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
TL;DR: In this article, the authors evaluate the validity of the test of individual equivalence ratios (TIER), a term coined by Anderson & Hauck (1990) and proved to be a valid test under a class of unimodal symmetric distributions.
Abstract: SUMMARY We evaluate the validity of the test of individual equivalence ratios (TIER), a term coined by Anderson & Hauck (1990). The test was also proposed in Wellek (1989). It is proved to be a valid test under a class of unimodal symmetric distributions; this includes normal distributions. Since most bioequivalence studies involve data which are normal, or at least unimodal and symmetric, the test is typically valid. It is, however, not always valid, as shown by counter examples.
9 citations
01 Jan 2017
TL;DR: A random set is a multivalued measurable function defined on a probability space that appears in random processes of sets (set-valued processes or random multivaluing functions) if thisMultivalued function depends on the second argument.
Abstract: A random set is a multivalued measurable function defined on a probability space. If this multivalued function depends on the second argument (e.g., time or space), then random processes of sets (set-valued processes or random multivalued functions) appear. Important examples are provided by growth processes, multivalued martingales and solutions of stochastic differential inclusions. If time is discrete, one deals with sequences of random closed sets.
9 citations
TL;DR: In this paper, a new stochastic characterization of the Loewner optimality design criterion is presented by proving a generalization to the well known corollary of Anderson's theorem.
Abstract: In this paper we present a new stochastic characterization of the Loewner optimality design criterion. The result is obtained by proving a generalization to the well known corollary of Anderson's theorem. Certain connections between the Loewner optimality and the stochastic distance optimality design criterion are showed. We also present applications and generalizations of the main result.
9 citations
Cites background from "The integral of a symmetric unimoda..."
...…isotonic relative to Loewner ordering (Liski et al. 1999) the result (5) is a direct consequence of a well-known corollary (see e.g. Perlman 1989, or Tong 1990, Theorem 4.2.5) from Anderson's theorem on the integral of a symmetric unimodal function over a symmetric convex set (see Anderson 1955)....
[...]
Posted Content•
TL;DR: In this article, a unified viewpoint on the implications of misspecification of OLS linear regression is presented, with the aim of making the various implications of missing covariates stand out.
Abstract: Ordinary least squares (OLS) linear regression is one of the most basic statistical techniques for data analysis. In the main stream literature and the statistical education, the study of linear regression is typically restricted to the case where the covariates are fixed, errors are mean zero Gaussians with variance independent of the (fixed) covariates. Even though OLS has been studied under misspecification from as early as the 1960's, the implications have not yet caught up with the main stream literature and applied sciences. The present article is an attempt at a unified viewpoint that makes the various implications of misspecification stand out.
9 citations
Cites background from "The integral of a symmetric unimoda..."
...(23) To show that the multiplier score bootstrap works, we need Anderson’s Lemma. Lemma 6.2 (Corollary 3, Anderson (1955))....
[...]
...This construction will be described in Section 5 (see Fahrmeir (1990, page 492), and also Bachoc et al. (2016) for an alternative proposal)....
[...]
...Lemma 6.2 (Corollary 3, Anderson (1955))....
[...]
TL;DR: In this article, the authors define a covariance-type operator on Wiener space, and prove corresponding non-Gaussian comparison inequalities for random fields and vectors, which extend the Sudakov-Fernique result on comparison of expected suprema of Gaussian fields.
9 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....
[...]