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Bayesian Multiple Testing under Sparsity for Exponential Distributions

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Title:
Bayesian Multiple Testing under Sparsity for Exponential Distributions
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english
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Li, Ke
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Ghosh, Malay
Committee Members:
Rosalsky, Andrew J
Bliznyuk, Nikolay A
Gezan, Salvador

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Subjects / Keywords:
asymptotic -- optimality
Statistics -- Dissertations, Academic -- UF
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Statistics thesis, Ph.D.
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theses   ( marcgt )
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Abstract:
Multiple testing problems are gaining increasing prominence in statistical research. The prime reason behind this is that there is a growing need for statisticians to analyze large data sets involving many parameters. Several methods to solve multiple testing problems have been introduced since the breakthrough paper by Benjamini and Hochberg. In this dissertation, we considers simultaneous testing of multiple exponential scale parameters under sparsity. Two general results provide thresholding conditions under which a given testing procedure achieves Bayesian optimality asymptotically. In particular, it is shown how multiple testing procedures of Benjamini and Hochberg, Efron and Tibshirani and Genovese and Wasserman can achieve this Bayesian optimality asymptotically. An alternative density for the scale parameter has also been considered.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Statement of Responsibility:
by Ke Li.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Ghosh, Malay.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

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MISSING IMAGE

Material Information

Title:
Bayesian Multiple Testing under Sparsity for Exponential Distributions
Physical Description:
1 online resource (51 p.)
Language:
english
Creator:
Li, Ke
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Statistics
Committee Chair:
Ghosh, Malay
Committee Members:
Rosalsky, Andrew J
Bliznyuk, Nikolay A
Gezan, Salvador

Subjects

Subjects / Keywords:
asymptotic -- optimality
Statistics -- Dissertations, Academic -- UF
Genre:
Statistics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Multiple testing problems are gaining increasing prominence in statistical research. The prime reason behind this is that there is a growing need for statisticians to analyze large data sets involving many parameters. Several methods to solve multiple testing problems have been introduced since the breakthrough paper by Benjamini and Hochberg. In this dissertation, we considers simultaneous testing of multiple exponential scale parameters under sparsity. Two general results provide thresholding conditions under which a given testing procedure achieves Bayesian optimality asymptotically. In particular, it is shown how multiple testing procedures of Benjamini and Hochberg, Efron and Tibshirani and Genovese and Wasserman can achieve this Bayesian optimality asymptotically. An alternative density for the scale parameter has also been considered.
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Ke Li.
Thesis:
Thesis (Ph.D.)--University of Florida, 2013.
Local:
Adviser: Ghosh, Malay.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2015-08-31

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2013
System ID:
UFE0045795:00001


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BAYESIANMULTIPLETESTINGUNDERSPARSITYFOREXPONENTIALDISTRIBUTIONSByKELIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2013

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c2013KeLi 2

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Youhavetoberst,best,ordifferent. 3

