Matrix-Exponential Distributions in Applied Probability

ISBN-13: 9781493983773 / Angielski / Miękka / 2018 / 736 str.

Mogens Bladt; Bo Friis Nielsen

Recenzje

"The book is useful not only for those who want to learn about ME distributions but also about renewal theory, random walks, Markov chains ... The book should be accessible to a beginning graduate student and many portions of it to undergraduates as well. ... Although this is a book geared towards practitioners, it also does a very good job in explaining some of the finest points of the theory." (Takis Konstantopoulos, Mathematical Reviews, May, 2018)

"This book may be used as a graduate-level textbook, and the authors provide outlines of several possible courses based on it, as well as exercises at the end of each chapter. ... this book is a very good introduction to phase-type and matrix-exponential distributions, which manages to effectively convey the scope of their applications across probability and statistics, and seems well suited to its intended graduate-level audience." (Fraser Daly, zbMATH 1375.60002, 2018)

Spis treści

Preface.- Notation.- Preliminaries on Stochastic Processes.- Martingales and More General Markov Processes.- Phase-type Distributions.- Matrix-exponential Distributions.- Renewal Theory.- Random Walks.- Regeneration and Harris Chains.- Multivariate Distributions.- Markov Additive Processes.- Markovian Point Processes.- Some Applications to Risk Theory.- Statistical Methods for Markov Processes.- Estimation of Phase-type Distributions.- Bibliographic Notes.- Appendix.

O autorze

Bo Friis Nielsen is an associate professor in the Department of Applied Mathematics and Computer Science at the Technical University of Denmark. 

Mogens Bladt is a researcher in the Department of Probability  and Statistics at the Institute for Applied Mathematics and Systems, National University of Mexico (UNAM).

Blurb

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.