Theory of Stochastic Objects: Probability, Stochastic Processes and Inference

409 Pages·2018·7.859 MB·English

by  Athanasios Christou Micheas

Theory of
Stochastic Objects
Probability, Stochastic Processes
and Inference
CHAPMAN & HALL/CRC
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Texts in Statistical Science
Theory of
Stochastic Objects
Probability, Stochastic Processes
and Inference
Athanasios Christou Micheas
Department of Statistics, University of Missouri, USA
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Library of Congress Cataloging-in-Publication Data
Names: Micheas, Athanasios Christou, author.
Title: Theory of stochastic objects : probability, stochastic processes and
inference / by Athanasios Christou Micheas.
Description: Boca Raton, Florida : CRC Press, [2018] | Includes
bibliographical references and index.
Identifiers: LCCN 2017043053| ISBN 9781466515208 (hardback) | ISBN
9781315156705 (e-book)
Subjects: LCSH: Point processes. | Stochastic processes.
Classification: LCC QA274.42 .M53 2018 | DDC 519.2/3--dc23
LC record available at https://lccn.loc.gov/2017043053
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To my family

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
List of Distribution Notations . . . . . . . . . . . . . . . . . . . . . . xxix
1 Rudimentary Models and Simulation Methods. . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 RudimentaryProbability . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 ProbabilityDistributions . . . . . . . . . . . . . . . . . . . 4
1.2.2 Expectation . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.3 MixturesofDistributions . . . . . . . . . . . . . . . . . . 9
1.2.4 TransformationsofRandomVectors . . . . . . . . . . . . . 11
1.3 TheBayesian Approach . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Conjugacy . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 General PriorSelection MethodsandProperties . . . . . . 14
1.3.3 Hierarchical Bayesian Models . . . . . . . . . . . . . . . . 15
1.4 SimulationMethods . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4.1 TheInverseTransformAlgorithm . . . . . . . . . . . . . . 17
1.4.2 TheAcceptance-Rejection Algorithm . . . . . . . . . . . . 18
1.4.3 TheCompositionMethodforGenerating Mixtures . . . . . 20
1.4.4 GeneratingMultivariateNormaland Related Distributions . 20
1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2 StatisticalInference . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 DecisionTheory . . . . . . . . . . . . . . . . . . . . . . . . . . 27
ix