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ACKNOWLEDGMENTS Iwouldliketothankmyadvisor,MalayGhosh.Ithasbeenmyhonortoworkwithsuchagreatmentorthathasdedicatedhisentirelifetoresearch,teachingandenrichingthelivesofstudents.Ihavebeengratefulfortheopportunitytolearnunderhimandseemydevelopmentinbothresearchandpersonalitygrowfromhisendlessknowledgenotonlyinallareasofstatisticsbutineverydaylife.HisenthusiamforlearningandworkinghasshapedwhoIamasapersonwhowillneverstopchasingwonderfuldreams.ThelifewithProfessorGhoshhasbeenthebestmemoryformeinthepastveyearsattheUniversityofFlorida.IwouldliketothankAndrewRosalskyforgivingmetheabilityofunderstandingandapplyingprobabilitytheory.Hisgreatteachinginprobabilitytheoryisbenetingmenotonlyincompletingmydissertationbutalsoinmyfutureresearch.IwouldalsoliketothankNikolayBliznyukandSalvadorGezanfortheirinsightthathasleadtomyworkimproving.Furthermore,Iwouldliketothankallofthefaculty,staff,students,andfriendsthatmadethisdissertationpossible.Finally,Iwouldliketothankmyparentsandmywife.Withouttheirconstantloveandsupport,IwouldnothavemadeittowhereIamtoday.IwouldnotbethepersonIamtodaywithoutmyparents'constantencouragementofscience,mathematicsandstatisticssinceIwasveryyoung. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 6 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 1.1MutipleTestingProblemsandSparsity .................... 8 1.2FamilyWiseErrorRateandFalseDiscoveryRate ............. 9 1.3ThresholdingEquivalenceoftheBenjamini-HochbergProcedure ..... 10 1.4ApproximateFixedThresholdingtheBenjamini-HochbergProcedure ... 11 1.5BayesianFalseDiscoveryRateandPositiveFalseDiscoveryRate .... 12 1.6AsymptoticBayesOptimalityunderSparsity ................. 13 1.7MultipleTestingforExponentialDistributions ................ 15 1.8ProposedResearch .............................. 16 2ASYMPTOTICBAYESOPTIMALITYUNDERSPARSITYFOREXPONENTIALDISTRIBUTION ................................... 17 3ASYMPTOTICBAYESOPTIMALITYUNDERSPARSITYOFSOMEPROCEDURES 27 3.1ConnectingtheFalseDiscoveryRate,theBayesianFalceDiscoveryRateandAsymptoticBayesOptimalityunderSparsity .............. 27 3.2AsymptoticBayesOptimalityunderSparsityoftheBenjamini-HochbergProcedure .................................... 31 3.3SimulationStudy ................................ 37 4ALTERNATIVECHOICEFORTHEDENSITYOFTHESCALEPARAMETER 39 5CONCLUSIONS ................................... 48 REFERENCES ....................................... 50 BIOGRAPHICALSKETCH ................................ 51 5

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LISTOFTABLES Table page 1-1Numberoferrorscommittedwhentestingnnullhypotheses ........... 10 1-2MatrixofLosses ................................... 13 2-1MatrixofLosses ................................... 18 3-1SumLossComparisonbetweenBayesOracleandBHProcedure ....... 36 4-1MatrixofLosses ................................... 40 6

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyBAYESIANMULTIPLETESTINGUNDERSPARSITYFOREXPONENTIALDISTRIBUTIONSByKeLiAugust2013Chair:MalayGhoshMajor:StatisticsMultipletestingproblemsaregainingincreasingprominenceinstatisticalresearch.Theprimereasonbehindthisisthatthereisagrowingneedforstatisticianstoanalyzelargedatasetsinvolvingmanyparameters.SeveralmethodstosolvemultipletestingproblemshavebeenintroducedsincethebreakthroughpaperbyBenjaminiandHochbergin1995.Inthisdissertation,weconsiderssimultaneoustestingofmultipleexponentialscaleparametersundersparsity.TwogeneralresultsprovidethresholdingconditionsunderwhichagiventestingprocedureachievesBayesianoptimalityasymptotically.Inparticular,itisshownhowmultipletestingproceduresofBenjaminiandHochberg's,EfronandTibshirani'sandGenoveseandWasserman'scanachievethisBayesianoptimalityasymptotically.Analternativedensityforthescaleparameterhasalsobeenconsidered. 7

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CHAPTER1INTRODUCTION 1.1MutipleTestingProblemsandSparsityMultipletestingproblemsaregainingincreasingprominenceinstatisticalresearch.Theprimereasonbehindthisisthatthereisagrowingneedforstatisticianstoanalyzelargedatasetsinvolvingmanyparameters.Tociteanexample,onemayrefertomicroarraydataanalysiswhereitisessentialtotestsimultaneouslyexpressionlevelsofthousandsofgenes.Particularlyintheeldofgeneticassociationstudies,onemaytestsimultaneouslyexpressionlevelsofthousandsofgenestoseetherelationshipbetweengenesand,e.g,adisease.Classicalmultiplecomparisonproceduresfocusonsimultaneouslyconsideringmoderatenumberofhypothesistests,ofteninananalysisofvariance.However,large-scalemultipletestingmayinvolvethousandsorevengreaternumbersoftests,onwhichthetraditionalmethodscannotworkwell.Thus,itisnecessarytodevelopothermethodstoeffectivelysolveproblemsofthistype.Inrecentyears,multipletestingproceduresundersparsityhavealsobecomeanimportanttopicofresearch.Inthiscase,sparsitymeansthattheproportionoftruealternativesamongallhypothesisisverysmall.TwoofthemostpopularmethodsforsolvingproblemsofthistypethatarecurrentlyinvoguearetheBonferroniapproachwhichcontrolsthefamilywiseerrorrate(FWER)andthe[ 1 ]Benjamini-Hochberg(1995)procedurewhichcontrolsthefalsediscoveryrate(FDR).Someotheroptimalitycriteriahavealsobeenproposedinthecontextofmultipletesting.Forinstance,intheclassicalNeyman-Pearsonspirit,onemaymaximizetheexpectednumberoftruediscoveries,whilekeepingxedoneoftheerrormeasuressuchasFWER,FDRorexpectednumberoffalsepositives.[ 9 ]Storey(2007)adoptedsuchanapproach.Hereferredtothisasanoracleproperty.Asecondoraclepropertyasdenedin[ 10 ]SunandCai(2007)assumedthatthedataaregeneratedaccordingto 8

