The dual multivariate Charlier and Edgeworth expansions

The Charlier differential series for distribution and density functions is the foundation for the Edgeworth expansions of distribution and density functions of sample estimators. Here, we give two forms of these expansions for multivariate distributions using multivariate Bell polynomials. Two forms arise because the multivariate Hermite polynomials have a dual form. These dual forms for the multivariate Charlier and Edgeworth expansions appear to be new.     

非线性随机效应模型的置信域

本文对非线性随机效应模型,建立了微分几何框架,推广了Bates＆Wates关于非线性模型几何结构．在吡基础上,我们导出了关于固定效应参数和子集参数的置信域的曲率表示,这些结果是BatesandWates(1980),Hamilton(1986)与Wei(1994)等的推广．     

A note about proving non-Γ under a finite non-microstates free Fisher information assumption

We prove that if X₁,…,Xn (n>1) are self-adjoints in a W^∗-probability space with finite non-microstates free Fisher information, then the von Neumann algebra W^∗(X₁,…,Xn) they generate doesn't have property Γ (especially is not amenable). This is an analog of a well-known result of Voiculescu for microstates free entropy. We also prove factoriality under finite non-microstates entropy.     

关于“SAS软件包中Fisher精确检验方法的分析”一文的争鸣意见

本文对ＳＡＳ软件包中精确概率的算法与一般统计学教科书有较大差别的问题提出了个人看法。认为在一般情况下根据ＳＡＳ软件包给出的双尾概率作出双侧检验结论时应取慎重态度     

Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada–Watanabe theorem (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain $D$ of the phase space $\mathbb{R}$ and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called free energy function on the density function space, we prove that the density functions, which are solutions of the Fokker–Planck equation, converge to the stationary density function exponentially under the Kullback–Leibler divergence, thus also in the total variation norm. The results show that there is a relation between the Bakry–Émery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher–Lamperti transformation. Several applications are discussed, including the Cox–Ingersoll–Ross model and the Ait-Sahalia model in finance and the Wright–Fisher model in population genetics.     

Uncertainty principle and quantum Fisher information

A family of inequalities, related to the uncertainty principle, has been recently proved by S. Luo, Z. Zhang, Q. Zhang, H. Kosaki, K. Yanagi, S. Furuichi and K. Kuriyama. We show that the inequalities have a geometric interpretation in terms of quantum Fisher information. Using this formulation one may naturally ask if this family of inequalities can be further extendend, for example to the RLD quantum Fisher information. We show that this is impossible by producing a family of counterexamples.     

A central limit theorem in nonlinear filtering

It is shown that the probability law of a diffusion process conditioned on weakly corrupted observations is asymptotically Gaussian when properly scaled. The method of proof involves Fisher information matrices and a Cramér-Rao inequality.     

Direct approach to quantum extensions of Fisher information

By manipulating classical Fisher information and employing various derivatives of density operators, and using entirely intuitive and direct methods, we introduce two families of quantum extensions of Fisher information that include those defined via the symmetric logarithmic derivative, via the right logarithmic derivative, via the Bogoliubov-Kubo-Mori derivative, as well as via the derivative in terms of commutators, as special cases. Some fundamental properties of these quantum extensions of Fisher information are investigated, a multi-parameter quantum Cramér-Rao inequality is established, and applications to characterizing quantum uncertainty are illustrated.      

H‐self‐adjoint matrices. Application to radiosity

In computer graphics, in the radiosity context, a linear system Φx=b must be solved and there exists a diagonal positive matrix H such that H Φ is symmetric. In this article, we extend this property to complex matrices: we are interested in matrices which lead to Hermitian matrices under premultiplication by a Hermitian positive‐definite matrix H. We shall prove that these matrices are self‐adjoint with respect to a particular innerproduct defined on ?ⁿ. As a result, like Hermitian matrices, they have real eigenvalues and they are diagonalizable. We shall also show how to extend the Courant–Fisher theorem to this class of matrices. Finally, we shall give a new preconditioning matrix which really improves the convergence speed of the conjugate gradient method used for solving the radiosity problem. Copyright © 2002 John Wiley & Sons, Ltd.     

THE ESTIMATION OF PRIOR FROM FISHER INFORMATION

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