Lecture Notes in Statistics

Their Use in Reliability and DNA Analysis

By: Vlad Stefan Barbu, Nikolaos Limnios

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Paperback | 28 August 2008

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240 Pages

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This book is concerned with the estimation of discrete-time semi-Markov and hidden semi-Markov processes. Semi-Markov processes are much more general and better adapted to applications than the Markov ones because sojourn times in any state can be arbitrarily distributed, as opposed to the geometrically distributed sojourn time in the Markov case. Another unique feature of the book is the use of discrete time, especially useful in some specific applications where the time scale is intrinsically discrete. The models presented in the book are specifically adapted to reliability studies and DNA analysis.

The book is mainly intended for applied probabilists and statisticians interested in semi-Markov chains theory, reliability and DNA analysis, and for theoretical oriented reliability and bioinformatics engineers. It can also serve as a text for a six month research-oriented course at a Master or PhD level. The prerequisites are a background in probability theory and finite state space Markov chains.

Industry Reviews

From the reviews:

"Overall, this is a stimulating book. As it says, the discrete-time framework has been underused in the past in the view of the fact that much real data in practice (I would even say most) is recorded in discrete time. Also, the applications listed, particularly the DNA sequencing, are important ad timely." (Martin Crowder, International Statistical Review 2009, 77, 2)

"Barbu and Limnios's goal is to present a complete picture of the basic theory of finite state space semi-Markov processes in discrete time, describe its applications to reliability and DNA analysis, and obtain estimation results for hidden semi-Markov models (HSSM). ... Each chapter contains several exercises. There are five appendices that render the book self-contained. ... I highly recommend this book for applied probabilists and statisticians interested in reliability and DNA analysis, and for theoretically oriented reliability and bioinformatics engineers." (P. R. Parthasarathy, ACM Computing Reviews, March, 2009)

"...this book will be useful as a reference for researchers interested in discrete-time semi-Markov and hidden semi-Markov processes..." (Biometrics 65, 1312, December 2009)

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Preface	p. VII
Introduction	p. 1
Object of the Study	p. 1
The Underlying Idea in Semi-Markov Models	p. 1
Discrete Time	p. 4
Discrete-Time Semi-Markov Framework	p. 5
Discrete-Time Renewal Processes	p. 5
Semi-Markov Chains	p. 7
Semi-Markov Chain Estimation	p. 10
Reliability Theory of Discrete-Time Semi-Markov Systems	p. 11
Hidden Semi-Markov Models	p. 13
Discrete-Time Renewal Processes	p. 17
Renewal Chains	p. 18
Limit Theorems	p. 26
Delayed Renewal Chains	p. 31
Alternating Renewal Chain	p. 37
Exercises	p. 40
Semi-Markov Chains	p. 43
Markov Renewal Chains and Semi-Markov Chains	p. 44
Markov Renewal Theory	p. 51
Limit Theorems for Markov Renewal Chains	p. 60
Periodic SMC	p. 67
Monte Carlo Methods	p. 67
Example: a Textile Factory	p. 68
Exercises	p. 73
Nonparametric Estimation for Semi-Markov Chains	p. 75
Construction of the Estimators	p. 76
Asymptotic Properties of the Estimators	p. 79
Example: a Textile Factory (Continuation)	p. 95
Exercises	p. 98
Reliability Theory for Discrete-Time Semi-Markov Systems	p. 101
Reliability Function and Associated Measures	p. 102
Reliability	p. 104
Availability	p. 106
Maintainability	p. 108
Failure Rates	p. 108
Mean Hitting Times	p. 111
Nonparametric Estimation of Reliability	p. 115
Example: a Textile Factory (Continuation)	p. 122
Exercises	p. 129
Hidden Semi-Markov Model and Estimation	p. 131
Hidden Semi-Markov Model	p. 132
Estimation of a Hidden Semi-Markov Model	p. 136
Consistency of Maximum-Likelihood Estimator	p. 137
Asymptotic Normality of Maximum-Likelihood Estimator	p. 144
Monte Carlo Algorithm	p. 150
EM Algorithm for a Hidden Semi-Markov Model	p. 151
Preliminaries	p. 151
EM Algorithm for Hidden Model SM-M0	p. 153
EM Algorithm for Hidden Model SM-M1	p. 158
Proofs	p. 163
Example: CpG Island Detection	p. 173
Exercises	p. 177
Lemmas for Semi-Markov Chains	p. 179
Lemmas for Hidden Semi-Markov Chains	p. 183
Lemmas for Hidden SM-M0 Chains	p. 183
Lemmas for Hidden SM-M1 Chains	p. 184
Some Proofs	p. 185
Markov Chains	p. 195
Definition and Transition Function	p. 195
Strong Markov Property	p. 197
Recurrent and Transient States	p. 197
Stationary Distribution	p. 199
Markov Chains and Reliability	p. 200
Miscellaneous	p. 203
Notation	p. 207
References	p. 211
Index	p. 221
Table of Contents provided by Ingram. All Rights Reserved.

Lecture Notes in Statistics

Their Use in Reliability and DNA Analysis

At a Glance

Paperback

Industry Reviews

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