The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities

doi:10.1090/S0002-9939-1955-0069229-1

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The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities

Journal Article•DOI•

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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Journal Article•DOI•

Cluster sets for a generalized law of the iterated logarithm in banach spaces

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Uwe Einmahl¹, J. Kuelbs•Institutions (1)

Vrije Universiteit Brussel¹

01 Oct 2001-Annals of Probability

TL;DR: In this article, the authors identify the possible cluster sets for a general law of the iterated logarithm in the Banach space and show that all the possible limit sets arise as cluster sets.

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Abstract: We identify the possible cluster sets for a general law of the iterated logarithm in the Banach space setting, and show that all the possible limit sets arise as cluster sets for some random vector in an arbitrary separable Banach space. This extends previous results obtained in finite dimensional Euclidean spaces.

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16 citations

Journal Article•DOI•

A New General Method for Constructing Confidence Sets in Arbitrary Dimensions: With Applications

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A. DasGupta, J. K. Ghosh, M. M. Zen

01 Aug 1995-Annals of Statistics

TL;DR: In this paper, the Camp-Meidell inequality on the Minkowski functional of an arbitrary star-shaped set $S$ is used to obtain size-efficient sets, and conditions are obtained under which this method reproduces a level (high density) set.

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Abstract: Let $\mathbf{X}$ have a star unimodal distribution $P_0$ on $\mathbb{R}^p$. We describe a general method for constructing a star-shaped set $S$ with the property $P_0(\mathbf{X} \in S) \geq 1 - \alpha$, where $0 < \alpha < 1$ is fixed. This is done by using the Camp-Meidell inequality on the Minkowski functional of an arbitrary star-shaped set $S$ and then minimizing Lebesgue measure in order to obtain size-efficient sets. Conditions are obtained under which this method reproduces a level (high density) set. The general theory is then applied to two specific examples: set estimation of a multivariate normal mean using a multivariate $t$ prior and classical invariant estimation of a location vector $\mathbf{\theta}$ for a mixture model. In the Bayesian example, a number of shape properties of the posterior distribution are established in the process. These results are of independent interest as well. A computer code is available from the authors for automated application. The methods presented here permit construction of explicit confidence sets under very limited assumptions when the underlying distributions are calculationally too complex to obtain level sets.

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16 citations

Journal Article•DOI•

Singular values of Gaussian matrices and permanent estimators

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Mark Rudelson¹, Ofer Zeitouni²•Institutions (2)

University of Michigan¹, Courant Institute of Mathematical Sciences²

01 Jan 2016-Random Structures and Algorithms

TL;DR: It is proved that for a large class of graphs satisfying an appropriate expansion property, the Barvinok–Godsil-Gutman estimator for the permanent achieves sub-exponential errors with high probability.

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Abstract: We present estimates on the small singular values of a class of matrices with independent Gaussian entries and inhomogeneous variance profile, satisfying a broad-connectedness condition. Using these estimates and concentration of measure for the spectrum of Gaussian matrices with independent entries, we prove that for a large class of graphs satisfying an appropriate expansion property, the Barvinok–Godsil-Gutman estimator for the permanent achieves sub-exponential errors with high probability. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 48, 183–212, 2016

