Matrix Variate Generalized Laplace Distributions PDF Download
Are you looking for read ebook online? Search for your book and save it on your Kindle device, PC, phones or tablets. Download Matrix Variate Generalized Laplace Distributions PDF full book. Access full book title Matrix Variate Generalized Laplace Distributions by Tomasz J. Kozubowski. Download full books in PDF and EPUB format.
Author: Tomasz J. Kozubowski
Publisher:
ISBN:
Category :
Languages : en
Pages :
Get Book Here
Book Description
The generalized asymmetric Laplace (GAL) distribution, also known as the variance/mean-gamma model, is a popular flexible class of distributions that can account for peakedness, skewness, and heavier than normal tails, often observed in financial or other empirical data. We consider extensions of the GAL distribution to the matrix variate case, which arise as covariance mixtures of matrix variate normal distributions. Two different mixing mechanisms connected with the nature of the random scaling matrix are considered, leading to what we term matrix variate GAL distributions of Type I and II. While Type I matrix variate GAL distribution has been studied before, there is no comprehensive account of Type II in the literature, except for their rather brief treatment as a special case of matrix variate generalized hyperbolic distributions. With this work we fill this gap, and present an account for basic distributional properties of Type II matrix variate GAL distributions. In particular, we derive their probability density function and the characteristic function, as well as provide stochastic representations related to matrix variate gamma distribution. We also show that this distribution is closed under linear transformations, and study the relevant marginal distributions. In addition, we also briefly account for Type I and discuss the connections with Type II. We hope that this work will be useful in the areas where matrix variate distributions provide an appropriate probabilistic tool for three-way or, more generally, panel data sets, which can arise across different applications.
Author: Tomasz J. Kozubowski
Publisher:
ISBN:
Category :
Languages : en
Pages :
Get Book Here
Book Description
The generalized asymmetric Laplace (GAL) distribution, also known as the variance/mean-gamma model, is a popular flexible class of distributions that can account for peakedness, skewness, and heavier than normal tails, often observed in financial or other empirical data. We consider extensions of the GAL distribution to the matrix variate case, which arise as covariance mixtures of matrix variate normal distributions. Two different mixing mechanisms connected with the nature of the random scaling matrix are considered, leading to what we term matrix variate GAL distributions of Type I and II. While Type I matrix variate GAL distribution has been studied before, there is no comprehensive account of Type II in the literature, except for their rather brief treatment as a special case of matrix variate generalized hyperbolic distributions. With this work we fill this gap, and present an account for basic distributional properties of Type II matrix variate GAL distributions. In particular, we derive their probability density function and the characteristic function, as well as provide stochastic representations related to matrix variate gamma distribution. We also show that this distribution is closed under linear transformations, and study the relevant marginal distributions. In addition, we also briefly account for Type I and discuss the connections with Type II. We hope that this work will be useful in the areas where matrix variate distributions provide an appropriate probabilistic tool for three-way or, more generally, panel data sets, which can arise across different applications.
Author: A K Gupta
Publisher: CRC Press
ISBN: 1351433008
Category : Mathematics
Languages : en
Pages : 382
Get Book Here
Book Description
Useful in physics, economics, psychology, and other fields, random matrices play an important role in the study of multivariate statistical methods. Until now, however, most of the material on random matrices could only be found scattered in various statistical journals. Matrix Variate Distributions gathers and systematically presents most of the recent developments in continuous matrix variate distribution theory and includes new results. After a review of the essential background material, the authors investigate the range of matrix variate distributions, including: matrix variate normal distribution Wishart distribution Matrix variate t-distribution Matrix variate beta distribution F-distribution Matrix variate Dirichlet distribution Matrix quadratic forms With its inclusion of new results, Matrix Variate Distributions promises to stimulate further research and help advance the field of multivariate statistical analysis.
