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Springer Texts in Statistics E. L. Lehmann Joseph P. Romano Testing Statistical Hypotheses Fourth Edition Springer Texts in Statistics SeriesEditors G.Allen,DepartmentofStatistics,Houston,TX,USA R.DeVeaux,DepartmentofMathematicsandStatistics,WilliamsCollege, Williamstown,MA,USA R.Nugent,DepartmentofStatistics,CarnegieMellonUniversity,Pittsburgh, PA,USA SpringerTextsinStatistics(STS)includesadvancedtextbooksfrom3rd-to4th-year undergraduatecoursesto1st-to2nd-yeargraduatecourses.Exercisesetsshouldbe included.TheserieseditorsarecurrentlyGeneveraI.Allen,RichardD.DeVeaux, and Rebecca Nugent. Stephen Fienberg, George Casella, and Ingram Olkin were editorsoftheseriesformanyyears. Moreinformationaboutthisseriesathttps://link.springer.com/bookseries/417 · E. L. Lehmann Joseph P. Romano Testing Statistical Hypotheses Fourth Edition E.L.Lehmann JosephP.Romano (Deceased)DepartmentofStatistics DepartmentofStatisticsandEconomics UniversityofCalifornia StanfordUniversity Berkeley,CA,USA Stanford,CA,USA ISSN 1431-875X ISSN 2197-4136 (electronic) SpringerTextsinStatistics ISBN 978-3-030-70577-0 ISBN 978-3-030-70578-7 (eBook) https://doi.org/10.1007/978-3-030-70578-7 1st–3rdedition:©SpringerScience+BusinessMedia,LLC2005,1986,1959 4thedition:©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringer NatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland DedicatedtotheMemoryof ErichL.Lehmann(1917–2009) Preface to the Fourth Edition Thefourtheditionof TestingStatisticalHypothesesprovidesasignificantupdateto thethirdedition,whichappearedin2005.Thestoriedhistoryofthefirsttwoeditions is recounted in DeGroot (1986) and Lehmann (1997), and an account of the third editionisgiveninRomano(2012). Inordertoaccommodatenewtopics,oneprincipalchangefromthethirdedition is the expansion of the book into two volumes. Volume I (Chapters 1–10) treats finite-sampletheory,whileVolumeII(Chapters11–18)treatsasymptotictheory.A majoradditiontothetreatmentoffinite-sampletheoryisamoreexpansivechapter (Chapter 9) on multiple hypothesis testing, including such topics as: the closure method, the false discovery rate, and other generalized error rates. A new section on the principle of monotonicity is included in Chapter 8. The asymptotic theory presentedinVolumeIIhasbeenreorganized.Chapter12coversasymptoticmethods for sums of dependent variables and their application to inference. In particular, Chapter12includes:limittheoremsforrandomsamplingwithoutreplacement;some theory of U-statistics; central limit theorems for stationary, mixing processes; and Stein’smethod.Chapter13includesanintroductiontohigh-dimensionaltesting;see Section13.5.Theasymptotictheoryofpermutationandrandomizationtestsisnow expanded to its own chapter, Chapter 17, largely driven by the resurgence of such methodology.Thereareover100newproblemsinthevolumes,bringingthetotalto around900. ExceptforsomeofthebasicresultsfromVolumeI,adetailedknowledgeoffinite- sampletheoryisnotrequiredforVolumeII.Inparticular,thenecessarybackground should include Chapter 3, Sections 3.1–3.5, 3.8–3.9; Chapter 4, Sections 4.1–4.4; Chapter5,Sections5.1–5.3;Chapter6,Sections6.1–6.2;Chapter7,Sections7.1– 7.2; Chapter 8, Sections 8.1–8.2, 8.4–8.5; and Chapter 9, Sections 9.1–9.3. These sectionscouldformthebasisofaone-semestercourseinhypothesistesting. A second course could begin with fundamental concepts of asymptotic theory presented in Chapter 11. For the reader with a background including a rigorous courseinprobabilitytheory,Chapter11servesasareviewofsomeimportanttheo- remsandtools.Fromthere,onecouldfocusonthelarge-samplepropertiesofsome fundamentalmethods,suchastheone-andtwo-sampleWilcoxontestsinChapter12, vii viii PrefacetotheFourthEdition ortherobustness ofsomeclassicalparametrictestsinChapter 13.Thesections in Chapters12and13arelargelyindependentofeachother.Chapter14requirescareful studysothattheconceptsandtheoremsdevelopedtherecanbeappliedtothetheory of large-sample optimality presented in Chapter 15. Chapters 17 and 18 provide the foundations for computer-intensive methods that remain vital in contemporary statisticalresearch. With great admiration, I dedicate the present volumes to the memory of Erich Lehmann. After working with Erich during the several years leading to the third edition, my memories of Erich can be summed up in some of his qualities that I mostrespect:humility,generosity,professionalism,andclarityofthought.Despite Erich’spassingin2009,thefourtheditionishighlyinfluencedbyErich’sphilosophy andrenowned scholarship.Every decisionon thefourthedition,including content as well as the decision to divide the work in two volumes, was based on what I believedErichwouldhavewanted.ThoseinterestedinErich’slifeshouldreadthe fine remembrances in Rojo (2011) and van Zwet (2011), as well as Erich’s own accountinLehmann(2008).Needlesstosay,Erichwillalwaysberememberedasa foundingfatherofmathematicalstatistics. Specialthanksgotothosewhoprovidedhelpfulsupportbyproofreadingdrafts andofferingconstructivesuggestions.Theyinclude:ThomasDiCiccio,WengeGuo, Bala Rajaratnam, David Ritzwoller (especially for help with the figures), Azeem Shaikh,MariusTirlea,MichaelWolf,andthemanystudentsatStanfordUniversity who proofread new sections and worked through many of the problems. Finally, heartfelt thanks go to friends and family who provided continual encouragement, especially my partner Frank Adair, Ann Marie and Mark Hodges, Anna and Kirk Warshaw,TheresaMontagna,DavidFogle,ScottMadover,TomNeville,andlastbut notleast,mymother. JosephP.Romano August2020 StanfordUniversity Stanford,CA USA Contents VolumeIFinite-SampleTheory 1 TheGeneralDecisionProblem ................................ 