Elements of distribution theory

529 Pages·2005·2.207 MB·English

by  Thomas A. Severini

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Elements of Distribution Theory
Thisdetailedintroductiontodistributiontheoryusesnomeasuretheory,
makingitsuitableforstudentsinstatisticsandeconometricsaswellasfor
researchers who use statistical methods. Good backgrounds in calculus
andlinearalgebraareimportantandacourseinelementarymathematical
analysisisuseful,butnotrequired.Anappendixgivesadetailedsummary
ofthemathematicaldefinitionsandresultsthatareusedinthebook.
Topicscoveredrangefromthebasicdistributionanddensityfunctions,
expectation, conditioning, characteristic functions, cumulants, conver-
gence in distribution, and the central limit theorem to more advanced
conceptssuchasexchangeability,modelswithagroupstructure,asymp-
totic approximations to integrals, orthogonal polynomials, and saddle-
point approximations. The emphasis is on topics useful in understand-
ing statistical methodology; thus, parametric statistical models and the
distribution theory associated with the normal distribution are covered
comprehensively.
ThomasA.SeverinireceivedhisPh.D.inStatisticsfromtheUniversityof
Chicago.HeisnowaProfessorofStatisticsatNorthwesternUniversity.
He has also written Likelihood Methods in Statistics. He has published
extensively in statistical journals such as Biometrika, Journal of the
American Statistical Association, and Journal of the Royal Statistical
Society. He is a member of the Institute of Mathematical Statistics and
theAmericanStatisticalAssociation.
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CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS
EditorialBoard:
R.Gill,DepartmentofMathematics,UtrechtUniversity
B.D.Ripley,DepartmentofStatistics,UniversityofOxford
S.Ross,EpsteinDepartmentofIndustrial&SystemsEngineering,Universityof
SouthernCalifornia
B.W.Silverman,St.Peter’sCollege,Oxford
M.Stein,DepartmentofStatistics,UniversityofChicago
This series of high-quality upper-division textbooks and expository monographs covers
all aspects of stochastic applicable mathematics. The topics range from pure and applied
statistics to probability theory, operations research, optimization, and mathematical pro-
gramming. The books contain clear presentations of new developments in the field and
alsoofthestateoftheartinclassicalmethods.Whileemphasizingrigoroustreatmentof
theoreticalmethods,andbooksalsocontainapplicationsanddiscussionsofnewtechniques
madepossiblebyadvancesincomputationalpractice.
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Elements of Distribution Theory
THOMAS A. SEVERINI
iii
CAMBRIDGEUNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB28RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521844727
© Cambridge University Press 2005
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2005
ISBN-13 978-0-511-34519-7 eBook (NetLibrary)
ISBN-10 0-511-34519-4 eBook (NetLibrary)
ISBN-13 978-0-521-84472-7 hardback
ISBN-10 0-521-84472-X hardback
Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
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ToMyParents
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Contents
Preface pagexi
1. PropertiesofProbabilityDistributions 1
1.1 Introduction 1
1.2 BasicFramework 1
1.3 RandomVariables 5
1.4 DistributionFunctions 8
1.5 QuantileFunctions 15
1.6 DensityandFrequencyFunctions 20
1.7 IntegrationwithRespecttoaDistributionFunction 26
1.8 Expectation 28
1.9 Exercises 34
1.10 SuggestionsforFurtherReading 38
2. ConditionalDistributionsandExpectation 39
2.1 Introduction 39
2.2 MarginalDistributionsandIndependence 39
2.3 ConditionalDistributions 46
2.4 ConditionalExpectation 53
2.5 Exchangeability 59
2.6 Martingales 62
2.7 Exercises 64
2.8 SuggestionsforFurtherReading 67
3. CharacteristicFunctions 69
3.1 Introduction 69
3.2 BasicProperties 72
3.3 FurtherPropertiesofCharacteristicFunctions 82
3.4 Exercises 90
3.5 SuggestionsforFurtherReading 93
4. MomentsandCumulants 94
4.1 Introduction 94
4.2 MomentsandCentralMoments 94
4.3 LaplaceTransformsandMoment-GeneratingFunctions 99
4.4 Cumulants 110
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4.5 MomentsandCumulantsoftheSampleMean 120
4.6 ConditionalMomentsandCumulants 124
4.7 Exercises 127
4.8 SuggestionsforFurtherReading 130
5. ParametricFamiliesofDistributions 132
5.1 Introduction 132
5.2 ParametersandIdentifiability 132
5.3 ExponentialFamilyModels 137
5.4 HierarchicalModels 147
5.5 RegressionModels 150
5.6 ModelswithaGroupStructure 152
5.7 Exercises 164
5.8 SuggestionsforFurtherReading 169
6. StochasticProcesses 170
6.1 Introduction 170
6.2 Discrete-TimeStationaryProcesses 171
6.3 MovingAverageProcesses 174
6.4 MarkovProcesses 182
6.5 CountingProcesses 187
6.6 WienerProcesses 192
6.7 Exercises 195
6.8 SuggestionsforFurtherReading 198
7. DistributionTheoryforFunctionsofRandomVariables 199
7.1 Introduction 199
7.2 FunctionsofaReal-ValuedRandomVariable 199
7.3 FunctionsofaRandomVector 202
7.4 SumsofRandomVariables 212
7.5 OrderStatistics 216
7.6 Ranks 224
7.7 MonteCarloMethods 228
7.8 Exercises 231
7.9 SuggestionsforFurtherReading 234
8. NormalDistributionTheory 235
8.1 Introduction 235
8.2 MultivariateNormalDistribution 235
8.3 ConditionalDistributions 240
8.4 QuadraticForms 244
8.5 SamplingDistributions 250
8.6 Exercises 253
8.7 SuggestionsforFurtherReading 256
9. ApproximationofIntegrals 257
9.1 Introduction 257
9.2 SomeUsefulFunctions 257
9.3 AsymptoticExpansions 264