Author: Tomasz J. Kozubowski
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Get Book Here
Book Description
There is considerable literature on matrix-variate gamma distributions, also known as Wishart distributions, which are driven by a shape parameter with values in the (Gindikin) set {i/2, i = 1, . . . , k-1}∪((k-1)/2, ∞). We provide an extension of this class to the case where the shape parameter may actually take on any positive value. In addition to the well-known singular Wishart as well as non-singular matrix-variate gamma distributions, the proposed class includes new singular matrix-variate distributions, with the shape parameter outside of the Gindikin set. This singular, non-Wishart case is no longer permutation invariant and derivation of its scaling properties requires special care. Among numerous newly established properties of the extended class are group-like relations with respect to the positive shape parameter. The latter provide a natural substitute for the classical convolution properties that are crucial in the study of infinite divisibility. Our results provide further clarification regarding the lack of infinite divisibility of Wishart distributions, a classical observation of Paul L'evy. In particular, we clarify why the row/column vectors in the off-diagonal blocks are infinitely divisible. A class of matrix-variate Laplace distributions arises naturally in this set-up as the distributions of the off-diagonal blocks of random gamma matrices. For the class of Laplace rectangular matrices, we obtain distributional identities that follow from the role they play in the structure of the matrix gamma distributions. We present several elegant and convenient stochastic representations of the discussed classes of matrix-valued distributions. In particular, we show that the matrix-variate gamma distribution is a symmetrization of the triangular Rayleigh distributed matrix - a new class of the matrix variables that naturally extend the classical univariate Rayleigh variables. Finally, a connection of the matrix-variate gamma distributions to matrix-valued L'evy processes of a vector argument is made. Namely, a L'evy process, termed a matrix gammaLaplace motion, is obtained by the subordination of the triangular Brownian motion of a vector argument to a vector-valued gamma motion of a vector argument. In this context, we introduce a triangular matrix-valued Rayleigh process, which, through symmetrization, leads to a new matrix-variate gamma process. This process when taken at a properly defined one-dimensional argument has the matrix gamma marginal distribution with the shape parameter equal to its argument.
Author: Samuel Kotz
Publisher: Springer Science & Business Media
ISBN: 146120173X
Category : Mathematics
Languages : en
Pages : 358
Get Book Here
Book Description
This book describes the inferential and modeling advantages that this distribution, together with its generalizations and modifications, offers. The exposition systematically unfolds with many examples, tables, illustrations, and exercises. A comprehensive index and extensive bibliography also make this book an ideal text for a senior undergraduate and graduate seminar on statistical distributions, or for a short half-term academic course in statistics, applied probability, and finance.
Author: Daya Krishna Nagar
Publisher: Universidad de Antioquia
ISBN: 9587142691
Category : Distribution (Probability theory)
Languages : en
Pages : 128
Get Book Here
Book Description
"Random matrices (real or complex) play an important role in the study of multivariate statistical methods. They have been found useful in physics, engineering, economics, psychology and other fields of investigation. Contributions to Complex Matrix Variate Distribution Theory gives a comprehensive coverage of complex random matrices, and defines a number of new complex matrix variate distributions. It also gathers and systematiclly [sic] presents several results on zonal polynomials, invariant polynomials and hypergeometric functions of Hermitian matrices which until now could only be found scattered in various mathematical or statistical journals. This book provides a compact self-contained introduction to the complex matrix variate distribution theory and includes new results that will be a useful source to all those working in the area, stimulate further research, and help advance this field. This book, valuable to researchers, graduate students, and instructors in multivariate statistical analysis, will also interest researchers in a variety of areas including physicists, engineers, psychometricians, and econometricians."--Back cover
Author: Mogens Bladt
Publisher: Springer
ISBN: 1493970496
Category : Mathematics
Languages : en
Pages : 749
Get Book Here
Book Description
This book contains an in-depth treatment of matrix-exponential (ME) distributions and their sub-class of phase-type (PH) distributions. Loosely speaking, an ME distribution is obtained through replacing the intensity parameter in an exponential distribution by a matrix. The ME distributions can also be identified as the class of non-negative distributions with rational Laplace transforms. If the matrix has the structure of a sub-intensity matrix for a Markov jump process we obtain a PH distribution which allows for nice probabilistic interpretations facilitating the derivation of exact solutions and closed form formulas. The full potential of ME and PH unfolds in their use in stochastic modelling. Several chapters on generic applications, like renewal theory, random walks and regenerative processes, are included together with some specific examples from queueing theory and insurance risk. We emphasize our intention towards applications by including an extensive treatment on statistical methods for PH distributions and related processes that will allow practitioners to calibrate models to real data. Aimed as a textbook for graduate students in applied probability and statistics, the book provides all the necessary background on Poisson processes, Markov chains, jump processes, martingales and re-generative methods. It is our hope that the provided background may encourage researchers and practitioners from other fields, like biology, genetics and medicine, who wish to become acquainted with the matrix-exponential method and its applications.
