Confidence, Likelihood, Probability This livelybook laysoutamethodology of confidence distributionsand putsthem through their paces. Among other merits, they lead to optimal combinations of confidence from different sources of information, and they can make complex models amenable to objective and indeed prior-free analysis for less subjectively inclined statisticians. The generous mixture of theory, illustrations, applications and exercises is suitable for statisticians at all levels of experience, as well as for data-orientedscientists. Some confidence distributions are less dispersed than their competitors. This concept leads to a theory of risk functions and comparisons for distributions of confidence. Neyman–Pearson type theorems leading to optimal confidence are developedandrichlyillustrated.Exactandoptimalconfidencedistributionsarethe goldstandardforinferredepistemicdistributionsinempiricalsciences. Confidencedistributionsandlikelihoodfunctionsareintertwined, allowingprior distributionstobemadepartofthelikelihood. Meta-analysisinlikelihoodtermsis developedandtakenbeyondtraditionalmethods,suitingitinparticulartocombining informationacrossdiversedatasources. TORE SCHWEDER isaprofessorofstatisticsintheDepartmentofEconomicsand attheCentreforEcologyandEvolutionarySynthesisattheUniversityofOslo. NILS LID HJORT is a professor of mathematical statistics in the Department of MathematicsattheUniversityofOslo. CAMBRIDGE SERIES IN STATISTICAL AND PROBABILISTIC MATHEMATICS EditorialBoard Z.Ghahramani(DepartmentofEngineering,UniversityofCambridge) R.Gill(MathematicalInstitute,LeidenUniversity) F.P.Kelly(DepartmentofPureMathematicsandMathematicalStatistics,UniversityofCambridge) B.D.Ripley(DepartmentofStatistics,UniversityofOxford) S.Ross(DepartmentofIndustrialandSystemsEngineering,UniversityofSouthernCalifornia) M.Stein(DepartmentofStatistics,UniversityofChicago) This series of high-quality upper-division textbooks and expository monographs covers all aspects of stochasticapplicablemathematics.Thetopicsrangefrompureandappliedstatisticstoprobabilitytheory, operationsresearch, optimizationandmathematicalprogramming. Thebookscontainclearpresentations ofnewdevelopmentsinthefieldandalsoofthestateoftheartinclassicalmethods.Whileemphasizing rigorous treatment of theoretical methods, the books also contain applications and discussions of new techniquesmadepossiblebyadvancesincomputationalpractice. Acompletelistofbooksintheseriescanbefoundatwww.cambridge.org/knowledge. Recenttitlesincludethefollowing: 15. MeasureTheoryandFiltering,byLakhdarAggounandRobertElliott 16. EssentialsofStatisticalInference,byG.A.YoungandR.L.Smith 17. ElementsofDistributionTheory,byThomasA.Severini 18. StatisticalMechanicsofDisorderedSystems,byAntonBovier 19. TheCoordinate-FreeApproachtoLinearModels,byMichaelJ.Wichura 20. RandomGraphDynamics,byRickDurrett 21. Networks,byPeterWhittle 22. SaddlepointApproximationswithApplications,byRonaldW.Butler 23. AppliedAsymptotics,byA.R.Brazzale,A.C.DavisonandN.Reid 24. RandomNetworksforCommunication,byMassimoFranceschettiandRonaldMeester 25. DesignofComparativeExperiments,byR.A.Bailey 26. SymmetryStudies,byMarlosA.G.Viana 27. ModelSelectionandModelAveraging,byGerdaClaeskensandNilsLidHjort 28. BayesianNonparametrics,editedbyNilsLidHjortetal. 29. FromFiniteSampletoAsymptoticMethodsinStatistics,byPranabK.Sen,JulioM.Singerand AntonioC.PedrosadeLima 30. BrownianMotion,byPeterMo¨rtersandYuvalPeres 31. Probability(FourthEdition),byRickDurrett 33. StochasticProcesses,byRichardF.Bass 34. RegressionforCategoricalData,byGerhardTutz 35. ExercisesinProbability(SecondEdition),byLo¨ıcChaumontandMarcYor 36. StatisticalPrinciplesfortheDesignofExperiments,byR.Mead,S.G.GilmourandA.Mead 37. QuantumStochastics,byMou-HsiungChang 38. NonparametricEstimationunderShapeConstraints,byPietGroeneboomandGeurtJongbloed 39. LargeSampleCovarianceMatrices,byJianfengYao,ZhidongBaiandShurongZheng 40. MathematicalFoundationsofInfinite-DimensionalStatisticalModels,byEvaristGine´and RichardNickl Confidence, Likelihood, Probability Statistical Inference with Confidence Distributions Tore Schweder UniversityofOslo Nils Lid Hjort UniversityofOslo 32AvenueoftheAmericas,NewYork,NY10013-2473,USA CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9780521861601 (cid:2)c ToreSchwederandNilsLidHjort2016 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2016 PrintedintheUnitedStatesofAmerica AcatalogrecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloginginPublicationData Schweder,Tore. Confidence,likelihood,probability:statisticalinferencewithconfidence distributions/ToreSchweder,UniversityofOslo,NilsLidHjort,UniversityofOslo. pages cm.