The Cauchy Problem for Shilov Parabolic Equations

We establish well-posed solvability of the Cauchy problem for the Shilov parabolic equations with time-dependent coefficients whose initial data are tempered distributions. For a certain class of equations we state necessary and sufficient conditions for unique solvability of the Cauchy problem whose properties with respect to the spatial variable are typical for a fundamental solution. The results are characterized only by the order and the parabolicity exponent.     

Multiplicity distributions in a two-component branching process

Probability (multiplicity) distributions and those densities (KNO scaling functions) are investigated in a two-component (charged and neutral) branching process. It is shown that the two-component KNO scaling functions depend effectively on one variable in two typical cases. A formula for multiplicity correlation between two components (charged and neutral particles) is formulated. It is applied to the analysis of experimental data.     

The shortest distance in random multi‐type intersection graphs

Using an associated branching process as the basis of our approximation, we show that typical inter‐point distances in a multi‐type random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions, the missing mass corresponding to the event that the vertices are not in the same component of the graph. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 179–209, 2011     

有限混合Gamma分布的拓扑稠密性证明

首先给出了有限混合Erlang分布在正实数轴上所有概率分布中稠密的理论证明,进而给出了混合Gamma分布具有稠密性的结论,说明有限混合Gamma分布具有广泛的适用性,可以用来刻画正实数轴上的任意随机变量.     

General Autoregressive Models with Long-Memory Noise

We give the limiting distribution of the least-squares estimator in the general autoregressive model driven by a long-memory process. We prove that with an appropriate normalization the estimation error converges, in distribution, to a random vector which contains: (1) a stochastic component, due to the presence of the unstable roots, which are multiple Wiener–Itô integrals and a non-linear functionals of stochastic integrals with respect to a Brownian motion; (2) a constant component due to the stable roots; (3) a stochastic component, due to the presence of the explosive roots, which is a mixture of normal distributions.     

Cauchy problem for a class of parabolic systems of Shilov type with variable coefficients

In the case of initial data belonging to a wide class of functions including distributions of Gelfand-Shilov type we establish the correct solvability of the Cauchy problem for a new class of Shilov parabolic systems of equations with partial derivatives with bounded smooth variable lower coefficients and nonnegative genus. We also investigate the conditions of local improvement of the convergence of a solution of this problem to its limiting value when the time variable tends to zero.     

Adaptive Markov Chain Monte Carlo for Bayesian Variable Selection

We describe adaptive Markov chain Monte Carlo (MCMC) methods for sampling posterior distributions arising from Bayesian variable selection problems. Point-mass mixture priors are commonly used in Bayesian variable selection problems in regression. However, for generalized linear and nonlinear models where the conditional densities cannot be obtained directly, the resulting mixture posterior may be difficult to sample using standard MCMC methods due to multimodality. We introduce an adaptive MCMC scheme that automatically tunes the parameters of a family of mixture proposal distributions during simulation. The resulting chain adapts to sample efficiently from multimodal target distributions. For variable selection problems point-mass components are included in the mixture, and the associated weights adapt to approximate marginal posterior variable inclusion probabilities, while the remaining components approximate the posterior over nonzero values. The resulting sampler transitions efficiently between models, performing parameter estimation and variable selection simultaneously. Ergodicity and convergence are guaranteed by limiting the adaptation based on recent theoretical results. The algorithm is demonstrated on a logistic regression model, a sparse kernel regression, and a random field model from statistical biophysics; in each case the adaptive algorithm dramatically outperforms traditional MH algorithms. Supplementary materials for this article are available online.     

Optimal Scaling of Random Walk Metropolis Algorithms with Non-Gaussian Proposals

The asymptotic optimal scaling of random walk Metropolis (RWM) algorithms with Gaussian proposal distributions is well understood for certain specific classes of target distributions. These asymptotic results easily extend to any light tailed proposal distribution with finite fourth moment. However, heavy tailed proposal distributions such as the Cauchy distribution are known to have a number of desirable properties, and in many situations are preferable to light tailed proposal distributions. Therefore we consider the question of scaling for Cauchy distributed proposals for a wide range of independent and identically distributed (iid) product densities. The results are somewhat surprising as to when and when not Cauchy distributed proposals are preferable to Gaussian proposal distributions. This provides motivation for finding proposal distributions which improve on both Gaussian and Cauchy proposals, both for finite dimensional target distributions and asymptotically as the dimension of the target density, d → ∞. Throughout we seek the scaling of the proposal distribution which maximizes the expected squared jumping distance (ESJD).     

