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 article, the power of the spacing test for least-angle regression (LARS) was investigated and it was shown that its power is always greater or equal to the significance level.
Abstract: Recent advances in Post-Selection Inference have shown that conditional testing is relevant and tractable in high-dimensions. In the Gaussian linear model, further works have derived unconditional test statistics such as the Kac-Rice Pivot for general penalized problems. In order to test the global null, a prominent offspring of this breakthrough is the spacing test that accounts the relative separation between the first two knots of the celebrated least-angle regression (LARS) algorithm. However, no results have been shown regarding the distribution of these test statistics under the alternative. For the first time, this paper addresses this important issue for the spacing test and shows that it is unconditionally unbiased. Furthermore, we provide the first extension of the spacing test to the frame of unknown noise variance.
More precisely, we investigate the power of the spacing test for LARS and prove that it is unbiased: its power is always greater or equal to the significance level $\alpha$. In particular, we describe the power of this test under various scenarii: we prove that its rejection region is optimal when the predictors are orthogonal; as the level $\alpha$ goes to zero, we show that the probability of getting a true positive is much greater than $\alpha$; and we give a detailed description of its power in the case of two predictors. Moreover, we numerically investigate a comparison between the spacing test for LARS and the Pearson's chi-squared test (goodness of fit).
7 citations
Cites background from "The integral of a symmetric unimoda..."
...Interestingly, our proof is based on Anderson’s inequality [1] for symmetric convex sets....
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...Lemma 4 (Anderson’s inequality for Gaussian measure [1])....
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TL;DR: In this article, a Markov's inequality for such spaces in P(X ≥ x) ≤ infG0E[g(X)]/g(x), where G0 is the class of nonnegative order-preserving functions on X such that, for each g∈G0, E[g (X)] is defined; and G1ņG0 be the subclass of concave functions.
7 citations
TL;DR: In this paper, the existence and uniqueness of the linear stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere S 2 was investigated and sample path regularity properties were established.
7 citations
Additional excerpts
...This and Anderson’s inequality (see [3]) imply...
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29 Sep 2014
TL;DR: In this paper, a survey of inequalities concerning bivariate and multivariate distributions in statistics is presented, as well as historical background, including inequalities arising through positive and negative dependence; Boole, Bonferroni, and Frechet inequalities; convex symmetric set inequalities.
Abstract: : Inequalities concerning bivariate and multivariate distributions in statistics are surveyed, as well as historical background. Subjects treated include inequalities arising through positive and negative dependence; Boole, Bonferroni, and Frechet inequalities; convex symmetric set inequalities; stochastic ordering; stochastic majorization and inequalities obtained by majorization; Chebyshev and Kolmogorov-type inequalities; multivariate moment inequalities; and applications to simultaneous inference, unbiased testing and reliability theory. (Author)
7 citations
Cites background from "The integral of a symmetric unimoda..."
...Ander- son (1955) showed that if X has a density g(x) symmetric around the origin and satisfying{(x:g(x) >t)}is a convex set for all t, 0 t -, then for any convex set C symmetric around the origin, PrCX+syLEC] is a monotone decreasing function of s, 0 s l, for every constant vector y. Khatri…...
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TL;DR: This paper studies the precise rates for @?"n"="1^~(logn)^bn^3^/^2E{@?S"n@?-@s@e2nloglogn}"+.
7 citations
Additional excerpts
...[2] T.W. Anderson, The integral of a summetric unimodal function over a symmetric convex set and some probability inequalities, Proc....
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...Applying the inequality of Anderson [2], we have that for any x ∈ R, P (∥∥∥∥∥ n∑ j=1 Ynj ∥∥∥∥∥ ≤ x ) ≥ P ( ‖Y‖ ≤ x √ n ) , n ≥ 1....
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...Applying the inequality of Anderson [2], we have that for any x ∈ R, P ∥∥∥∥ n ∑ j=1 Ynj ∥∥∥∥ ≤ x ) ≥ P ( ‖Y‖ ≤ x √ n ) , n ≥ 1....
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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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