The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities
T. W. Anderson
- Vol. 6, Iss: 2, pp 170-176
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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.read more
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Journal ArticleDOI
Non-mean-square error criteria
TL;DR: It is shown that in the case of Gaussian processes all non-mean-square error criteria given by three recent textbooks yield the same predictor as the linear minimum mean-square predictor of Wiener.
Inequalitites on the probability content of convex regions for elliptically contoured distributions
TL;DR: The work in this article was supported in part by the National Science Foundation Grant No. 17172 at Stanford University, Grants No. 11021, 9593, and 21074 at the University of Minnesota and grant No. 25911 at University of Chicago; by the U.S. Navy, Air Force and NASA under a contract (N00014-67-4-0097-0014, Task Number NR-042-242) administered by the Office of Naval Research at Yale University, by the Army under a Contract (DA-ARO-D31
Proceedings ArticleDOI
Correlated jamming on MIMO Gaussian fading channels
TL;DR: A zero-sum mutual information game on MIMO Gaussian Rayleigh fading channels is considered, and it is proved that the knowledge of the channel input is useless to the jammer.
Journal ArticleDOI
Majorization in Multivariate Distributions
Albert W. Marshall,Ingram Olkin +1 more
TL;DR: In this paper, it was shown that the convolution of Schur-concave functions is SchurConcave, which implies that the Schurconcavity of the convolutions implies the existence of exchangeability.
Journal ArticleDOI
Minimax risk of matrix denoising by singular value thresholding
David L. Donoho,Matan Gavish +1 more
Abstract: An unknown $m$ by $n$ matrix $X_0$ is to be estimated from noisy measurements $Y=X_0+Z$, where the noise matrix $Z$ has i.i.d. Gaussian entries. A popular matrix denoising scheme solves the nuclear norm penalization problem $\operatorname {min}_X\|Y-X\|_F^2/2+\lambda\|X\|_*$, where $\|X\|_*$ denotes the nuclear norm (sum of singular values). This is the analog, for matrices, of $\ell_1$ penalization in the vector case. It has been empirically observed that if $X_0$ has low rank, it may be recovered quite accurately from the noisy measurement $Y$. In a proportional growth framework where the rank $r_n$, number of rows $m_n$ and number of columns $n$ all tend to $\infty$ proportionally to each other ($r_n/m_n\rightarrow \rho$, $m_n/n\rightarrow \beta$), we evaluate the asymptotic minimax MSE $\mathcal {M}(\rho,\beta)=\lim_{m_n,n\rightarrow \infty}\inf_{\lambda}\sup_{\operatorname {rank}(X)\leq r_n}\operatorname {MSE}(X_0,\hat{X}_{\lambda})$. Our formulas involve incomplete moments of the quarter- and semi-circle laws ($\beta=1$, square case) and the Marcenko-Pastur law ($\beta<1$, nonsquare case). For finite $m$ and $n$, we show that MSE increases as the nonzero singular values of $X_0$ grow larger. As a result, the finite-$n$ worst-case MSE, a quantity which can be evaluated numerically, is achieved when the signal $X_0$ is "infinitely strong." The nuclear norm penalization problem is solved by applying soft thresholding to the singular values of $Y$. We also derive the minimax threshold, namely the value $\lambda^*(\rho)$, which is the optimal place to threshold the singular values. All these results are obtained for general (nonsquare, nonsymmetric) real matrices. Comparable results are obtained for square symmetric nonnegative-definite matrices.
References
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Asymptotic Theory of Certain "Goodness of Fit" Criteria Based on Stochastic Processes
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.
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Theorie der Konvexen Körper
T. Bonnesen,W. Fenchel +1 more
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.
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The Cramer-Smirnov Test in the Parametric Case
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].