Harnack and functional inequalities for generalized Mehler semigroups

Harnack inequalities are established for a class of generalized Mehler semigroups, which in particular imply upper bound estimates for the transition density. Moreover, Poincaré and log-Sobolev inequalities are proved in terms of estimates for the square field operators. Furthermore, under a condition, well-known in the Gaussian case, we prove that generalized Mehler semigroups are strong Feller. The results are illustrated by concrete examples. In particular, we show that a generalized Mehler semigroup with an α-stable part is not hyperbounded but exponentially ergodic, and that the log-Sobolev constant obtained by our method in the special Gaussian case can be sharper than the one following from the usual curvature condition. Moreover, a Harnack inequality is established for the generalized Mehler semigroup associated with the Dirichlet heat semigroup on (0,1). We also prove that this semigroup is not hyperbounded.     

Dispersion estimates for the two-dimensional Vlasov-Yukawa system with small data

We present dispersion estimates for the two-dimensional Vlasov-Yukawa system with small data. When the initial data are sufficiently regular and small, we show that the local mass density and the Yukawa force field decay to zero algebraically fast in time. These dispersion estimates are not known for the two-dimensional Vlasov-Poisson system. For the dispersion estimates, we effectively use the short-range character of the Yukawa potential and the optimal gradient estimates introduced by Hwang, Rendall and Velázquez for the three-dimensional Vlasov-Poisson system.     

Estimates for Correlation in Dynamical Systems: From Hölder Continuous Functions to General Observables

For many dynamical systems that are popular in applications, estimates are known for the decay of correlation in the case of Hölder continuous functions. In the present article, we suggest an approach that allows us to obtain estimates for correlation in dynamical systems in the case of arbitrary functions. This approach is based on approximation and estimates are obtained with the use of known estimates for Hölder continuous functions. We apply our approach to transitive Anosov diffeomorphisms and derive the central limit theorem for the characteristic functions of certain sets with boundary of zero measure.     

Error estimates for a finite-difference approximation of a mean field model of superconducting vortices in one-dimension

Error estimates for a semi-implicit finite-difference approximationof a mean field model of superconducting vortices are obtained.The L(L¹) error between the approximate and the exact superconductingvortex density of the model is of order h^1/3.     

Prediction and Conditional Simulation of a 2D Lognormal Diffusion Random Field

This paper describes techniques for estimation, prediction and conditional simulation of two-parameter lognormal diffusion random fields which are diffusions on each coordinate and satisfy a particular Markov property. The estimates of the drift and diffusion coefficients, which characterize the lognormal diffusion random field under certain conditions, are used for obtaining kriging predictors. The conditional simulations are obtained using the estimates of the drift and diffusion coefficients, kriging prediction and unconditional simulation for the lognormal diffusion random field.      

Gaussian estimates for the density of the non-linear stochastic heat equation in any space dimension

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the non-linear stochastic heat equation in any space dimension. The driving perturbation is a Gaussian noise which is white in time with some spatially homogeneous covariance. These estimates are obtained using tools of the Malliavin calculus. The most challenging part is the lower bound, which is obtained by adapting a general method developed by Kohatsu-Higa to the underlying spatially homogeneous Gaussian setting. Both lower and upper estimates have the same form: a Gaussian density with a variance which is equal to that of the mild solution of the corresponding linear equation with additive noise.     

Solution of Kirkwood-Salsburg equations for a one-dimensional lattice gas

Expressions are obtained for the lowest correlation functions directly from the Kirkwood-Salsburg equations for an infinite system of particles on a one-dimensional lattice with two-body nearest-neighbor interaction in certain external fields. The problem of finding the external field that makes the density oscillation near the wall uniform is considered.State University, Tyumen'. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 102, No. 3, pp. 463–469, March, 1995.     

多组随机变量相关度量的几个估计

本文首先给出了矩阵行列式的几个不等式,进一步,给出了多组随机变量及其子系统的相关度量的几个估计。     

Inference in canonical correlation analysis

The asymptotic behavior, for large sample size, is given for the distribution of the canonical correlation coefficients. The result is used to examine the Bartlett-Lawley test that the residual population canonical correlation coefficients are zero. A marginal likelihood function for the population coefficients is obtained and the maximum marginal likelihood estimates are shown to provide a bias correction.     

On the mean square error of maximum likelihood estimates of the distribution density of sufficient statistics of the multivariate normal distribution

The extensive use of maximum likelihood estimates underscores the importance of the problem of statistical estimation of their errors. These estimates are of utmost importance in cases where the family of normal distributions and the families related to the normal distributions are considered [1, 2, 4]. The mean square errors of the maximum likelihood estimates of the normal density were investigated in the author's paper [3]. The mean square errors of statistical estimates of some families of densities related to the normal distributions were considered in the papers [4–6]. In the present paper, we obtain an asymptotic expansion of the mean square error of the maximum likelihood estimates of the densities of the joint distribution of sufficient statistics of the family of multivariate normal distributions. The results obtained allow us to construct the mean square errors of the maximum likelihood estimates for the chi-square density and Wishart's density. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 4–11, Perm. 1990.     