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atwo-componentmixturemodel.Theymaximizedthemarginalfalsenondiscoveryrate,whilecontrollingthemarginalfalsediscoveryrateataxedlevel.Theoptimalitywasachievedasymptoticallyforanyxed(thoughunknown)proportionofalternatives.Incontrastto[ 10 ]SunandCai(2007),[ 2 ]Bogdan,Chakrabarti,FrommletandGhosh(henceforth,referredtoasBCFG)(2011)tookadecisiontheoreticpointofview,whereforeachtest,theyassignedxedlossesfortypeIandtypeIIerrors.Theyalsobeganwithatwo-componentmixturemodel,andconsideredafullyasymptoticsetupunderwhichtheproportionpoftruealternativesamongalltestsconvergestozeroasthenumberoftestsgoestoinnity.Withanadditivelossstructure,theyconsideredanasymptoticallyoptimalBayesianprocedureundersparsity.Intheprocess,theycouldalsodemonstratetheasymptoticoptimalityoftheBHprocedureundersparsityfromapurelyBayesianperspective.[ 6 ]NeuvialandRoquain(2012)extendedBCFG'sworktotheexponentialpowerfamilyofdistributions,includingthedoubleexponentialdistributionasaspecialcase.Thisclassofdistributionsarisesascertainscalemixturesofthenormaldistribution. 1.2FamilyWiseErrorRateandFalseDiscoveryRateFWERisquiteadequatefordetectingsignicantalternativesinthemultipletestingcontextwithsmallormoderatenumberofhypotheses.However,inhighdimensionalproblems,forexampleinanalyzingmicroarraydata,FWERmayfailtodetectmostofthesignicantalternatives,thusFWERcannotbeusedforexploratorystudy.Ontheotherhand,foranexploratorystudy,orifsignicantresultscaneasilybere-testedinanindependentstudy,controloftheFDRisoftenprefered.TheFDRallowsresearcherstoidentifyasetofcandidatepositives,ofwhichahighproportionarelikelytobetrue.Thefalsepositiveswithinthecandidatesetcanthenbeidentiedinafollow-upstudy.Inabreakthroughpaper,[ 1 ]BenjaminiandHochberg(1995)consideredtheproblemoftestingsimultaneouslynhypotheses,ofwhichn0aretrue.Table 1-1 summarizesthesituation.Thespecicnhypothesesareassumedtobeknown.R 9

