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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 first-order Edgeworth expansion of a functionals with skew-symmetric influence curve is asymptotically minimax.
Abstract: Let $X_1, X_2, \cdots, X_n$ be i.i.d random variables with d.f. $F$. Suppose the $\{\hat{T}_n = \hat{T}_n(X_1, X_2, \cdots, X_n); n \geq 1\}$ are real-valued statistics and the $\{T_n(F); n \geq 1\}$ are centering functionals such that the asymptotic distribution of $n^{1/2}\{\hat{T}_n - T_n(F)\}$ is normal with mean zero. Let $H_n(x, F)$ be the exact d.f. of $n^{1/2}\{\hat{T}_n - T_n(F)\}$. The problem is to estimate $H_n(x, F)$ or functionals of $H_n(x, F)$. Under regularity assumptions, it is shown that the bootstrap estimate $H_n(x, \hat{F}_n)$, where $\hat{F}_n$ is the sample d.f., is asymptotically minimax; the loss function is any bounded monotone increasing function of a certain norm on the scaled difference $n^{1/2}\{H_n(x, \hat{F}_n) - H_n(x, F)\}$. The estimated first-order Edgeworth expansion of $H_n(x, F)$ is also asymptotically minimax and is equivalent to $H_n(x, \hat{F}_n)$ up to terms of order $n^{- 1/2}$. On the other hand, the straightforward normal approximation with estimated variance is usually not asymptotically minimax, because of bias. The results for estimating functionals of $H_n(x, F)$ are similar, with one notable difference: the analysis for functionals with skew-symmetric influence curve, such as the mean of $H_n(x, F)$, involves second-order Edgeworth expansions and rate of convergence $n^{-1}$.

125 citations

Journal ArticleDOI
TL;DR: In this paper, Siotani et al. improved the Bonferroni inequality in the context of normal variates and obtained shorter confidence bounds on variances and on a given set of linear functions of location parameters when this set is previously chosen for study.
Abstract: In practical situations, one is generally faced with multivariate problems in the form of testing the hypotheses or obtaining a set of simultaneous confidence bounds on certain parameters of interest. We shall consider here the variates under study to be normally distributed. A lot of work on the univariate and multivariate normal populations for the simultaneous confidence bounds on the location and scale parameters has been done, (see references, not necessarily exhaustive). Establishing certain inequalities for normal variates, we try to give shorter confidence bounds on variances and on a given set of linear functions of location parameters when this set is previously chosen for study. For the univariate case, Dunn [6], [8] using the Bonferroni inequality, obtained shorter confidence bounds when the number of linear functions is not too large. We may note that Nair [12], David [5], Dunn [6], [7], [8] and Siotani [22], [24] have studied the closeness of the Bonferroni inequality while deriving the percentage points of certain statistics in univariate and multivariate normal cases. In this paper, we improve the Bonferroni inequality in all the situations considered by Siotani [22], [23], [24] and Dunn [6], [7], [8], and point out various uses of these results in obtaining simultaneous confidence bounds on variances and on linear functions of means (or location parameters) with confidence greater than or equal to $1 - \alpha$ where $\alpha$ is the size of the test. We mention our main results in Section 2 for those who are interested in results and not in proofs. Since our results are extensions of Dunn [6], [8], Siotani [22], [24] and Banerjee [2], [3], their comments on the shortness of the confidence bounds apply to our cases too.

125 citations

Journal ArticleDOI
TL;DR: A general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems.
Abstract: We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call \emph{Optimal Uncertainty Quantification} (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions they have finite-dimensional reductions. As an application, we develop \emph{Optimal Concentration Inequalities} (OCI) of Hoeffding and McDiarmid type. Surprisingly, these results show that uncertainties in input parameters, which propagate to output uncertainties in the classical sensitivity analysis paradigm, may fail to do so if the transfer functions (or probability distributions) are imperfectly known. We show how, for hierarchical structures, this phenomenon may lead to the non-propagation of uncertainties or information across scales. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact and on the seismic safety assessment of truss structures, suggesting the feasibility of the framework for important complex systems. The introduction of this paper provides both an overview of the paper and a self-contained mini-tutorial about basic concepts and issues of UQ.

120 citations

Journal ArticleDOI
TL;DR: Theorem 2. as mentioned in this paper is a generalization of Anderson's Theorem and has been used for many applications in probability and statistics, e.g. to give different statistical applications.
Abstract: T. W. Anderson [1] has proved the following theorem and has given applications to probability and statistics. THEOREM 1. Let $E$ be a convex set in $n$-space, symmetric about the origin. Let $f(x) \geqq 0$ be a function such that i) $f(x) = f(-x)$, ii) $\{x |f(x) \geqq u\} = K_u$ is convex for every $u (0 \geqq u \geqq \infty)$ and iii) $\int_E f(x) dx < \infty$, then$ \begin{equation*}\tag{(1)}\int_E f(x + ky) dx \geqq \int_E f(x + y) dx\quad for 0 \leqq k \leqq 1.$ The purpose of this paper is to prove what can be considered a generalization of Anderson's Theorem and to give different statistical applications. Functions in $L_1$ satisfying the hypothesis were called unimodal by Anderson and he noted in [1] that if we let $\varphi(y)$ be equal to the right hand side of (1) then $\varphi$ is not unimodal in his sense insofar as it does not necessarily satisfy $ii$ (i.e., there exist $f, E$, and $u$ such that $\{x\mid \varphi(x) \geqq u\}$ is not convex). His example is the case where $n = 2$ and \begin{equation*}f(x) = \begin{cases}3,\quad\|x_1\| \leqq 1,\quad\|x_2\| \leqq 1, \\ 2,\quad\|x_1\| \leqq 1,\quad 1 < \|x_2\| \leqq 5, \\ 0,\quad {other} x,\end{cases}\end{equation*} where $x_1, x_2$ are the components of $x$ relative to rectangular cartesian coordinate system. Let $E$ be the set of vectors where $|x_1| \leqq 1, |x_2| \leqq 1$. The set $\{x \mid \varphi(x) \geqq 6\}$ is not convex since for $x = (.5,4)$ and $x = (1,0), \varphi(x) = 6$, while for $x = (.75.2), \varphi (x) < 6$. The point of departure of this paper is to see what can be said about $\varphi$. This is achieved in Theorem 2, giving a stronger and more symmetrical statement than Anderson makes (but one which does not yield more information for his applications). The main Lemma, presented below, proceeds along the line of his argument but squeezes out additional information (convexity of level lines) under a weaker hypothesis (no symmetry assumptions) than Anderson uses at the corresponding stage of his argument.

120 citations

01 Jan 2010
TL;DR: In this paper, a survey of the theory of γ-radonifying operators and its applications to stochastic integration in Banach spaces is presented, with a focus on the application of the γradonification operator to the integration problem.
Abstract: We present a survey of the theory of γ-radonifying operators and its applications to stochastic integration in Banach spaces.

118 citations


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

  • ...A Banach space E is said to have type p ∈ [1, 2] if there exists a constant Cp > 0 such that for all finite sequences x1, ....

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  • ...6(1) can be given for Banach spaces E having type p ∈ [1, 2]....

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  • ...Theorem 8.2 (Anderson)....

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  • ...As an application of Anderson’s inequality we have the following comparison result for E-valued Gaussian random variables (see Neidhardt [93, Lemma 28])....

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  • ...Hence, by Fubini’s theorem and Anderson’s inequality, P{X2 ∈ C} = P̃{X̃1 + X̃3 ∈ C} 6 P{X1 ∈ C}....

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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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