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, it was shown that if q = [ q 1, …, q n ]′ majorizes p = [ p 1, p 1, …, p n ]', then q is more peaked about μ than p in the sense of Birnbaum (1948) and p is peaked about σ 2.
12 citations
Posted Content•
TL;DR: An efficient procedure is developed that achieves weak recovery down to the information-theoretically optimal threshold of the minimax risk in the high-dimensional limit of m,n\to-infty.
Abstract: We consider the problem of estimating an unknown matrix $\boldsymbol{X}\in {\mathbb R}^{m\times n}$, from observations $\boldsymbol{Y} = \boldsymbol{X}+\boldsymbol{W}$ where $\boldsymbol{W}$ is a noise matrix with independent and identically distributed entries, as to minimize estimation error measured in operator norm. Assuming that the underlying signal $\boldsymbol{X}$ is low-rank and incoherent with respect to the canonical basis, we prove that minimax risk is equivalent to $(\sqrt{m}\vee\sqrt{n})/\sqrt{I_W}$ in the high-dimensional limit $m,n\to\infty$, where $I_W$ is the Fisher information of the noise. Crucially, we develop an efficient procedure that achieves this risk, adaptively over the noise distribution (under certain regularity assumptions).
Letting $\boldsymbol{X} = \boldsymbol{U}{\boldsymbol{\Sigma}}\boldsymbol{V}^{\sf T}$ --where $\boldsymbol{U}\in {\mathbb R}^{m\times r}$, $\boldsymbol{V}\in{\mathbb R}^{n\times r}$ are orthogonal, and $r$ is kept fixed as $m,n\to\infty$-- we use our method to estimate $\boldsymbol{U}$, $\boldsymbol{V}$. Standard spectral methods provide non-trivial estimates of the factors $\boldsymbol{U},\boldsymbol{V}$ (weak recovery) only if the singular values of $\boldsymbol{X}$ are larger than $(mn)^{1/4}{\rm Var}(W_{11})^{1/2}$. We prove that the new approach achieves weak recovery down to the the information-theoretically optimal threshold $(mn)^{1/4}I_W^{1/2}$.
12 citations
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TL;DR: This paper first proves that post-selection inference is equivalent to simultaneous inference and then construct valid post- selection confidence regions which are computationally simple and apply to independent as well as dependent random variables without the requirement of correct distributional assumptions.
Abstract: Construction of valid statistical inference for estimators based on data-driven selection has received a lot of attention in the recent times. Berk et al. (2013) is possibly the first work to provide valid inference for Gaussian homoscedastic linear regression with fixed covariates under arbitrary covariate/variable selection. The setting is unrealistic and is extended by Bachoc et al. (2016) by relaxing the distributional assumptions. A major drawback of the aforementioned works is that the construction of valid confidence regions is computationally intensive. In this paper, we first prove that post-selection inference is equivalent to simultaneous inference and then construct valid post-selection confidence regions which are computationally simple. Our construction is based on deterministic inequalities and apply to independent as well as dependent random variables without the requirement of correct distributional assumptions. Finally, we compare the volume of our confidence regions with the existing ones and show that under non-stochastic covariates, our regions are much smaller.
12 citations
Cites background from "The integral of a symmetric unimoda..."
...Hence by Anderson’s Lemma (Corollary 3 of Anderson (1955)), for all A P Asre, P ` SG‹n P A ˘ ď P ` SGn P A ˘ ....
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TL;DR: In this article, conditions are given such that and/or obtain these conditions imply that chi-squared random variables defined from a multivariate normal distribution are always positively dependent and nonnegatively correlated.
Abstract: Suppose Y′ = (Y′ 1, …, Y′ k ) possesses a multivariate normal distribution with mean vector 0 and positive semidefinite covariance matrix Σ If Ci ⊂ Rpi denote convex regions symmetric about the origin, then conditions are given such that and/or obtain These conditions imply that chi-squared random variables defined from a multivariate normal distribution are always positively dependent and nonnegatively correlated Other applications involve conservative simultaneous confidence regions in a multivariate regression setting
11 citations
Cites background from "The integral of a symmetric unimoda..."
...Moreover, since the distribution of Q is invariant if multiplied on the left by a fixed orthogonal matrix, Q must possess the Haar invariant distribution discussed in Anderson (1958). Since the conditional distribution of Y'2Y2 is (except for a constant) a noncentral chi-squared distribution with noncentrality parameter I Z 12, and hence free of Q, we may express the quantity in (2....
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...This lemma is from Anderson (1955) and is well known....
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TL;DR: In this article, the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics for spherical mixed p-spin disordered mean-field models, starting uniformly within one of the spherical bands on which the Gibbs measure concentrates at low temperature for the pure P-spin models and mixed perturbations of them, was derived.
Abstract: We derive the thermodynamic limit of the empirical correlation and response functions in the Langevin dynamics for spherical mixed p-spin disordered mean-field models, starting uniformly within one of the spherical bands on which the Gibbs measure concentrates at low temperature for the pure p-spin models and mixed perturbations of them. We further relate the large time asymptotics of the resulting coupled non-linear integro-differential equations, to the geometric structure of the Gibbs measures (at low temperature), and derive their FDT solution (at high temperature).
11 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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