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, a probability bound that follows from Anderson's Theorem is derived for distributions whose covariance matrices are ordered with respect to Lowner partial ordering, a relation that is based on whether the difference between two matrices is positive definite.
6 citations
01 Oct 2007
TL;DR: Simulated GPS clock phase and frequency deviations, and simulated GPS pseudo-range measurements, are used to understand GPS Time in terms of Kalman filter estimation errors.
Abstract: The GPS Composite Clock defines GPS Time, the timescale used today in GPS operations. GPS Time is illuminated by examination of its role in the complete estimation and control problem relative to UTC/TAI. Simulated GPS clock phase and frequency deviations, and simulated GPS pseudo-range measurements, are used to understand GPS Time in terms of Kalman filter estimation errors.
6 citations
Cites methods from "The integral of a symmetric unimoda..."
...Kalman’s MU [9] derives from Sherman’s theorem [11, 15, 16], Sherman’s theorem derives from Anderson’s theorem [1], and Anderson’s theorem derives from the Brunn-Minkowki inequality theorem [5, 17]....
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TL;DR: In this article, Cela nous permet d'etudier le comportement asymptotique de E ϕ(t), quand t→0 +,+∞, ou j 0 est le premier zero de la fonction de Bessel J 0.
Abstract: Resume Soient C la cellule typique de la mosaique de Poisson–Voronoi standard de R d , d⩾2 , et ϕ(t)=∑ n⩾1 e −λ n t , t>0 , la fonction spectrale du Laplacien avec condition de Dirichlet au bord de C . Nous montrons que l'esperance E ϕ(t) , t>0 , est une fonctionnelle de l'enveloppe convexe d'un pont brownien d -dimensionnel standard. Cela nous permet d'etudier le comportement asymptotique de E ϕ(t) , quand t→0 + ,+∞ . En particulier, nous prouvons qu'en dimension d=2 , la loi de la premiere valeur propre λ 1 de C satisfait la relation asymptotique ln E e −tλ 1 ∼−t 1/2 4 π j 0 , quand t→+∞ , ou j 0 est le premier zero de la fonction de Bessel J 0 .
6 citations
6 citations
TL;DR: In this paper, the median cross validation criterion is employed to select the smoothing parameter for the nonparametric regression model, and the upper and lower bounds of the convergence rate and the weak consistency of median cross-validated estimate are obtained under mild regularity conditions.
Abstract: Consider the nonparametric regression model
$$Y_{ni} = g\left( {x_{ni} } \right) + e_{ni} ,1 \leqslant i \leqslant n$$
, whereg is an unknown function to be estimated on [0, 1],
$$x_{ni} \left( {1 \leqslant i \leqslant n} \right)$$
are the fixed design points in the interval [0, 1] and
$$\{ e_{ni} ,1 \leqslant i \leqslant n\} $$
is a triangular array of row iid random variables having median zero. The nearest neighbor median estimator
$$\hat g_{n,h} \left( {x_{ni} } \right) = \hat m\left( {Y_{i(l)}^n , \cdots ,Y_{i(h)}^n } \right)$$
is taken as the estimator of the unknown functiong(x). Median cross validation (mev) criterion is employed to select the smoothing parameterh. Leth
π
*
be the smoothing parameter chosen by mev criterion. Under mild regularity conditions, the upper and lower bounds ofh
π
*
, the rate of convergence and the weak consistency of the median cross-validated estimate
$$\hat g_{n,h_n^ * } \left( {x_{ni} } \right)$$
are obtained.
6 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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