A review on consistency and robustness properties of support vector machines for heavy-tailed distributions

Support vector machines (SVMs) belong to the class of modern statistical machine learning techniques and can be described as M-estimators with a Hilbert norm regularization term for functions. SVMs are consistent and robust for classification and regression purposes if based on a Lipschitz continuous loss and a bounded continuous kernel with a dense reproducing kernel Hilbert space. For regression, one of the conditions used is that the output variable Y has a finite first absolute moment. This assumption, however, excludes heavy-tailed distributions. Recently, the applicability of SVMs was enlarged to these distributions by considering shifted loss functions. In this review paper, we briefly describe the approach of SVMs based on shifted loss functions and list some properties of such SVMs. Then, we prove that SVMs based on a bounded continuous kernel and on a convex and Lipschitz continuous, but not necessarily differentiable, shifted loss function have a bounded Bouligand influence function for all distributions, even for heavy-tailed distributions including extreme value distributions and Cauchy distributions. SVMs are thus robust in this sense. Our result covers the important loss functions ${\epsilon}$ -insensitive for regression and pinball for quantile regression, which were not covered by earlier results on the influence function. We demonstrate the usefulness of SVMs even for heavy-tailed distributions by applying SVMs to a simulated data set with Cauchy errors and to a data set of large fire insurance claims of Copenhagen Re.     

Total population size in critical branching processes in a random environment

For a critical branching process evolving in a randomenvironment and having geometric distributions of offspring sizes, we study the tail behavior of the distributions of the total size of the population and the maximal number of particles in a generation up to the moment of extinction of the process.     

Normed-convergence theory for supercritical branching processes

A proof is given of the basic normed-convergence theorem for the ordinary supercritical Bienaymé-Galton-Watson process with finite mean. Part of it is adapted to obtain an analogous result for inhomogeneous supercritical processes (i.e. branching processes in varying environment). This is used in part to give a detailed discussion on the normed- convergence behaviour of the ordinary process in the ‘explosive’ case (i.e with infinite mean); and rather pathological limit behaviour is found to obtain.     

Sampling Bias Correction in the Model of Mixtures with Varying Concentrations

Model of mixture with varying concentrations is a generalization of the classical finite mixture model in which the mixing probabilities (concentrations) vary from observation to observation. We consider the case when the concentrations of the mixture components are known, but no assumptions on the distributions of the observed variable are made. The problem is to estimate the moments of the components’ distributions. We propose a modification of the Horvitz-Thompson weighting for moments estimation by observations from mixture with varying concentrations in presence of sampling bias. Consistency of obtained estimators is demonstrated. Results of simulations are presented.     

Robust Hierarchical Bayes Estimation of Small Area Characteristics in the Presence of Covariates and Outliers

A robust hierarchical Bayes method is developed to smooth small area means when a number of covariates are available. The method is particularly suited when one or more outliers are present in the data. It is well known that the regular Bayes estimators of small. area means, under normal prior distribution, perform poorly in presence of even one extreme observation. In this case the Bayes estimators collapse to the direct survey estimators. This paper introduces a general theory for robust hierarchical Bayes estimation procedure using a fairly rich class of scale mixtures of normal prior distributions. To retain maximum benefit from combining information from related sources, we suggest to use Cauchy prior distribution for the outlying areas and an appropriate scale mixture of normal prior whose tail is lighter than the Cauchy prior for the rest of the areas. It is shown that, unlike the hierarchical Bayes estimator under a normal prior, our estimator has more protection against outlying observations.     

