Saddlepoint approximations with applications

578 Pages·2007·3.86 MB·English

by  Ronald W. Butler

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Saddlepoint Approximations with Applications
Modernstatisticalmethodsusemodelsthatrequirethecomputationofprobabilities
fromcomplicateddistributions,whichcanleadtointractablecomputations.Saddle-
point approximations can be the answer. Written from the user’s point of view, this
bookexplainsinclear,simplelanguagehowsuchapproximateprobabilitycomputa-
tionsaremade,takingreadersfromtheverybeginningstocurrentapplications.
Thebookaimstomakethesubjectaccessibletothewidestpossibleaudienceby
usinggraduatedlevelsofdifficultyinwhichthecorematerialispresentedinchapters
1–6atanelementarymathematicallevel.Readersareguidedinapplyingthemethods
in various computations that will build their skills and deepen their understanding
when later complemented with discussion of theoretical aspects. Chapters 7–9 ad-
dressthe p∗andr∗formulasofhigherorderasymptoticinference,developedthrough
the Danish approach to the subject by Barndorff-Nielsen and others. These provide
areadablesummaryoftheliteratureandanoverviewofthesubjectbeginningwith
theoriginalworkofFisher.Laterchaptersaddressspecialtopicswheresaddlepoint
methodshavehadsubstantialimpactthroughparticularapplications.Theseinclude
applications in multivariate testing, applications to stochastic systems and applied
probability, bootstrap implementation in the transform domain, and Bayesian com-
putationandinference.
Nopreviousbackgroundintheareaisrequiredasthebookintroducesthesubject
fromtheverybeginning.Manydataexamplesfromrealapplicationsshowthemeth-
ods at work and demonstrate their practical value. Ideal for graduate students and
researchers in statistics, biostatistics, electrical engineering, econometrics, applied
mathematics, and other fields where statistical and probabilistic modeling are used,
thisisbothanentry-leveltextandavaluablereference.
Ronald W. Butler is the C. F. Frensley Professor of Mathematical Sciences
andProfessorofStatisticalScienceatSouthernMethodistUniversity,Dallas,Texas.
CAMBRIDGE SERIES IN STATISTICAL AND
PROBABILISTIC MATHEMATICS
EditorialBoard
R.Gill(DepartmentofMathematics,LeidenUniversity)
B.D.Ripley(DepartmentofStatistics,UniversityofOxford)
S.Ross(DepartmentofIndustrial&SystemsEngineering,UniversityofSouthernCalifornia)
B.W.Silverman(St.Peter’sCollege,Oxford)
M.Stein(DepartmentofStatistics,UniversityofChicago)
This series of high-quality upper-division textbooks and expository monographs covers all
aspectsofstochasticapplicablemathematics.Thetopicsrangefrompureandappliedstatistics
toprobabilitytheory,operationsresearch,optimization,andmathematicalprogramming.The
books contain clear presentations of new developments in the field and also of the state of
theartinclassicalmethods.Whileemphasizingrigoroustreatmentoftheoreticalmethods,the
booksalsocontainapplicationsanddiscussionsofnewtechniquesmadepossiblebyadvances
incomputationalpractice.
Alreadypublished
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5. BayesianMethods,byThomasLeonardandJohnS.J.Hsu
6. EmpiricalProcessesinM-Estimation,bySaravandeGeer
7. NumericalMethodsofStatistics,byJohnF.Monahan
8. AUser’sGuidetoMeasureTheoreticProbability,byDavidPollard
9. TheEstimationandTrackingofFrequency,byB.G.QuinnandE.J.Hannan
10. DataAnalysisandGraphicsusingR,byJohnMaindonaldandJohnBraun
11. StatisticalModels,byA.C.Davison
12. SemiparametricRegression,byD.Ruppert,M.P.Wand,R.J.Carroll
13. ExercisesinProbability,byLoicChaumontandMarcYor
14. StatisticalAnalysisofStochasticProcessesinTime,byJ.K.Lindsey
15. MeasureTheoryandFiltering,byLakhdarAggounandRobertElliott
16. EssentialsofStatisticalInference,byG.A.Young,R.L.Smith
17. ElementsofDistributionTheory,byThomasA.Severini
18. StatisticalMechanicsofDisorderedSystems,byAntonBovier
19. RandomGraphDynamics,byRickDurrett
Saddlepoint Approximations
with Applications
Ronald W. Butler
DepartmentofStatisticalScience,
SouthernMethodistUniversity
CAMBRIDGEUNIVERSITYPRESS
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/9780521872508
© Ronald W. Butler 2007
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 2007
ISBN-13 978-0-511-34239-4 eBook (NetLibrary)
ISBN-10 0-511-34239-X eBook (NetLibrary)
ISBN-13 978-0-521-87250-8 hardback