On Burr XII Distribution Analysis Under Progressive Type-II Hybrid Censored Data

In the current paper, based on progressive type-II hybrid censored samples, the maximum likelihood and Bayes estimates for the two parameter Burr XII distribution are obtained. We propose the use of expectation-maximization (EM) algorithm to compute the maximum likelihood estimates (MLEs) of model parameters. Further, we derive the asymptotic variance-covariance matrix of the MLEs by applying the missing information principle and it can be utilized to construct asymptotic confidence intervals (CIs) for the parameters. The Bayes estimates of the unknown parameters are obtained under the assumption of gamma priors by using Lindley’s approximation and Markov chain Monte Carlo (MCMC) technique. Also, MCMC samples are used to construct the highest posterior density (HPD) credible intervals. Simulation study is conducted to investigate the accuracy of the estimates and compare the performance of CIs obtained. Finally, one real data set is analyzed for illustrative purposes.     

Strong solutions to the three-dimensional compressible viscoelastic fluids

The existence and uniqueness of the local strong solution to the three-dimensional compressible viscoelastic fluids near the equilibrium are established. In addition to the uniform estimates on the velocity, some essential uniform estimates on the density and the deformation gradient are also obtained.     

Weighted estimating equation: modified GEE in longitudinal data analysis

The method of generalized estimating equations （GEE） introduced by K. Y. Liang and S. L. Zeger has been widely used to analyze longitudinal data. Recently, this method has been criticized for a failure to protect against misspecification of working correlation models, which in some cases leads to loss of efficiency or infeasibility of solutions. In this paper, we present a new method named as ＇weighted estimating equations （WEE）＇ for estimating the correlation parameters. The new estimates of correlation parameters are obtained as the solutions of these weighted estimating equations. For some commonly assumed correlation structures, we show that there exists a unique feasible solution to these weighted estimating equations regardless the correlation structure is correctly specified or not. The new feasible estimates of correlation parameters are consistent when the working correlation structure is correctly specified. Simulation results suggest that the new method works well in finite samples.     

Dissipation of one-species VMB system with non-uniform ionic background

The Vlasov–Maxwell–Boltzmann system is a system that describes the dynamics of charged particles when there is electromagnetic field. There have been extensive studies on the energy estimates and global existence of solutions to the one- and two-species Vlasov–Maxwell–Boltzmann system when the background density is constant. In this work, we first deal with the energy estimates of the one-species Vlasov–Maxwell–Boltzmann system when the background density is a small perturbation from the stationary state. Then, we handle the dissipation rate for the linearized one-species Vlasov–Maxwell–Boltzmann system with the background density having a small perturbation around the stationary state.     

A Multiresolution Tree-Structured Spatial Linear Model

Multiresolution spatial models are able to capture complex dependence correlation in spatial data and are excellent alternatives to the traditional random field models for mapping spatial processes. Because of the multiresolution structures, spatial process prediction can be obtained by direct and fast computation algorithms. However, the existing multiresolution models usually assume a simple constant mean structure, which may not be suitable in practice. In this article, we focus on a multiresolution tree-structured spatial model and extend the model to incorporate a linear regression mean. We explore the properties of the multiresolution tree-structured spatial linear model in depth and estimate the parameters in the linear regression mean and the spatial-dependence structure simultaneously. An expectation-maximization algorithm is adopted to obtain the maximum likelihood estimates of the model parameters and the corresponding information matrix. Given the estimated parameters, a one-pass change-of-resolution Kalman filter algorithm is implemented to obtain the best linear unbiased predictor of the true underlying spatial process. For illustration, the methodology is applied to optimally map crop yield in a Wisconsin field, after accounting for the field conditions by a linear regression.     

Multivariate k-nearest neighbor density estimates

Under appropriate assumptions, expressions describing the asymptotic behavior of the bias and variance of k-nearest neighbor density estimates with weight function w are obtained. The behavior of these estimates is compared with that of kernel estimates. Particular attention is paid to the properties of the estimates in the tail.     

GLOBAL EXISTENCE AND CONVERGENCE RATES OF SMOOTH SOLUTIONS FOR THE 3-D COMPRESSIBLE MAGNETOHYDRODYNAMIC EQUATIONS WITHOUT HEAT CONDUCTIVITY

In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations without heat conductivity, which is a hyperbolic-parabolic system. The global solutions are obtained by combining the local existence and a priori estimates if H3-norm of the initial perturbation around a constant states is small enough and its L1-norm is bounded. A priori decay-in-time estimates on the pressure, velocity and magnetic field are used to get the uniform bound of entropy. Moreover, the optimal convergence rates are also obtained.     

Asymptotics for a Curve with a Given Distribution Density of the First Exit Position for a Wiener Process

The following inverse first exit problem for a Wiener process is considered: to find an upper class with a given distribution of the first exit point from the domain bounded by this curve. Some estimates are obtained for a curve with a given density at zero. Bibliography: 2 titles.     

Asymptotic lower bounds for the risk of estimators of the value of a spectral density function

Summary Discrete parameter stationary processes with joint Gaussian distributions are considered. Loss is measured by squared error. It is shown that when estimating the value of g(x ₀) of a spectral density function g the risk has an asymptotic lower bound similar to those previously obtained by Farrell [6] for the problem of density function estimation. A similar result is obtained for estimates of an entire function when risk is measured by integrated square error.     

Estimation of the distribution of random shifts deformation

Consider discrete values of functions shifted by unobserved translation effects, which are independent realizations of a random variable with unknown distribution μ modeling the variability in the response of each individual. Our aim is to construct a nonparametric estimator of the density of these random translation deformations using semiparametric preliminary estimates of the shifts. Based on the results of Dalalyan et al. [7], semiparametric estimators are obtained in our discrete framework and their performance studied. From these estimates we construct a nonparametric estimator of the target density. Both rates of convergence and an algorithm to construct the estimator are provided.