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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, a Chung-type law of the iterated logarithm for a large class of weight functions was proved for the case where ψ is a weight function and ψ(t) is a standard Brownian motion.

2 citations

01 Jan 2007
TL;DR: In this paper, the authors propose a solution to solve the problem of the problem: this paper.v.v.s.q.vq.qqq q.
Abstract: v

2 citations


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

  • ...Then for any fixed r > 0, n‖θ̂r,n − θ0‖ = Op(1), n →∞....

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  • ...→ convergence almost surely p → convergence in probability d → convergence in distribution Θ parameter space Z observations space 1A the indicator function of the set A P probability measure Pn empirical probability measure of observations, Pn(A) = 1 n ∑n i=1 1{Zi ∈ A} E expectation with respect to P Zn = Op(1) if for all 2 > 0 there exist a constant M2 for which P (|Zn| ≥ M2) < 2 Zn = op(1) if Zn p → 0...

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  • ...If we take δ = 1/Rn, then the solution of R 2 n( 1 Rn ) ≤ √n, is Rn ≤ n, and we conclude that n‖θ̂r,n − θ0‖ = Op(1)....

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01 Jan 1979
TL;DR: A new method for estimating yield and the gradient of yield is developed based upon a combination of the Simplicial Approximation technique of Director and Hachtel and the yield estimation procedure of Bandler and Abdel-Malek.
Abstract: A number of recent papers have described circuit optimization methods y in which maximizing yield was the sole design criterion. However, in actual practice there are many competing design criterion such as minimizing power, y maximizing speed, area, etc., as well as maximizing yield. In this paper we use the techniques of Multiple Criterion Optimization (MCO) to provide a framework within which to consider all of these objectives simultaneously. Towards this end we develop a new method for estimating yield and the gradient of yield. This method is based upon a combination of the Simplicial Approximation technique of Director and Hachtel and the yield estimation procedure of Bandler and Abdel-Malek. The ideas of MCO and the new yield estimation procedure are applied to the design of a two-input MOSFET NAND gate.

2 citations

Dissertation
24 Sep 2018
TL;DR: This thesis investigates the problem of the inference of the structure of an undirected graphical model by a multiple testing procedure, and presents some multiple testing procedures, well-suited for correlation tests, which provide an asymptotic control of the FWER.
Abstract: This thesis is motivated by the analysis of the functional magnetic resonance imaging (fMRI). The need for methods to build such structures from fMRI data gives rise to exciting new challenges for mathematics. In this regards, the brain connectivity networks are modelized by a graph and we study some procedures that allow us to infer this graph.More precisely, we investigate the problem of the inference of the structure of an undirected graphical model by a multiple testing procedure. The structure induced by both the correlation and the partial correlation are considered. The statistical tests based on the latter are known to be highly dependent and we assume that they have an asymptotic Gaussian distribution. Within this framework, we study some multiple testing procedures that allow a control of false edges included in the inferred graph.First, we theoretically examine the False Discovery Rate (FDR) control of Benjamini and Hochberg’s procedure in Gaussian setting for non necessary positive dependent statistical tests. Then, we explore both the FDR and the Family Wise Error Rate (FWER) control in asymptotic Gaussian setting. We present some multiple testing procedures, well-suited for correlation (resp. partial correlation) tests, which provide an asymptotic control of the FWER. Furthermore, some first theoretical results regarding asymptotic FDR control are established.Second, the properties of the multiple testing procedures that asymptotically control the FWER are illustrated on a simulation study, for statistical tests based on correlation. We finally conclude with the extraction of cerebral connectivity networks on real data set.

2 citations


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

  • ...The Šidák’s inequality relies on a theorem of Anderson (Corollary 2 in [73]), which generalizes to the dimension m the following intuitive result: Let f : R→ R be a density function such that 1....

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Journal ArticleDOI
TL;DR: In this article, the estimation problem for a non-linear stochastic model with control is studied, where the drift of the model depends on the value of an unknown parameter and a control which is a function of the preceding observations.
Abstract: The estimation problem for a non-linear stochastic model with control is studied. We suppose that the drift of the model depends on the value of an unknown parameter and a control which is a function of the preceding observations. The goal is to construct strategies including a control rule, a stopping time and an estimator of the parameter that are optimal in the minimax sense. The exhaustive solution of this problem is given for the particular case when the drift is a linear function of the parameter. For the general case the asymptotic approach is used when the average observation time tends to infinity. Lower bounds of asymptotic minimax risk are stated for arbitrary sequential strategies. Then an explicit construction of asymptotically optimal strategies is given. These are so called “two-stage strategies”. At the first stage one can choose any non-degenerate plan: the purpose of the first stage is only to construct a preliminary consistent estimator. A design for the second stage (taking most of the...

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....

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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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