ISBN-10 0-521-87250-2 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.
Contents
Preface pageix
1 Fundamentalapproximations 1
1.1 Univariatedensitiesandmassfunctions 1
1.2 Univariatecumulativedistributionfunctions 12
1.3 Failure(hazard)rateapproximation 28
1.4 Finalremarks 30
1.5 Computationalnotes 30
1.6 Exercises 31
2 Propertiesandderivations 38
2.1 Simplepropertiesoftheapproximations 38
2.2 Saddlepointdensity 41
2.3 SaddlepointCDFapproximation 49
2.4 Furthertopics 54
2.5 Appendix 66
2.6 Exercises 70
3 Multivariatedensities 75
3.1 Saddlepointdensityandmassfunctions 75
3.2 Developmentofthesaddlepointdensity 83
3.3 Propertiesofmultivariatesaddlepointdensities 91
3.4 Furtherexamples 93
3.5 MultivariateCDFs 101
3.6 Exercises 102
4 Conditionaldensitiesanddistributionfunctions 107
4.1 Conditionalsaddlepointdensityandmassfunctions 107
4.2 Conditionalcumulativedistributionfunctions 113
4.3 Furtherexamples:Linearcombinationsofindependentvariables 123
4.4 Furthertopics 126
4.5 Appendix 132
4.6 Exercises 136
v
vi Contents
5 Exponentialfamiliesandtilteddistributions 145
5.1 Regularexponentialfamilies 145
5.2 Edgeworthexpansions 151
5.3 Tiltedexponentialfamiliesandsaddlepointapproximations 156
5.4 Saddlepointapproximationinregularexponentialfamilies 158
5.5 Exercises 179
6 Furtherexponentialfamilyexamplesandtheory 183
6.1 LogisticregressionandLD50estimation 183
6.2 Commonoddsratioin2×2tables 193
6.3 Timesseriesanalysisoftruncatedcountdata 208
6.4 ExponentialfamiliesofMarkovprocesses 209
6.5 Truncation 212
6.6 Exercises 213
7 Probabilitycomputationwith p∗ 219
7.1 The p∗ densityinregularexponentialfamilies 219
7.2 Conditionalinferenceand p∗ ingrouptransformationmodels 225
7.3 Approximateconditionalinferenceand p∗ incurvedexponentialfamilies 230
7.4 Appendix 250
7.5 Exercises 254
8 Probabilitieswithr∗-typeapproximations 259
8.1 Notation,models,andsamplespacederivatives 259
8.2 Scalarparameterapproximations 260
8.3 Examples 261
8.4 Derivationof(8.1) 265
8.5 Otherversionsofuˆ 266
8.6 Numericalexamples 271
8.7 Properties 278
8.8 Appendix 279
8.9 Exercises 282
9 Nuisanceparameters 285
9.1 Approximationwithnuisanceparameters 285
9.2 Examples 286
9.3 Derivationof(9.3)and(9.4) 291
9.4 Exactandapproximatesamplespacederivatives 292
9.5 Numericalexamples 294
9.6 Variationindependence,conditionallikelihood,andmarginallikelihood 297
9.7 Examples 304
9.8 Propertiesof(9.3) 314
9.9 Exercises 315
10 Sequentialsaddlepointapplications 323
10.1 Sequentialsaddlepointapproximation 323
Contents vii
10.2 Comparisontothedouble-saddlepointapproach 324
10.3 Examples 325
10.4 P-valuesfortheBartlett–Nanda–Pillaitracestatistic 330
10.5 Exercises 334
11 Applicationstomultivariatetesting 341
11.1 P-valuesinMANOVA 342
11.2 P-valuesintestsofcovariance 348
11.3 Powerfunctionsformultivariatetests 355
11.4 Somemultivariatesaddlepointdensities 363
11.5 Appendix 365
11.6 Exercises 366
12 Ratiosandrootsofestimatingequations 374
12.1 Ratios 375
12.2 Univariaterootsofestimatingequations 384
12.3 Distributionsforvectorratios 392
12.4 Multivariaterootsofestimatingequations 401
12.5 TheconditionalCDFof Rm given R1,...,Rm−1 411
12.6 Appendix 420
12.7 Exercises 422
13 Firstpassageandtimetoeventdistributions 430
13.1 Semi-MarkovandMarkovprocesseswithfinitestatespace 430
13.2 Passagetimeswithasingledestinationstate 435
13.3 Passagetimeswithmanypossibledestinationstates 452
13.4 Birthanddeathprocessesandmodularsystems 454
13.5 Markovprocesses 461
13.6 Aredundantandrepairablesystem 462
13.7 Appendix 466
13.8 Exercises 469
14 Bootstrappinginthetransformdomain 474
14.1 Saddlepointapproximationtothesinglebootstrapdistribution 475
14.2 Saddlepointapproximationsfordoublebootstrapconfidencebands 482
14.3 Semiparametricbootstrap 487
14.4 Indirectsaddlepointapproximation 494
14.5 Empiricalsaddlepointapproximations 500
14.6 Appendix 500
14.7 Exercises 502
15 Bayesianapplications 506
15.1 Bayesianpredictionwithintractablepredictanddistribution 507
15.2 PassagetimeexamplesforMarkovprocesses 510
15.3 Passagetimeexamplesforsemi-Markovprocesses 517
viii Contents
15.4 Conclusions,applicability,andalternativemethods 522
15.5 Computationalalgorithms 524
15.6 Exercises 525
16 Nonnormalbases 528
16.1 Nonnormal-basedsaddlepointexpressions 528
16.2 Choiceofbasedistribution 532
16.3 Conditionaldistributionapproximation 538
16.4 Examples 539
16.5 Exercises 545
References 548
Index 560