Why certain discrete phase type representations have numerically stable spectral decomposition

The discrete-time phase type (PH) distributions are used in the numerical solution of many problems. The representation of a PH distribution consists of a vector and a substochastic matrix. The common feature of the PH distributions is that they have rational moment generating function. The moment generating function depends on the eigendecomposition of the transition probability matrix of the PH distribution, but the eigenvalues and eigenvectors are important in other cases as well. Due to the finite precision of the numerical calculations or other reasons, a representation may contain errors that change the eigendecomposition. This paper presents upper bounds on the change of the eigendecomposition when the PH representation is perturbed. Since these bounds do not depend on a particular representation, but they hold for all, the analysed PH representations have uniformly stable spectral decompositions.      

The skew elliptical distributions and their quadratic forms

In this paper, a family of the skew elliptical distributions is defined and investigated. Some basic properties, such as stochastic representation, marginal and conditional distributions, distribution under linear transformations, moments and moment generating function are derived. The joint distribution of several quadratic forms is obtained. An example is given to show that the distributions of some statistics as the functions of the quadratic forms can be derived for various applications.     

On powers of Stieltjes moment sequences,II

We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, and prove an integral representation of the logarithm of the moment sequence in analogy to the Lévy–Khintchine representation. We use the result to construct product convolution semigroups with moments of all orders and to calculate their Mellin transforms. As an application we construct a positive generating function for the orthonormal Hermite polynomials.     

Perturbed Laguerre unitary ensembles,Hankel determinants,and information theory

This article investigates a key information‐theoretic performance metric in multiple‐antenna wireless communications, the so‐called outage probability. The article is partly a review, with the methodology based mainly on [10], while also presenting some new results. The outage probability may be expressed in terms of a moment generating function, which involves a Hankel determinant generated from a perturbed Laguerre weight. For this Hankel determinant, we present two separate integral representations, both involving solutions to certain non‐linear differential equations. In the second case, this is identified with a particular σ‐form of Painlevé V. As an alternative to the Painlevé V, we show that this second integral representation may also be expressed in terms of a non‐linear second order difference equation. Copyright © 2014 John Wiley & Sons, Ltd.     

MOMENT GENERATING FUNCTIONS OF RANDOM VARIABLES AND ASYMPTOTIC BEHAVIOUR FOR GENERALIZED FELLER OPERATORS

1. Introduction and NotationsThe generalized Feller operators which include many famous operators, such ajsBernstein, Szasz-Mirakjan, BaskakoV, Meyer--K5nig and Zeller operators, can be constructed by making use of the probabilistic method. In the paper [1][2], Xu JihuaPr(--lvjdetl a general scheme f(,r its construction, and Zhao Jillghui showed that theFeller type operators are of good approximations f'or unbounded functions.Our purpose is to present representation of moment generating …     

A new construction for skew multivariate distributions

This paper considers a new approach to develop a very general class of skew multivariate distributions. The approach is based on a linear combination of an elliptically distributed random variable with a linear constraint. Using this approach two different classes of multivariate distributions are constructed based on original distribution. These new classes include different types of skew normal (type A and type B) and other skew elliptical distributions, exist in the literature. We also derive the moment generating function, marginal and conditional density of our proposed classes of distributions. Straightforward explanations are applied to demonstrate the relationships among previous approaches by others with our proposed class of skew distributions.     

Testing skew normality via the moment generating function

In this paper, goodness-of-fit tests are constructed for the skew normal law. The proposed tests utilize the fact that the moment generating function of the skew normal variable satisfies a simple differential equation. The empirical counterpart of this equation, involving the empiricalmoment generating function, yields appropriate test statistics. The consistency of the tests is investigated under general assumptions, and the finite-sample behavior of the proposed method is investigated via a parametric bootstrap procedure.     

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本文研究了一种可用于分析非负数据的新的分布族. 它用一个随机表示来定义, 是一个半正态随机变量与一个指数随机变量的幂的混合. 其密度函数和性质, 包括风险函数, 矩和矩母函数被导出. 该分布族的实用性和灵活性通过一个实例用最大似然程序被阐明.     

Local linear regression for functional data

We study a non-linear regression model with functional data as inputs and scalar response. We propose a pointwise estimate of the regression function that maps a Hilbert space onto the real line by a local linear method and derive its asymptotic mean square error. Computations involve a linear inverse problem as well as a representation of the small ball probability of the data and are based on recent advances in this area.     

Extremes of Normed Empirical Moment Generating Function Processes

We present a method for deriving the limiting distribution of the maximum of a normed empirical moment generating function process indexed by one parameter. We first extend slightly the results of Csörg et al. (1986b) to provide the rate of convergence for a Gaussian approximation to a non-Donsker empirical process. In cases we consider, the maximum tends to infinity in probability, but when appropriately scaled has a limiting Gumbel extreme value distribution.AMS 2000 Subject Classification Primary—62E20, 60G70*Author for correspondence: School of Mathematics and Statistics, University of Sydney, New South Wales 2006, Australia.     

