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
More filters
15 Sep 2003
TL;DR: It is shown that the expected log determinant of a complex noncentral Wishart matrix is an increasing function of the noncentrality parameter and it is demonstrated that the mutual information corresponding to an isotropically distributed Gaussian input to a multiantenna Ricean fading channel is nondecreasing in the line-of-sight component.
Abstract: In this paper, we show that the expected log determinant of a complex noncentral Wishart matrix is an increasing function of the noncentrality parameter. This demonstrates that the mutual information corresponding to an isotropically distributed Gaussian input to a multiantenna Ricean fading channel is nondecreasing in the line-of-sight component.
50 citations
TL;DR: It is proved that the outage probability and the mutual information induced by a multivariate circularly symmetric Gaussian input with any covariance matrix are monotonic in the LOS matrix D, or more precisely, monotony in D/sup /spl dagger//D in the sense of the Loewner partial order.
Abstract: The dependence of the Gaussian input information rate on the line-of-sight (LOS) matrix in multiple-input multiple-output (MIMO) coherent Rician fading channels is explored. It is proved that the outage probability and the mutual information induced by a multivariate circularly symmetric Gaussian input with any covariance matrix are monotonic in the LOS matrix D, or more precisely, monotonic in D/sup /spl dagger//D in the sense of the Loewner partial order. Conversely, it is also demonstrated that this ordering on the LOS matrices is a necessary condition for the uniform monotonicity over all input covariance matrices. This result is subsequently applied to prove the monotonicity of the isotropic Gaussian input information rate and channel capacity in the singular values of the LOS matrix. Extensions to multiple-access channels (MAC) are also provided.
49 citations
TL;DR: In this paper, the problem of hypotheses testing where equivalence is stated as the alternative hypothesis is investigated, and the power problem is completely solved, among other things, for tests based on the range statistic.
49 citations
01 Jan 1991
TL;DR: In this article, three different types of stochastic dominance relations, namely set dominance, kernel dominance and higher degree dominance, are considered: shifts and dilations, respectively, or inequalities for distribution functions, and, in the second case, inequalities for integrals of the distribution functions.
Abstract: Three different types of stochastic dominance relations are considered: set dominance, kernel dominance and higher degree dominance. The connections between these definitions are examined. Preservation results are given and implications between joint and marginal dominance are studied in the finite and infinite dimensional setting. 1. Introduction. For distributions on the real line, two basic stochastic orderings have been of interest to researchers in many fields: stochastic dominance with respect to all increasing functions and stochastic dominance with respect to all convex functions. These orderings can be characterized by shifts and dilations, respectively, or, in the first case, by inequalities for distribution functions, and, in the second case, by inequalities for integrals of the distribution functions. Beginning with these characterizations several attempts have been made to unify the theory of stochastic dominance relations in d-dimensional and more general spaces (Brumelle and Vickson (1975), Fishburn and Vickson (1978), Stoyan (1977) (1983), Mosler (1982)). Based on this tradition, the primary aim of this paper is to investigate three different ways by which stochastic dominance relations on several spaces may be characterized. The first one is the characterization via probability inequalities for certain families of sets. Orderings which allow for this kind of characterization are named set dominance orderings. The second approach employs Markov kernels to define a stochastic ordering. These orderings are called kernel dominance orderings. Third, inequalities on integrals of distribution functions are
48 citations
Cites background from "The integral of a symmetric unimoda..."
...T? is the cone of central unimodal functions (see Anderson (1955), Sherman (1955), and for recent results Dharmadhikari and Joag-Dev (1988), Bergmann (1991), Eaton and Perlman (1991)). Example 2.7. Let O be a normed space and let ?? be the class of sets Aa = {x : \\x\\ < a}. Then ?? is the cone of decreasing in norm functions (R?schendorf (1981))....
[...]
...T? is the cone of central unimodal functions (see Anderson (1955), Sherman (1955), and for recent results Dharmadhikari and Joag-Dev (1988), Bergmann (1991), Eaton and Perlman (1991))....
[...]
TL;DR: In this paper, a probabilistic interpretation of a class of natural capacities on Euclidean space in terms of the level sets of a suitably chosen multiparameter additive Levy process X is provided.
Abstract: We provide a probabilistic interpretation of a class of natural capacities on Euclidean space in terms of the level sets of a suitably chosen multiparameter additive Levy process X. We also present several probabilistic applications of the aforementioned potential-theoretic connections. They include areas such as intersections of Levy processes and level sets, as well as Hausdorff dimension computations.
47 citations
References
More filters
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....
[...]
...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)....
[...]
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....
[...]
...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....
[...]