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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TL;DR: In this article, the authors studied the monotonicity of the integral of a symmetric, unimodal density over translates of a convex set and obtained similar results in the case of unknown variances.
Abstract: Anderson studied the monotonicity of the integral of a symmetric, unimodal density over translates of a symmetric convex set. Restricting attention to elliptically contoured, unimodal densities, Mukerjee, Robertson and Wright weakened the assumption of symmetry on the set and obtained monotonicity properties of power functions, including unbiasedness, for some likelihood ratio tests in order restricted inference for the variance-known case. For elliptically contoured, unimodal densities, we weaken the assumption of convexity to obtain similar results in the case of unknown variances. The results apply to situations in which the null hypothesis is a linear space and the alternative is a closed, convex cone.
14 citations
TL;DR: In this paper, a Bayesian test for the point null testing problem in multivariate case is developed and a procedure to get the mixed distribution using the prior density is suggested for comparisons between the Bayesian and classical approaches, lower bounds on posterior probabilities of the null hypothesis are computed and compared with the p-value of the classical test.
Abstract: A Bayesian test for the point null testing problem in the multivariate case is developed A procedure to get the mixed distribution using the prior density is suggested For comparisons between the Bayesian and classical approaches, lower bounds on posterior probabilities of the null hypothesis, over some reasonable classes of prior distributions, are computed and compared with the p-value of the classical test With our procedure, a better approximation is obtained because the p-value is in the range of the Bayesian measures of evidence
14 citations
Cites background from "The integral of a symmetric unimoda..."
...These distributions are called elliptical and are unimodal in the sense of [17]....
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TL;DR: In this paper, the convergence rate of the tail probabilities of both the maximum sum and the maximum partial sum was studied and sufficient and necessary conditions for a kind of limit theorems to hold on the convergence of both of them were given.
Abstract: Let $X$, $X_1$, $X_2$, $...$ be i.i.d. random variables, and let $S_n=X_1+... + X_n$ be the partial sums and $M_n=\max_{k\le n}|S_k|$ be the maximum partial sums. We give the sufficient and necessary conditions for a kind of limit theorems to hold on the convergence rate of the tail probabilities of both $S_n$ and $M_n$. These results are related to the law of the iterated logarithm. The results of Gut and Spataru (2000) are special cases of ours.
13 citations
Cites background from "The integral of a symmetric unimoda..."
...P { ‖Y ‖ ≥ σ √ 2 log log n(2 + an(2)) } ∼ 2Aσ(2) ( σ ( 2 + an(2) )√ 2 log log n )d−2 exp{− log log(2 + an(2))(2)} ∼ 2Aσ (( 2 + an(2) )√ 2 log log n )d−2 exp{−22 log log n} exp{−22an(2) log log n} ∼ 2Aσd(2 √ 2 log log n )d−2 exp{−22 log log n} exp{−22an(2) log log n} as n → ∞, uniformly in 2 ∈ (√1 + a,√1 + a + δ) for some δ > 0, where A is as in Proposition 2....
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...n=1 (log n)(log log n) n P { ‖Sn‖ ≥ σφ(n)(2 + an(2)) } = Γ−1(d/2)K(Σ)(1 + a) d−2 2 Γ(b + d/2) exp{−2τ√1 + a}, (1....
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...1 Let a > −1 and b > −d/2 and let an(2) be a function of 2 satisfying (1....
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...n=1 (log n)(log log n) n ·P { ‖Y ‖ ≥ σ √ 2 log log n(2 + an(2)) } = Γ−1(d/2)K(Σ)(1 + a) d−2 2 Γ(b + d/2) exp{−2τ√1 + a}, (2....
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...1 Let a > −1 and b > −d/2 and let an(2) be a function of 2 such that an(2) log log n → τ as n →∞ and 2 ↘ √ 1 + a....
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TL;DR: In this article, a general framework for unifying modes of inference for sharp and weak nulls is proposed, where a single procedure simultaneously delivers exact inference for strong nulls and asymptotically valid inference for weaknulls.
Abstract: In finite population causal inference exact randomization tests can be constructed for sharp null hypotheses, i.e. hypotheses which fully impute the missing potential outcomes. Oftentimes inference is instead desired for the weak null that the sample average of the treatment effects takes on a particular value while leaving the subject-specific treatment effects unspecified. Without proper care, tests valid for sharp null hypotheses may be anti-conservative should only the weak null hold, creating the risk of misinterpretation when randomization tests are deployed in practice. We develop a general framework for unifying modes of inference for sharp and weak nulls, wherein a single procedure simultaneously delivers exact inference for sharp nulls and asymptotically valid inference for weak nulls. To do this, we employ randomization tests based upon prepivoted test statistics, wherein a test statistic is first transformed by a suitably constructed cumulative distribution function and its randomization distribution assuming the sharp null is then enumerated. For a large class of commonly employed test statistics, we show that prepivoting may be accomplished by employing the push-forward of a sample-based Gaussian measure based upon a suitably constructed covariance estimator. In essence, the approach enumerates the randomization distribution (assuming the sharp null) of a P-value for a large-sample test known to be valid under the weak null, and uses the resulting randomization distribution to perform inference. The versatility of the method is demonstrated through a host of examples, including rerandomized designs and regression-adjusted estimators in completely randomized designs.
13 citations
Cites background from "The integral of a symmetric unimoda..."
...The following result is a straightforward corollary of Anderson’s (1955) theorem for multivariate Gaussians; see also Theorem 4....
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TL;DR: In this paper, a block-diagonal, hence quasidiagonal, operator T on H was constructed, which is not a norm limit of a sequence of algebraically m-normal operators.
13 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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