Maximum entropy interpretation of autoregressive spectral densities

A new proof is given of the maximum entropy characterization of autoregressive spectral densities as models for the spectral density of a stationary time series. The new proof is presented in parallel with a proof of the maximum entropy characterization of exponential models for probability densities. Concepts of entropy, cross-entropy and information divergence are defined for probability densities and for spectral densities.     

A study on densities of sets of points and lines

In this paper, firstly, our purpose is to give the relationship among the densities of the sets of collinear points, the relationship among the densities of the sets of noncollinear points, and the relationship among the densities of the sets of the intersecting lines in Euclidean plane, respectively. In addition to that, we define the density formulas for the sets of points and lines under the two‐parameter planar Euclidean motion. By means of these results, we obtain essential properties that explain the connection among the densities of the sets of points and among the densities of the sets of intersecting lines under the two‐parameter planar Euclidean motion.     

Stokes-Einstein Behaviour for Liquids with Bounded Potentials at High Densities

By means of molecular dynamics simulations we investigate the Stokes–Einstein behaviour of the Gaussian Core Model liquid at high densities. From a recent analysis at moderately high densities, we found that the Stokes–Einstein factor grows linearly with the viscosity. We show that at high densities the behaviour is similar, but there are slight departures at lower temperatures and higher densities. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generating M-Indeterminate Probability Densities by Way of Quantum Mechanics

Probability densities that are not uniquely determined by their moments are said to be “moment-indeterminate,” or “M-indeterminate.” Determining whether or not a density is M-indeterminate, or how to generate an M-indeterminate density, is a challenging problem with a long history. Quantum mechanics is inherently probabilistic, yet the way in which probability densities are obtained is dramatically different in comparison with standard probability theory, involving complex wave functions and operators, among other aspects. Nevertheless, the end results are standard probabilistic quantities, such as expectation values, moments and probability density functions. We show that the quantum mechanics procedure to obtain densities leads to a simple method to generate an infinite number of M-indeterminate densities. Different self-adjoint operators can lead to new classes of M-indeterminate densities. Depending on the operator, the method can produce densities that are of the Stieltjes class or new formulations that are not of the Stieltjes class. As such, the method complements and extends existing approaches and opens up new avenues for further development. The method applies to continuous and discrete probability densities. A number of examples are given.

     

Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasmas

The existence of global smooth solutions to the multi-dimensional hydrodynamic model for plasmas of electrons and positively charged ions is shown under the assumption that the initial densities are close to a constant. The model consists of the conservation laws for the particle densities and the current densities, coupled to the Poisson equation for the electrostatic potential. Furthermore, it is proved that the particle densities converge exponentially fast to the (constant) steady state. The proof uses a higher-order energy method inspired from extended thermodynamics.     

Sampling Truncated Normal,Beta, and Gamma Densities

We consider the Bayesian analysis of constrained parameter and truncated data problems within a Gibbs sampling framework and concentrate on sampling truncated densities that arise as full conditional densities within the context of the Gibbs sampler. In particular, we restrict attention to the normal, beta, and gamma densities. We demonstrate that, in many instances, it is possible to introduce a latent variable which facilitates an easy solution to the problem. We also discuss a novel approach to sampling truncated densities via a “black-box” algorithm, based on the latent variable idea, valid outside of the context of a Gibbs sampler.     

Optimal Scaling for Random Walk Metropolis on Spherically Constrained Target Densities

We consider the problem of optimal scaling of the proposal variance for multidimensional random walk Metropolis algorithms. It is well known, for a wide range of continuous target densities, that the optimal scaling of the proposal variance leads to an average acceptance rate of 0.234. Therefore a natural question is, do similar results hold for target densities which have discontinuities? In the current work, we answer in the affirmative for a class of spherically constrained target densities. Even though the acceptance probability is more complicated than for continuous target densities, the optimal scaling of the proposal variance again leads to an average acceptance rate of 0.234.     

Analyses and simulation of anisotropic fractal surfaces

The spectral densities for an anisotropic fractal surfaces are investigated. Since there is no general definition for anisotropic fractal surface, the profiles of anisotropic fractal surfaces are assumed to be fractal in two main axes. Then, the possible forms of the surface spectral densities are proposed. By using the inverse Fast Fourier Transform, anisotropic fractal surfaces can be simulated from the spectral densities.     