Existence of Bayesian Estimates for the Polychotomous Quantal Response Models

This paper investigates the existence of Bayesian estimates for polychotomous quantal response models using a uniform improper prior distribution on the regression parameters. Necessary and sufficient conditions for the propriety of the posterior distribution with a general link function are established. In addition, the sufficient conditions for the existence of the posterior moments and the posterior moment generating function are obtained. It is also found that the propriety guarantees the existence of the maximum likelihood estimate.     

Group performance evaluation, an application of data envelopment analysis

The contribution of this paper is to provide an approach for evaluating the performance of a group of decision making units (DMUs) based on the production technology. Group evaluation is an application of data envelopment analysis (DEA). DEA uses linear programming to provide a suitable technique to estimate a multiple-input/multiple-output empirical efficient function. This paper applies group evaluation to evaluate the performance of Iranian commercial banks.     

Some results on co-recursive associated Meixner and Charlier polynomials

An explicit representation of the co-recursive associated Meixner polynomials is given in terms of hypergeometric functions. This representation allows to derive a generating function, the Stieltjes transform of the orthogonality measure and the fourth-order difference equation verified by these polynomials. Special attention is given to some simple limiting cases ocurring in the solution of the Chapman-Kolmogorov equation of linear birth and death processes.     

Generalized moment estimation of stochastic differential equations

We study the semiparametric estimation of stochastic differential equations employing methods based on moment conditions, comparing the finite sample and robustness properties of generalized method of moments, empirical likelihood and minimum contrast methods using unconditional and conditional formulations of moment conditions. The results obtained indicate that the estimators proposed, particularly, the estimators based on exponential tilting, obtain better results than those of the generalized methods of moments normally used to estimate stochastic differential equations. This conclusion is mainly derived from the robustness properties of this method in the presence of problems of incorrect specification.     

Approximate Bayesian shrinkage estimation

A Bayesian shrinkage estimate for the mean in the generalized linear empirical Bayes model is proposed. The posterior mean under the empirical Bayes model has a shrinkage pattern. The shrinkage factor is estimated by using a Bayesian method with the regression coefficients to be fixed at the maximum extended quasi-likelihood estimates. This approach develops a Bayesian shrinkage estimate of the mean which is numerically quite tractable. The method is illustrated with a data set, and the estimate is compared with an earlier one based on an empirical Bayes method. In a special case of the homogeneous model with exchangeable priors, the performance of the Bayesian estimate is illustrated by computer simulations. The simulation result shows as improvement of the Bayesian estimate over the empirical Bayes estimate in some situations.     

The perturbed compound Poisson risk model with linear dividend barrier

In this paper, we consider a diffusion perturbed classical compound Poisson risk model in the presence of a linear dividend barrier. Partial integro-differential equations for the moment generating function and the nth moment of the present value of all dividends until ruin are derived. Moreover, explicit solutions for the nth moment of the present value of dividend payments are obtained when the individual claim size distribution is exponential. We also provided some numerical examples to illustrate the applications of the explicit solutions. Finally we derive partial integro-differential equations with boundary conditions for the Gerber-Shiu function.     

Simulations of full multivariate Tweedie with flexible dependence structure

We employ a variables-in-common method for constructing multivariate Tweedie distributions, based on linear combinations of independent univariate Tweedie variables. The method lies on the convolution and scaling properties of the Tweedie laws, using the cumulant generating function for characterization of the distributions and correlation structure. The routine allows the equivalence between independence and zero correlation and gives a parametrization through given values of the mean vector and dispersion matrix, similarly to the Gaussian vector. Our approach leads to a matrix representation of multivariate Tweedie models, which permits the simulations of many known distributions, including Gaussian, Poisson, non-central gamma, gamma, and inverse Gaussian, both positively or negatively correlated.     

A Poisson bridge between fractional Brownian motion and stable Lévy motion

We study a non-Gaussian and non-stable process arising as the limit of sums of rescaled renewal processes under the condition of intermediate growth. The process has been characterized earlier by the cumulant generating function of its finite-dimensional distributions. Here, we derive a more tractable representation for it as a stochastic integral of a deterministic function with respect to a compensated Poisson random measure. Employing the representation we show that the process is locally and globally asymptotically self-similar with fractional Brownian motion and stable Lévy motion as its tangent limits.     

Exponential Hilbert series and the Stirling numbers of the second kind

We consider the exponential generating function whose coefficients encode the dimensions of irreducible highest weight representations which lie on a given ray in the dominant chamber of the weight lattice. This formal power series can be considered as an exponential version of the Hilbert series of a flag variety. In this context, we compute a simple closed form for the exponential generating function in terms of finitely many differential operators and the Stirling polynomials. We prove that this series converges to a product of a rational polynomial and an exponential, and that, by summing the constant term and linear coefficient of this polynomial, we recover the dimension of the representation.