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: A new deconvolution-based method for estimating unknown sparsity, which has wider applicability and sharper theoretical guarantees is offered, which handles measurement noise with infinite variance and yields confidence intervals for s_q(x) with asymptotically exact coverage probability.
Abstract: The theory of Compressed Sensing asserts that an unknown signal $x\in\mathbb{R}^p$ can be accurately recovered from an underdetermined set of $n$ linear measurements with $n\ll p$, provided that $x$ is sufficiently sparse. However, in applications, the degree of sparsity $\|x\|_0$ is typically unknown, and the problem of directly estimating $\|x\|_0$ has been a longstanding gap between theory and practice. A closely related issue is that $\|x\|_0$ is a highly idealized measure of sparsity, and for real signals with entries not exactly equal to 0, the value $\|x\|_0=p$ is not a useful description of compressibility. In our previous conference paper that examined these problems, Lopes 2013, we considered an alternative measure of "soft" sparsity, $\|x\|_1^2/\|x\|_2^2$, and designed a procedure to estimate $\|x\|_1^2/\|x\|_2^2$ that does not rely on sparsity assumptions.
The present work offers a new deconvolution-based method for estimating unknown sparsity, which has wider applicability and sharper theoretical guarantees. Whereas our earlier work was limited to estimating the quantity $\|x\|_1^2/\|x\|_2^2$, the current paper introduces a family of entropy-based sparsity measures $s_q(x):=\big(\frac{\|x\|_q}{\|x\|_1}\big)^{\frac{q}{1-q}}$ parameterized by $q\in[0,\infty]$. Two other main advantages of the new approach are that it handles measurement noise with infinite variance, and that it yields confidence intervals for $s_q(x)$ with asymptotically exact coverage probability (whereas our previous intervals were conservative). In addition to confidence intervals, we also analyze several other aspects of our proposed estimator $\hat{s}_q(x)$ and show that randomized measurements are an essential aspect of our procedure.
3 citations
Cites background from "The integral of a symmetric unimoda..."
...e variable jS 1 + ˆ q1j. Consequently, it follows from a standard that median(F n) ! P median(F), and then mb q qkxk q ! P median(F) =: 1=c1 >0: (124) (Note that it follows from Anderson’s Lemma [And55] that median(F) median(jS1j), which is clearly positive.) Altogether, we have veried that btinitial = 1=mb q satises btinitial qkxk q ! P c1. Combining the previous limit with line (41) and Theorem...
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TL;DR: In this article, it was shown that the product-convolutions of unimodal distributions are not unimmodal either, and an analogue of Wintner's result based on the relatively recent notion of R-symmetry was offered by showing that R-Symmetry is R-smooth.
Abstract: Gnedenko and Kolmogorov (1949) in their acclaimed monograph. Limit Distributions for Sums of Independent Random Variables, claimed that the convolutions of unimodal distributions are unimodal. Kai Lai Chung, in an appendix of his English translation of the monograph, by a counterexample, refuted the claim and further noted Wintner's (1938) result that the convolutions of symmetric unimodal distributions are symmetric unimodal. In this note, it is shown that the product-convolutions of unimodal distributions are not unimodal either. Furthermore, an analogue of Wintner's result based on the relatively recent notion of R-symmetry (Mudholkar and Wang, 2007) is offered by showing that the product-convolutions of R-symmetric unimodal distributions are R-symmetric unimodal.
3 citations
TL;DR: In this article, the authors prove that a centered real-valued operator-scaling Gaussian random field with stationary increments satisfies a form of strong local nondeterminism and establish its exact uniform and local moduli of continuity.
Abstract: Let $X=\{X(t),t\in\mathrm{R}^N\}$ be a centered real-valued operator-scaling Gaussian random field with stationary increments, introduced by Bierme, Meerschaert and Scheffler (Stochastic Process. Appl. 117 (2007) 312-332). We prove that $X$ satisfies a form of strong local nondeterminism and establish its exact uniform and local moduli of continuity. The main results are expressed in terms of the quasi-metric $\tau_E$ associated with the scaling exponent of $X$. Examples are provided to illustrate the subtle changes of the regularity properties.
3 citations
Cites background from "The integral of a symmetric unimoda..."
...Thus by the fact that the conditional distributions of the Gaussian process is almost surely Gaussian, and by Anderson’s inequality (see Anderson [1]) and the definition of C8, we obtain P2(k)≤ P ( N(0,1)≤ (1− μ) √ 2(a1 − δ) log(1 + r k ) ) ,...
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...Thus by the fact that the conditional distributions of the Gaussian process is almost surely Gaussian, and by Anderson’s inequality (see Anderson [1]) and the definition of C8, we obtain P2(k)≤ P ( N(0,1)≤ (1− µ) √ 2(a1 − δ) log(1 + r−1k ) ) , where N(0,1) denotes a standard normal random variable....
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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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