Deconvolving compactly supported densities

This paper addresses the statistical problem of density deconvolution under the condition that the density to be estimated has compact support. We introduce a new estimation procedure, which establishes faster rates of convergence for smooth densities as compared to the optimal rates for smooth densities with unbounded support. This framework also allows us to relax the usual condition of known error density with non-vanishing Fourier transform, so that a nonparametric class of densities is valid; therefore, even the shape of the noise density need not be assumed. These results can also be generalized for fast decaying densities with unbounded support. We prove optimality of the rates in the underlying experiment and study the practical performance of our estimator by numerical simulations.      

基于MCMC的贝叶斯变结构金融时序GARCH模型研究

针对变结构GARCH模型没有解析形式的条件后验分布的问题。借助辅助变量把没有具体解析形式的后验分布转化为一系列完全条件分布,实现了变结构GARCH模型参数的贝叶斯估计。中国外汇市场波动性的实证研究,表明了辅助变量-Gibbs抽样有效的解决了贝叶斯变结构GARCH模型中的高维数值计算问题,并发现其波动持续性是由时间序列的状态转移引起的。     

On convergence of extremes under power normalization

In this note, we discuss two aspects of convergence of extremes under power normalization: convergence of moments and convergence of densities. The moments convergence is established for four p-max-stable laws according to conditions imposed on the considered distributions or on the parameter of the p-max-stable laws. For densities convergence, local uniform convergence of the densities is shown to coincide with some von Mises conditions.     

Some Results on the Upper Convex Densities of the Self-Similar Sets at the Contracting-Similarity Fixed Points

In this paper, some results on the upper convex densities of self-similar sets at the contracting-similarity fixed points are discussed. Firstly, a characterization of the upper convex densities of self-similar sets at the contracting-similarity fixed points is given. Next, under the strong separation open set condition, the existence of the best shape for the upper convex densities of self-similar sets at the contracting-similarity fixed points is proven. As consequences, an open problem and a conjecture, which were posed by Zhou and Xu, are answered.     

The eigenvalue gap for vibrating strings with symmetric densities

We establish several comparison results on the eigenvalue gap for vibrating strings with symmetric single-well densities or symmetric double-well densities.      

On nonlinear cross-diffusion systems: an optimal transport approach

We study a nonlinear, degenerate cross-diffusion model which involves two densities with two different drift velocities. A general framework is introduced based on its gradient flow structure in Wasserstein space to derive a notion of discrete-time solutions. Its continuum limit, due to the possible mixing of the densities, only solves a weaker version of the original system. In one space dimension, we find a stable initial configuration which allows the densities to be segregated. This leads to the evolution of a stable interface between the two densities, and to a stronger convergence result to the continuum limit. In particular derivation of a standard weak solution to the system is available. We also study the incompressible limit of the system, which addresses transport under a height constraint on the total density. In one space dimension we show that the problem leads to a two-phase Hele-Shaw type flow.     

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Methods for Evaluating Density Functions of Exponential Functionals Represented as Integrals of Geometric Brownian Motion

The purpose of this paper is to present a survey on Yor's formula on the probability densities of the exponential functionals represented as integrals in time of geometric Brownian motions and to present results on numerical computations for the densities. We perform the computations via another formula for the densities obtained by Dufresne and we show numerically the desired coincidence in some cases. As an application, we compute the price of an Asian option. AMS 2000 Subject Classification: 65C50, 60J65     

Empirical Bayes predictive densities for high-dimensional normal models

This paper addresses the problem of estimating the density of a future outcome from a multivariate normal model. We propose a class of empirical Bayes predictive densities and evaluate their performances under the Kullback–Leibler (KL) divergence. We show that these empirical Bayes predictive densities dominate the Bayesian predictive density under the uniform prior and thus are minimax under some general conditions. We also establish the asymptotic optimality of these empirical Bayes predictive densities in infinite-dimensional parameter spaces through an oracle inequality.