–(Cambridgeseriesinstatisticalandprobabilisticmathematics) Includesbibliographicalreferencesandindex. ISBN978-0-521-86160-1(hardback) 1. Mathematicalstatistics. 2. Probability. 3. Appliedstatistics. I. Hjort,NilsLid. II. Title. QA277.5.S39 2016 519.2–dc23 2015016878 ISBN978-0-521-86160-1Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyInternetWebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchWebsitesis,orwillremain, accurateorappropriate. Tomyfourchildren –T.S. Tomyfourbrothers –N.L.H. Contents Preface pagexiii 1 Confidence,likelihood,probability:Aninvitation 1 1.1 Introduction 1 1.2 Probability 4 1.3 Inverseprobability 6 1.4 Likelihood 7 1.5 Frequentism 8 1.6 Confidenceandconfidencecurves 10 1.7 Fiducialprobabilityandconfidence 14 1.8 WhynotgoBayesian? 16 1.9 Notesontheliterature 19 2 Inferenceinparametricmodels 23 2.1 Introduction 23 2.2 Likelihoodmethodsandfirst-orderlarge-sampletheory 24 2.3 Sufficiencyandthelikelihoodprinciple 30 2.4 Focusparameters,pivotsandprofilelikelihoods 32 2.5 Bayesianinference 40 2.6 Relatedthemesandissues 42 2.7 Notesontheliterature 48 Exercises 50 3 Confidencedistributions 55 3.1 Introduction 55 3.2 Confidencedistributionsandstatisticalinference 56 3.3 Graphicalfocussummaries 65 3.4 Generallikelihood-basedrecipes 69 3.5 Confidencedistributionsforthelinearregressionmodel 72 3.6 Contingencytables 78 3.7 Testinghypothesesviaconfidenceforalternatives 80 3.8 Confidencefordiscreteparameters 83 vii viii Contents 3.9 Notesontheliterature 91 Exercises 92 4 Furtherdevelopmentsforconfidencedistribution 100 4.1 Introduction 100 4.2 Boundedparametersandboundedconfidence 100 4.3 Randomandmixedeffectsmodels 107 4.4 TheNeyman–Scottproblem 111 4.5 Multimodality 115 4.6 Ratiooftwonormalmeans 117 4.7 Hazardratemodels 122 4.8 ConfidenceinferenceforMarkovchains 128 4.9 Timeseriesandmodelswithdependence 133 4.10 Bivariatedistributionsandtheaverageconfidencedensity 138 4.11 Devianceintervalsversusminimumlengthintervals 140 4.12 Notesontheliterature 142 Exercises 144 5 Invariance,sufficiencyandoptimalityforconfidence distributions 154 5.1 Confidencepower 154 5.2 Invarianceforconfidencedistributions 157 5.3 Lossandriskfunctionsforconfidencedistributions 161 5.4 Sufficiencyandriskforconfidencedistributions 165 5.5 Uniformlyoptimalconfidenceforexponentialfamilies 173 5.6 Optimalityofcomponentconfidencedistributions 177 5.7 Notesontheliterature 179 Exercises 180 6 Thefiducialargument 185 6.1 Theinitialargument 185 6.2 Thecontroversy 188 6.3 Paradoxes 191 6.4 FiducialdistributionsandBayesianposteriors 193 6.5 Coherencebyrestrictingtherange:Invarianceorirrelevance? 194 6.6 Generalisedfiducialinference 197 6.7 Furtherremarks 200 6.8 Notesontheliterature 201 Exercises 202 7 Improvedapproximationsforconfidencedistributions 204 7.1 Introduction 204 7.2 Fromfirst-ordertosecond-orderapproximations 205 Contents ix 7.3 Pivottuning 208 7.4 Bartlettcorrectionsforthedeviance 210 7.5 Median-biascorrection 214 7.6 Thet-bootstrapandabc-bootstrapmethod 217 7.7 Saddlepointapproximationsandthemagicformula 219 7.8 Approximationstothegoldstandardintwotestcases 222 7.9 Furtherremarks 227 7.10 Notesontheliterature 228 Exercises 229 8 Exponentialfamiliesandgeneralisedlinearmodels 233 8.1 Theexponentialfamily 233 8.2 Applications 235 8.3 AbivariatePoissonmodel 241 8.4 Generalisedlinearmodels 246 8.5 Gammaregressionmodels 249 8.6 Flexibleexponentialandgeneralisedlinearmodels 252 8.7 Strauss,Ising,Potts,Gibbs 256 8.8 Generalisedlinear-linearmodels 260 8.9 Notesontheliterature 264 Exercises 266 9 Confidencedistributionsinhigherdimensions 274 9.1 Introduction 274 9.2 Normallydistributeddata 275 9.3 Confidencecurvesfromdeviancefunctions 278 9.4 Potentialbiasandthemarginalisationparadox 279 9.5 Productconfidencecurves 280 9.6 Confidencebandsforcurves 284 9.7 Dependenciesbetweenconfidencecurves 291 9.8 Notesontheliterature 292 Exercises 292 10 Likelihoodsandconfidencelikelihoods 295 10.1 Introduction 295 10.2 Thenormalconversion 298 10.3 Exactconversion 301 10.4 Likelihoodsfrompriordistributions 302 10.5 Likelihoodsfromconfidenceintervals 305 10.6 Discussion 311 10.7 Notesontheliterature 312 Exercises 313