL;DR: In this article , the authors present a simple formula to calculate the K-factor for any value of integrity risk and time interval, under the hypothesis of Gaussian error distribution, without any assumption on the correlation structure of the successive position estimates.
Abstract: The aviation Minimum Operational Performance Standard defines the SBAS protection levels as the product of the estimated standard deviation of the positioning error and a scaling factor called K-factor. The K-factor depends on the time window of interest and on the correlation between errors in the time window. The K-factors defined in aviation are difficult to generalize to other specifications in other domains, such as rail and maritime applications. This article presents a simple formula to calculate the K-factor for any value of integrity risk and time interval. The resulting K-factor is shown to be mathematically rigorous under the hypothesis of Gaussian error distribution but without any assumption on the correlation structure of the successive position estimates. The Gaussian assumption can be relaxed and replaced by overbounding with a Gaussian distribution with a very good approximation. This formula can be used in any GNSS application where integrity is needed.
TL;DR: In this article , an information-theoretic approach to study the Kneser-Poulsen conjecture in discrete geometry was developed, which leads to a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a 1-Lipschitz map.
Abstract: We develop an information‐theoretic approach to study the Kneser–Poulsen conjecture in discrete geometry. This leads us to a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a 1‐Lipschitz map. We answer this question affirmatively in various cases.
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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