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16 citations

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

...Lemma 4.3. Let m,n ∈ N, n ≤ 2m. Let A,B be (possibly random) n × m matrices, and set W = A ⊙ G + B, where G is the standard n× m Gaussian matrix, independent of A,B. Assume that, a.s., (1) ai,j ∈ {0}∪[r,1] for some constant r > 0 and all i,j; (2) the graph ΓAT is broadly connected. Then for any c0 and any u,v > 0, such that u ≥ c0 and (1+κ/2)u < 1, and for any z ∈ Rn P ∃x ∈ Comp((1+κ/2)u,(v/K)...
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...nd a set of rows of a ﬁxed matrix with big ℓ2 norms, provided that the graph of the matrix has a large minimal degree. Lemma 6.1. Let k < n, and let A be an n ×n matrix. Assume that (1) ai,j ∈ {0}∪[r,1] for some constant r > 0 and all i,j; (2) the graph ΓA satisﬁes deg(j) ≥ δn for all j ∈ [n]. Then for any J ⊂ [n] there exists a set I ⊂ [n] of cardinality |I| ≥ (r2δ/2)n, such that for any i ∈ I X...
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... ball estimate. Lemma 4.1. Let m,n ∈ N. Let A,B be (possibly random) n × m matrices, and let W = A ⊙ G + B, where G is the n × m Gaussian matrix, independent of A,B. Assume that, a.s., (1) ai,j ∈ {0}∪[r,1] for some constant r > 0 and all i,j; (2) the graph ΓA satisﬁes deg(j) ≥ δn for all j ∈ [m]. Then for any x ∈ Sm−1, z ∈ Rn and for any t > 0 P(kWx −zk2 ≤ t √ n) ≤ (Ct)cn. Proof. Let x ∈ Sm−1. Se...
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... ≤ n ≤ 2m. Let A,B be an n×m matrices, and set W = A ⊙ G + B, where G is the standard n × m Gaussian matrix, independent of A,B. Assume that SINGULAR VALUES AND PERMANENT ESTIMATORS 15 (1) ai,j ∈ {0}∪[r,1] for some constant r > 0 and all i,j; (2) the graph ΓAT is broadly connected. Then for all z ∈ Rn P ∃x ∈ Comp(1−κ/2,K−C) : kWx− zk2 ≤ K −C √ n and kWk ≤ K √ n ≤ e cn. Proof. Set u0 = c0, v0 = c...
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...t will be useful for the application of the theorem in the proof of Theorem 2.7 Theorem 2.3. Let W be an n × n matrix with independent normal entries wi,j ∼ N(bi,j,a2 i,j). Assume that (1) ai,j ∈ {0}∪[r,1] for some constant r > 0 and all i,j; (2) the graph ΓA is broadly connected; (3) kEWk ≤ K √ n for some K ≥ 1. Then for any t > 0 P(sn(W) ≤ ctK−Cn−1/2) ≤ t+e−c ′n. Theorem 2.4. Let n/2 < m ≤ n...
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Journal Article•DOI•

Optimality of the least squares estimator

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Robert H. Berk¹, Jiunn T Hwang¹•Institutions (1)

Rutgers University¹

01 Aug 1989-Journal of Multivariate Analysis

TL;DR: In this paper, the optimality of the least squares estimator under assuptions stronger than those for the Gauss-Markov theorem is explored. And the importance of these results for risk functions is also discussed.

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16 citations

Journal Article•DOI•

A new probabilistic approach to the path criticality in stochastic PERT

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Davaadorjin Monhor¹•Institutions (1)

University of West Hungary¹

01 Dec 2011-Central European Journal of Operations Research

TL;DR: A probabilistic background theory for univariate and bivariate marginal distributions of path durations of stochastic PERT whose joint path d duration are modelled by multivariate normal distribution is developed and based on this comparison the concept of probabilistically critical path as a Stochastic counterpart of the deterministic critical path is defined.

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Abstract: The notion of critical path is a key issue in the temporal analysis of project scheduling in deterministic setting. The very essence of the CPM consists in identifying the critical path, i.e., the longest path in a project network, because this path conveys information on how long it should take to complete the project to the project manager. The problem how can a stochastic counterpart of the deterministic critical path be defined is an important question in stochastic PERT. However, in the literature of stochastic PERT this question has so far almost been ignored, and the research into the random nature of a project duration has mainly been concentrated on the completion time in stochastic PERT in which any concrete special path is not specified. In the present paper we attempt to take first steps to fill this gap. We first developed a probabilistic background theory for univariate and bivariate marginal distributions of path durations of stochastic PERT whose joint path durations are modelled by multivariate normal distribution. Then, a new probabilistic approach to the comparison of path durations is introduced, and based on this comparison we define the concept of probabilistically critical path as a stochastic counterpart of the deterministic critical path. Also, an illustrative simple example of PCP and numerical results on the established probability bounds are presented.

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16 citations

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

...Furthermore, as an one dimensional special case of the celebrated theorem of Anderson (Anderson 1955) on multidimensional convex symmetric set, the probability of any interval which is symmetric about the expectation is increases as variance decreases....
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...We do not need to prove it, because it is well-known from probability theory and theory of multivariate statistical analysis (for example, Anderson 1986)....
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References

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Book•

Stochastic processes

[...]

J. L. Doob, Joseph L. Doob

01 Jan 1953

10,512 citations

Journal Article•DOI•

Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes

[...]

T. W. Anderson, Donald Allan Darling

01 Jan 1952-Annals of Mathematical Statistics

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.

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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

...read moreread less

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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Journal Article•DOI•

Table for Estimating the Goodness of Fit of Empirical Distributions

[...]

N. Smirnov

01 Jun 1948-Annals of Mathematical Statistics

1,660 citations

Book•DOI•

Theorie der Konvexen Körper

[...]

T. Bonnesen, W. Fenchel

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.

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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.

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927 citations

Journal Article•DOI•

The Cramer-Smirnov Test in the Parametric Case

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D. A. Darling

01 Mar 1955-Annals of Mathematical Statistics

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].

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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.

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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....
[...]
...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....
[...]