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
29 Sep 2014
TL;DR: In this paper, a multivariate normal distribution is used to model and analyze multivariate data, which is often central to the modeling and analysis of such data, and empirical evidence often points towards the normality of multivariate datasets, and many normal-theory procedures remain exact for many non-normal multivariate distributions exhibiting suitable symmetries.
Abstract: Biometric data typically entail observations on multiple characteristics for each experimental subject. Multivariate normal distributions are often central to the modeling and analysis of such data. Empirical evidence often points towards the normality of multivariate data. Biometric measurements, especially, may emerge as the result of many small increments due to heredity and environment, so that the approximate multivariate normality of such data rests on multidimensional central limit theory. In addition, many normal-theory procedures, both univariate and multivariate, remain exact for many nonnormal multivariate distributions exhibiting suitable symmetries.
Keywords:
multinormal distributions;
bivariate correlation analysis;
multiple correlation coefficient
Posted Content•
TL;DR: In this paper, the existence and uniqueness of the solution of the stochastic heat equation in certain Sobolev space is investigated and sample path regularity properties are established, in particular, the exact uniform modulus of continuity in time/spatial variable is derived.
Abstract: We study the stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere $\mathbb{S}^{2}$. The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution in time/spatial variable is derived.
TL;DR: In this paper, it was shown that for b > −1, for a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator Σ, the largest eigenvalue of Σ is σ2.
Abstract: Let {X, Xn; n ≥ 1} be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with covariance operator Σ. Set Sn = X1 + X2 + ... + Xn, n ≥ 1. We prove that, for b > −1,
$$ \mathop {\lim }\limits_{\varepsilon \searrow 0} \varepsilon ^{2(b + 1)} \sum\limits_{n = 1}^\infty {\frac{{(logn)^b }} {{n^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-
ulldelimiterspace} 2}} }}} E\{ \left\| {S_n } \right\| - \sigma \varepsilon \sqrt {nlogn} \} _ + = \frac{{\sigma ^{ - 2(b + 1)} }} {{^{(2b + 3)(b + 1)} }}E\left\| Y \right\|^{2b + 3} $$
holds if EX = 0, and E‖X‖2(log ‖X‖)3b∨(b+4) < ∞, where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator Σ, and σ2 denotes the largest eigenvalue of Σ.
Cites methods from "The integral of a symmetric unimoda..."
...Applying Corollary 3 of Anderson [ 16 ], we have that, for any x ∈ R,...
[...]
...Applying Corollary 3 of Anderson [ 16 ] again, we have that, for any x ∈ R,...
[...]
TL;DR: In this paper, the authors proposed a new Bayesian quantile regression estimator using conditional empirical likelihood as the working likelihood function, which has low computation cost since the posterior distribution function has explicit form.
Abstract: In this paper, we propose a new Bayesian quantile regression estimator using conditional empirical likelihood as the working likelihood function. We show that the proposed estimator is asymptotically efficient and the confidence interval constructed is asymptotically valid. Our estimator has low computation cost since the posterior distribution function has explicit form. The finite sample performance of the proposed estimator is evaluated through Monte Carlo studies.
Cites background from "The integral of a symmetric unimoda..."
...Especially when the loss function is symmetric, the minimizer x∗ assumed in Assumption 2c is 0 by Anderson (1955)....
[...]
TL;DR: In this article, a short and direct proof that the maximum likelihood estimator is a maximum probability estimator with respect to a certain sequence of convex and bounded sets in R(k) that are symmetric about the origin is given.
Abstract: For multiparameter exponential models a short and direct proof is given that the maximum likelihood estimator is a maximum probability estimator with respect to a certain sequence of convex and bounded sets inR(k) that are symmetric about the origin; asymptotically these sets are allowed to be unbounded.
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....
[...]