3 1.1 StatisticalInferenceandStatisticalDecisions ................ 3 1.2 SpecificationofaDecisionProblem ........................ 4 1.3 Randomization;ChoiceofExperiment ..................... 8 1.4 OptimumProcedures .................................... 9 1.5 InvarianceandUnbiasedness .............................. 11 1.6 BayesandMinimaxProcedures ........................... 14 1.7 MaximumLikelihood .................................... 16 1.8 CompleteClasses ....................................... 17 1.9 SufficientStatistics ...................................... 19 1.10 Problems ............................................... 22 1.11 Notes .................................................. 28 2 TheProbabilityBackground .................................. 29 2.1 ProbabilityandMeasure .................................. 29 2.2 Integration ............................................. 32 2.3 StatisticsandSubfields ................................... 36 2.4 ConditionalExpectationandProbability .................... 38 2.5 ConditionalProbabilityDistributions ....................... 43 2.6 CharacterizationofSufficiency ............................ 47 2.7 ExponentialFamilies .................................... 49 2.8 Problems ............................................... 54 2.9 Notes .................................................. 60 3 UniformlyMostPowerfulTests ................................ 61 3.1 StatingtheProblem ...................................... 61 3.2 TheNeyman–PearsonFundamentalLemma ................. 64 3.3 p-values ............................................... 69 3.4 DistributionswithMonotoneLikelihoodRatio .............. 72 3.5 ConfidenceBounds ...................................... 79 ix x Contents 3.6 AGeneralizationoftheFundamentalLemma ............... 88 3.7 Two-SidedHypotheses ................................... 92 3.8 LeastFavorableDistributions ............................. 95 3.9 ApplicationstoNormalDistributions ....................... 99 3.9.1 UnivariateNormalModels ........................ 99 3.9.2 MultivariateNormalModels ....................... 102 3.10 Problems ............................................... 105 3.11 Notes .................................................. 123 4 Unbiasedness:TheoryandFirstApplications ................... 125 4.1 UnbiasednessforHypothesisTesting ....................... 125 4.2 One-ParameterExponentialFamilies ....................... 126 4.3 SimilarityandCompleteness .............................. 130 4.4 UMPUnbiasedTestsforMultiparameterExponential Families ............................................... 135 4.5 ComparingTwoPoissonorBinomialPopulations ............ 141 4.6 TestingforIndependenceina2×2Table .................. 144 4.7 AlternativeModelsfor2×2Tables ........................ 147 4.8 SomeThree-FactorContingencyTables .................... 150 4.9 TheSignTest ........................................... 153 4.10 Problems ............................................... 157 4.11 Notes .................................................. 168 5 Unbiasedness: Applications to Normal Distributions; ConfidenceIntervals .......................................... 171 5.1 StatisticsIndependentofaSufficientStatistic ............... 171 5.2 TestingtheParametersofaNormalDistribution ............. 175 5.3 ComparingtheMeansandVariancesofTwoNormal Distributions ............................................ 180 5.4 ConfidenceIntervalsandFamiliesofTests .................. 185 5.5 UnbiasedConfidenceSets ................................ 187 5.6 Regression ............................................. 193 5.7 BayesianConfidenceSets ................................ 196 5.8 PermutationTests ....................................... 200 5.9 MostPowerfulPermutationTests .......................... 202 5.10 RandomizationasaBasisForInference .................... 206 5.11 PermutationTestsandRandomization ...................... 210 5.12 RandomizationModelandConfidenceIntervals ............. 214 5.13 Testing for Independence in a Bivariate Normal Distribution ............................................ 217 5.14 Problems ............................................... 220 5.15 Notes .................................................. 239