一类变系数线性波动方程初值问题的求解

本文讨论了一类变系数线性波动方程初值问题的求解，获得较为理想的结果．     

Invariants for feedback equivalence and Cauchy characteristic multifoliations of nonlinear control systems

Linearization of a nonlinear feedback control system under nonlinear feedback is treated as a problem of equivalence-under the Lie pseudogroup of feedback transformations-of distributions on the product manifold of the state and control variables. The new feature of this paper is that it introduces the Cauchy characteristic sub-distributions of these distributions and their derived distributions. These Cauchy characteristic distributions are involutive and nested, hence define a Multifoliate Structure. A necessary condition for feedback equivalence of two nonlinear control systems is that these multifoliations be transformed under the feedback pseudogroup. For linear systems, this Cauchy characteristic multifoliate structuee is readily computed in terms of the (A, B)-matrix that defines the linear system. Assuming that the conditions for local feedback linearization are satisfied, the existence of a global feedback linearizing transformation is dependent on computing an element of the first cohomology group of the space with coefficients in the sheaf of groupoid of infinitesimal feedback automorphisms of the linear system. The theorem quoted above about the Cauchy characteristic multifoliations provides some information about this groupoid. It is computed explicitly and directly for control systems with one- or two-state dimensions. Finally, these Cauchy characteristic sub-distributions must inevitably play a role in the numerical or symbolic computational analysis of the Hunt-Su partial differential equations for the feedback-linearizing transformation.Senior Research Associate of the National Research Council at the Ames Research Center of NASA.     

The sector constants of continuous state branching processes with immigration

Continuous state branching processes with immigration are studied. We are particularly concerned with the associated (non-symmetric) Dirichlet form. After observing that gamma distributions are only reversible distributions for this class of models, we prove that every generalized gamma convolution is a stationary distribution of the process with suitably chosen branching mechanism and with continuous immigration. For such non-reversible processes, the strong sector condition is discussed in terms of a characteristic called the Thorin measure. In addition, some connections with notion from non-commutative probability theory will be pointed out through calculations involving the Stieltjes transform.     

Two-Type Branching Processes with Subexponential Life-Spans and SIR Epidemic Models

Abstract

We consider a model of age-dependent branching stochastic process that takes into account the incubation period of the life of individuals. We demonstrate that such processes may be treated as a two-type branching process with a periodic mean matrix. In the case when the Malthusian parameter does not exist study of the process requires additional restrictions on the life and incubation time distributions which define so called subexponential family (Athreya, K. 1972. Branching Processes, Springer, New York). We obtain certain new properties of subexponential distributions, in particular, describe a subclass, which is closed with respect to convolution. Using these results we derive asymptotic behavior of the first and second moments and of the probability of nonextinction. We also prove a limit theorem for the process conditioned on nonextinction.     

出生率与年龄段有关的生灭分枝树

本文基于生灭过程的生灭演化机理, 将生物繁衍过程描述为有向随机图过程-随机分枝树, 建立了出生率与年龄段有关的生灭分枝树演化模型. 本文研究了任一节点在不同年龄及临死时刻的出度分布、虚出度分布和拟出度分布, 并证明了拟出度过程是随机时刻终止的Poisson过程, 讨论了首生年龄及相对出生年龄的分布, 给出了任一节点成为孤立节点的概率.     

Modeling of the microwave drying process of aqueous dielectrics

We propose a thermodynamic framework for describing the microwave drying process of aqueous dielectrics based on Maxwell-Lorentz Field equations and mixture theory. Several issues are discussed such as the form of entropy equation; the constitutive relations for the macroscopic electric polarization vectors, Cauchy stresses, heat fluxes, internal momentum supplies, etc., for each component of the mixture: porous solid, water and gas in different regions; and the interfacial jump conditions between different regions in the mixture. A brief examination of the status of material frame indifference within the context of our framework is presented.     

A new class of bivariate distributions and its mixture

A new class of bivariate distributions is presented in this paper. The procedure used in this paper is based on a latent random variable with exponential distribution. The model introduced here is of Marshall-Olkin type. A mixture of the proposed bivariate distributions is also discussed. The results obtained here generalize those of the bivariate exponential distribution present in the literature.