Statistical Inference for Piecewise-deterministic Markov Processes E-Book
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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Piecewise-deterministic Markov processes form a class of stochastic models with a sizeable scope of applications: biology, insurance, neuroscience, networks, finance... Such processes are defined by a deterministic motion punctuated by random jumps at random times, and offer simple yet challenging models to study. Nevertheless, the issue of statistical estimation of the parameters ruling the jump mechanism is far from trivial.
Responding to new developments in the field as well as to current research interests and needs, Statistical inference for piecewise-deterministic Markov processes offers a detailed and comprehensive survey of state-of-the-art results. It covers a wide range of general processes as well as applied models. The present book also dwells on statistics in the context of Markov chains, since piecewise-deterministic Markov processes are characterized by an embedded Markov chain corresponding to the position of the process right after the jumps.
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Seitenzahl: 411
Veröffentlichungsjahr: 2018
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Table of Contents
Cover
Preface
Introduction
I.1. Piecewise-deterministic Markov processes
I.2. Connection with other stochastic processes
I.3. Book contents
I.4. Bibliography
1 Statistical Analysis for Structured Models on Trees
1.1. Introduction
1.2. Size-dependent division rate
1.3. Estimating the age-dependent division rate
1.4. Bibliography
2 Regularity of the Invariant Measure and Non-parametric Estimation of the Jump Rate
2.1. Introduction
2.2. Absolute continuity of the invariant measure
2.3. Estimation of the spiking rate in systems of interacting neurons
2.4. Bibliography
3 Level Crossings and Absorption of an Insurance Model
3.1. An insurance model
3.2. Some results about the crossing and absorption features
3.3. Inference for the absorption features of the process
3.4. Inference for the average number of crossings
3.5. Some additional proofs
3.6. Bibliography
4 Robust Estimation for Markov Chains with Applications to Piecewise-deterministic Markov Processes
4.1. Introduction
4.2. (Pseudo)-regenerative Markov chains
4.3. Robust functional parameter estimation for Markov chains
4.4. Central limit theorem for functionals of Markov chains and robustness
4.5. A Markov view for estimators in PDMPs
4.6. Robustness for risk PDMP models
4.7. Simulations
4.8. Bibliography
5 Numerical Method for Control of Piecewise-deterministic Markov Processes
5.1. Introduction
5.2. Simulation of piecewise-deterministic Markov processes
5.3. Optimal stopping
5.4. Exit time
5.5. Numerical example
5.6. Conclusion
5.7. Bibliography
6 Rupture Detection in Fatigue Crack Propagation
6.1. Phenomenon of crack propagation
6.2. Modeling crack propagation
6.3. PDMP models of propagation
6.4. Rupture detection
6.5. Conclusion and perspectives
6.6. Bibliography
7 Piecewise-deterministic Markov Processes for Spatio-temporal Population Dynamics
7.1. Introduction
7.2. Stratified dispersal models
7.3. Metapopulation epidemic model
7.4. Stochastic approaches for modeling spatial trajectories
7.5. Conclusion
7.6. Bibliography
List of Authors
Index
End User License Agreement
List of Tables
5 Numerical Method for Control of Piecewise-deterministic Markov Processes
Table 5.1. Numerical values of the parameters of the corrosion model
Table 5.2. Numerical results for the calculation of the value function for the corrosion process
Table 5.3. Approximation results for the distribution of the exit time.
6 Rupture Detection in Fatigue Crack Propagation
Table 6.1. Notations
Table 6.2. Statistics concerning the crack length at transition, the transition times in terms of number of cycles and the corresponding stress intensity factor range
Table 6.3. Normalized distance D and crack position at the end of the propagation d
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as a function of the type of crack
Table 6.4. Statistics concerning the crack at transition, the transition times in terms of the number of cycles and the corresponding stress intensity factor range
Guide
Cover
Table of Contents
Begin Reading
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Chinese proverb promotingcooperation over solitary work
Series Editor
Nikolaos Limnios
Statistical Inference for Piecewise-deterministic Markov Processes
Edited by
Romain Azaïs
Florian Bouguet
First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUK
www.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USA
www.wiley.com
© ISTE Ltd 2018
The rights of Romain Azaïs and Florian Bouguet to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2018944661
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-302-8
Preface
The idea for this book stems from the organization of a workshop that took place in Nancy in February 2017. Our motivation was to bring together the French community of statisticians – and a few probability researchers – working directly or indirectly on piecewise-deterministic Markov processes (PDMPs). Thanks to the impetus and advice of Prof. Nikolaos Limnios, we were able to convert this short manifestation into a lasting project, this book.
Since PDMPs form a class of stochastic model with a large scope of applications, many mathematicians have come to work on this subject, sometimes without even realizing it. Although these stochastic models are rather simple, the issue of statistical estimation of the parameters ruling the jump mechanism is far from trivial. The aim of this book is to offer an overview of state-of-the-art methods developed to tackle this issue. Thus, we invited our orators and their co-authors to participate in this project and tried to keep the style of the various authors while providing a homogeneous work with consistent notation and goals.
Statistical Inference for Piecewise-deterministic Markov Processes consists of a general introduction and seven autonomous chapters that reflect the research work of their respective authors, with distinct interests and methods. Nevertheless, they can be investigated according to two reading grids corresponding to the application domains (biology in Chapters 1, 2 and 7, reliability in Chapters 5 and 6, and risk and insurance in Chapters 3 and 4) or to the statistical issues (non-parametric jump rate estimation in Chapters 1 and 2, estimation problems related to level crossing in Chapters 3, 4 and 5, and parametric estimation from partially observed trajectories in Chapters 6 and 7).
The production of this book and of the workshop it originates from would not have been possible without the direct support of the Inria Nancy–Grand Est research center, the Institut Élie Cartan de Lorraine and grants from the French institutions Centre National de la Recherche Scientifique and Agence Nationale de la Recherche.
This adventure started in Nancy, but we write these opening lines half a world apart, each of us far from Lorraine. We want to dedicate this book to all the friends and colleagues we have there. We sincerely thank all the authors, and also the orators of the workshop who did not participate in the writing of this book but nevertheless contributed to a delightful colloquium. Last but not least, warm thanks are due to Marine and Élodie, who constantly encouraged and supported us during this project.
Romain AZAÏS & Florian BOUGUETMay 2018
List of Acronyms
a.s.
almost surely
c.d.f.
cumulative distribution function
càdlàg
right continuous with left limits
CL
Cramér–Lundberg
CLT
central limit theorem
CLVQ
competitive learning vector quantization
DP
diffusion process
EM
expectation maximization
FCP
fatigue crack propagation
i.i.d.
independent and identically distributed
KDEM
kinetic dietary exposure model
MC
Markov chain
MCMC
Markov chain Monte Carlo
ODE
ordinary differential equation
PDE
partial differential equation
PDMP
piecewise-deterministic Markov process
r.v.
random variable
RP
renewal process
SA
Sparre–Andersen
SDE
stochastic differential equation