Author: Joel Tropp
Publisher:
ISBN: 9781601988386
Category : Computers
Languages : en
Pages : 256
Get Book Here
Book Description
Random matrices now play a role in many areas of theoretical, applied, and computational mathematics. It is therefore desirable to have tools for studying random matrices that are flexible, easy to use, and powerful. Over the last fifteen years, researchers have developed a remarkable family of results, called matrix concentration inequalities, that achieve all of these goals. This monograph offers an invitation to the field of matrix concentration inequalities. It begins with some history of random matrix theory; it describes a flexible model for random matrices that is suitable for many problems; and it discusses the most important matrix concentration results. To demonstrate the value of these techniques, the presentation includes examples drawn from statistics, machine learning, optimization, combinatorics, algorithms, scientific computing, and beyond.
Author: Marc S. Paolella
Publisher: John Wiley & Sons
ISBN: 9780470035054
Category : Mathematics
Languages : en
Pages : 430
Get Book Here
Book Description
Intermediate Probability is the natural extension of the author's Fundamental Probability. It details several highly important topics, from standard ones such as order statistics, multivariate normal, and convergence concepts, to more advanced ones which are usually not addressed at this mathematical level, or have never previously appeared in textbook form. The author adopts a computational approach throughout, allowing the reader to directly implement the methods, thus greatly enhancing the learning experience and clearly illustrating the applicability, strengths, and weaknesses of the theory. The book: Places great emphasis on the numeric computation of convolutions of random variables, via numeric integration, inversion theorems, fast Fourier transforms, saddlepoint approximations, and simulation. Provides introductory material to required mathematical topics such as complex numbers, Laplace and Fourier transforms, matrix algebra, confluent hypergeometric functions, digamma functions, and Bessel functions. Presents full derivation and numerous computational methods of the stable Paretian and the singly and doubly non-central distributions. A whole chapter is dedicated to mean-variance mixtures, NIG, GIG, generalized hyperbolic and numerous related distributions. A whole chapter is dedicated to nesting, generalizing, and asymmetric extensions of popular distributions, as have become popular in empirical finance and other applications. Provides all essential programming code in Matlab and R. The user-friendly style of writing and attention to detail means that self-study is easily possible, making the book ideal for senior undergraduate and graduate students of mathematics, statistics, econometrics, finance, insurance, and computer science, as well as researchers and professional statisticians working in these fields.
Author: Arak M Mathai
Publisher: World Scientific Publishing Company
ISBN: 9813105070
Category : Mathematics
Languages : en
Pages : 449
Get Book Here
Book Description
This book concentrates on the topic of evaluation of Jacobians in some specific linear as well as nonlinear matrix transformations, in the real and complex cases, which are widely applied in the statistical, physical, engineering, biological and social sciences. It aims to develop some techniques systematically so that anyone with a little exposure to multivariable calculus can easily follow the steps and understand the various methods by which the Jacobians in complicated matrix transformations are evaluated. The material is developed slowly, with lots of worked examples, aimed at self-study. Some exercises are also given, at the end of each section.The book is a valuable reference for statisticians, engineers, physicists, econometricians, applied mathematicians and people working in many other areas. It can be used for a one-semester graduate level course on Jacobians and functions of matrix argument.
Author: Alexander Zeifman
Publisher: MDPI
ISBN: 3036504524
Category : Mathematics
Languages : en
Pages : 370
Get Book Here
Book Description
The aim of this Special Issue of Mathematics is to commemorate the outstanding Russian mathematician Vladimir Zolotarev, whose 90th birthday will be celebrated on February 27th, 2021. The present Special Issue contains a collection of new papers by participants in sessions of the International Seminar on Stability Problems for Stochastic Models founded by Zolotarev. Along with research in probability distributions theory, limit theorems of probability theory, stochastic processes, mathematical statistics, and queuing theory, this collection contains papers dealing with applications of stochastic models in modeling of pension schemes, modeling of extreme precipitation, construction of statistical indicators of scientific publication importance, and other fields.
