REMARKS ON A SEMICIRCULAR PERTURBATION OF THE FREE FISHER INFORMATION

2008 ◽  
Vol 11 (01) ◽  
pp. 97-108 ◽  
Author(s):  
HIROAKI YOSHIDA
Keyword(s):  
Fisher Information ◽  
Conditional Variance ◽  
Information Inequality ◽  
Free Entropy ◽  
Variance Formula ◽  
The Mean ◽  
Power Inequality ◽  
Free Fisher Information ◽  
Special Case

In this paper, we shall give the representation of a semicircular perturbation of the free Fisher information [Formula: see text] by the mean of the conditional variance [Formula: see text], where S is a standard semicircular element freely independent of X, and ε > 0. Using this representation, we will give alternative proofs of the free Fisher information inequality, in which the free analogue of Stam's inequality can be obtained as a special case, and of the free entropy power inequality in an infinitesimal approach to the free entropy.

A Free Logarithmic Sobolev Inequality on the Circle

2006 ◽  
Vol 49 (3) ◽  
pp. 389-406 ◽  
Author(s):  
Fumio Hiai ◽  
Dénes Petz ◽  
Yoshimichi Ueda
Keyword(s):  
Fisher Information ◽  
Sobolev Inequality ◽  
Large Deviation ◽  
Logarithmic Sobolev Inequality ◽  
Approximation Procedure ◽  
Matrix Approximation ◽  
Unitary Matrices ◽  
Free Entropy ◽  
Large Deviation Result ◽  
Free Fisher Information

AbstractFree analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

Portfolio selection and matching: a synthesis

1992 ◽  
Vol 119 (1) ◽  
pp. 87-105 ◽  
Author(s):  
M. Sherris
Keyword(s):  
Portfolio Selection ◽  
Life Insurance ◽  
General Framework ◽  
Pension Fund ◽  
Central Value ◽  
Efficient Portfolios ◽  
The Mean ◽  
Mean Variance ◽  
Special Case ◽  
Selection Of

AbstractThis paper considers a general framework for the selection of assets to meet the liabilities of a life insurance or pension fund. This general framework contains the mean-variance efficient portfolios of modern portfolio theory as a special case. The paper also demonstrates how the portfolio selection and matching approach of Wise (1984a, 1984b, 1987a, 1987b) and Wilkie (1985) fits into this general framework. The matching portfolio is derived as a special case, and is also shown to have implications for determining the central value of the liabilities.

Connections of Gini, Fisher, and Shannon by Bayes risk under proportional hazards

2017 ◽  
Vol 54 (4) ◽  
pp. 1027-1050 ◽  
Author(s):  
Majid Asadi ◽  
Nader Ebrahimi ◽  
Ehsan S. Soofi
Keyword(s):  
Fisher Information ◽  
Shannon Entropy ◽  
Equilibrium Distribution ◽  
Proportional Hazards ◽  
Random Variable ◽  
Bayes Risk ◽  
Entropy Functionals ◽  
Renewal Time ◽  
Potential Applications ◽  
The Mean

Abstract The proportional hazards (PH) model and its associated distributions provide suitable media for exploring connections between the Gini coefficient, Fisher information, and Shannon entropy. The connecting threads are Bayes risks of the mean excess of a random variable with the PH distribution and Bayes risks of the Fisher information of the equilibrium distribution of the PH model. Under various priors, these Bayes risks are generalized entropy functionals of the survival functions of the baseline and PH models and the expected asymptotic age of the renewal process with the PH renewal time distribution. Bounds for a Bayes risk of the mean excess and the Gini's coefficient are given. The Shannon entropy integral of the equilibrium distribution of the PH model is represented in derivative forms. Several examples illustrate implementation of the results and provide insights for potential applications.

scholarly journals A note on the ruled surface

1937 ◽  
Vol 30 ◽  
pp. i-ii
Author(s):  
R. Wilson
Keyword(s):  
Mean Curvature ◽  
Geodesic Curvature ◽  
Ruled Surface ◽  
Local Geometry ◽  
Parametric Curves ◽  
The Mean ◽  
Curvature And Torsion ◽  
Asymptotic Lines ◽  
Special Case ◽  
Elegant Form

The generators and their orthogonal trajectories form, perhaps, the most useful set of parametric curves for the study of the local geometry of a ruled surface. It is not generally realised, however, that the fundamental quantities of the surface can be expressed quite simply in terms of the geodesic curvature, the geodesic torsion and the normal curvature of the directrix, that particular orthogonal trajectory which is chosen as base curve. Certain of the results are similar in form to those arising in the special case of a surface which is generated by the principal normals to a given curve, except that the curvature and torsion are geodetic. In addition it is possible to obtain in an elegant form the differential equation of the curved asymptotic lines and the expression for the mean curvature.

The Largest Missing Value in a Sample of Geometric Random Variables

2014 ◽  
Vol 23 (5) ◽  
pp. 670-685 ◽  
Author(s):  
MARGARET ARCHIBALD ◽  
ARNOLD KNOPFMACHER
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Missing Value ◽  
Asymptotic Expressions ◽  
Mean And Variance ◽  
The Mean ◽  
Special Case

We consider samples of n geometric random variables with parameter 0 < p < 1, and study the largest missing value, that is, the highest value of such a random variable, less than the maximum, that does not appear in the sample. Asymptotic expressions for the mean and variance for this quantity are presented. We also consider samples with the property that the largest missing value and the largest value which does appear differ by exactly one, and call this the LMV property. We find the probability that a sample of n variables has the LMV property, as well as the mean for the average largest value in samples with this property. The simpler special case of p = 1/2 has previously been studied, and verifying that the results of the present paper coincide with those previously found for p = 1/2 leads to some interesting identities.

Unsteady plane flow past curved obstacles with infinite wakes

1955 ◽  
Vol 229 (1177) ◽  
pp. 152-180 ◽  
Keyword(s):  
Unsteady Flow ◽  
Classical Theory ◽  
Steady Motion ◽  
Uniform Motion ◽  
Plane Flow ◽  
Flow Past ◽  
Long Duration ◽  
Infinite Extent ◽  
The Mean ◽  
Special Case

A theory of unsteady flow about obstacles behind which are wakes or cavities of infinite extent is developed for the case when the velocities and displacements of the unsteady perturbations about the mean steady motion are small. Unsteady Helmholtz flows (constant wake pressure) receive detailed attention both for general non-uniform motion and for the special case of harmonic motions of long duration. A number of possible applications of the theory to aerodynamic problems are indicated, the most important being the flutter of a stalled aerofoil. The classical theory of unsteady aerofoik motion is shown to be a special case of the theory given in this paper.

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