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isthenumberofhypothesesrejectedwhichisanobservablerandomvariable,whileU,V,SandTareunobservablerandomvariables. Table1-1. Numberoferrorscommittedwhentestingnnullhypotheses Declarednon-signicantDeclaredsignicant TruenullsUVn0Non-truenullsTSn)]TJ /F3 11.955 Tf 11.96 0 Td[(n0n)]TJ /F3 11.955 Tf 11.95 0 Td[(RRn TheFalseDiscoveryRate(FDR)isdenedasFDR=EV RwhereVisthenumberoffalserejectionsanditisassumedthatV R=0whenR=0.TwosimplepropertiesofFDRwereshown.Therstoneisthatifallnullhypothesesaretrue,theFDRisequivalenttotheFWER.Thesecondoneisthatwhenn0
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observable.ManyothermultipletestingcriteriauseobservablerandomvariablesXni'sinsteadofusingcorrespondingp-value's.[ 4 ]EfronandTibshirani(2002)provedanequivalencetheoremfortheBenjamini-Hochbergprocedure.Theyconsideredarandomthreshold^CBHn=infy:PH0n1(Xn1>y) 1)]TJ /F6 11.955 Tf 13.18 2.66 Td[(^Fn(y)6,whereistheFDRleveland^Fn(y)istheempericalcdfwhichisdenedas1)]TJ /F6 11.955 Tf 13.34 2.65 Td[(^Fn(y)=#fXni>yg n.TheequivalencetheoremsaysthattheBHprocedurerejectsthenullhypothesisH0niwhenXni>^CBHn.EfronandTibshirani'sequivalencetheoremgivesatoolforpeopletocompareotherthresholdingmultipletestingcriterionswiththeBHprocedure.WealsoneedutilizethisequivalencetheoremtoprovesomeresultsinChapter2. 1.4ApproximateFixedThresholdingtheBenjamini-HochbergProcedureForthesamestatisticalmodelasgiveninSection1.3,[ 5 ]GenoveseandWasserman(2002)investigatedtheasymptoticbehavioroftherandomthresholdforBenjamini-Hochbergprocedureproposedintheequivalencetheorem.Theyprovedthatwhilethenumberofteststendstoinnityandthefractionoftruealternativesremainsxed,therandomthresholdofaBenjamini-HochbergprocedurecontrollingFDRatthelevelcanbeapproximatedbyaxedthresholdCGWnwithtypeIerrortGW1nandtypeIIerrortGW2nwhichsatises=tGW1n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)tGW1n+pn(1)]TJ /F3 11.955 Tf 11.95 0 Td[(tGW2n).InSection1.3,wehaveseentherelationshipbetweentheBenjamini-HochbergprocedureandtherandomthresholdingprocedureusingXni's.GenoveseandWasserman'sresultconstructsanotherbridgewhichconnectstheBenjamini-Hochbergprocedure,therandomthresholdingprocedureandthexedthresholdingprocedure. 11

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1.5BayesianFalseDiscoveryRateandPositiveFalseDiscoveryRateAstheFalseDiscoveryRateplaysanimportantroleinmultipletestingproblemresearch,manyothermeasureshavealsobeenproposedbypeople.OnefamousexampleistheBayesianFalseDiscoveryRate.InadditiontothestatisticalmodelinSection1.3,assumethatH0niandHAnioccurwithprobabilitiespnand1)]TJ /F3 11.955 Tf 12.55 0 Td[(pnrespectively.Consideraxedthresholdingprocedure(i.e.thesamethresholdforeachindividualtest).Denotebyt1nandt2nrespesctivelytheprobabilitiesoftypeIerrorandtypeIIerror.[ 4 ]EfronandTibshirani(2002)denedtheBayesianFalseDiscoveryRate(BFDR)asBFDR=P(H0niistruejH0niisrejected)=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n+pn(1)]TJ /F3 11.955 Tf 11.96 0 Td[(t2n).TounderstandtherelationshipbetweentheFalseDiscoveryRateandtheBayesianFalseDiscoveryRate,wemayrsthavealookatanothermeasurenamedthePositiveFalseDiscoveryRatewhichisdenedby[ 8 ]Storey(2003)aspFDR=EV RR>0.Foratwo-groupmodel,StoreyprovedatheoremwhichshowstheequivalencebetweentheBayesianFalseDiscoveryRateandthePositiveFalseDiscoveryRate,i.e.BFDR=pFDR.Inaddition,acorrespondingcorollarysaysthattheyarebothequaltoE(V) E(R)bynotingthatE(V)=nP(H0niistrue)P(H0niisrejectedjH0niistrue)andE(R)=nP(H0niisrejected). 12

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Table1-2. MatrixofLosses chooseH0nichooseHAni H0nitrue00nHAnitrueAn0 Nowitiseasilyseenthat,comparingtoFDR=E)]TJ /F8 7.97 Tf 6.67 -4.97 Td[(V R,theBFDR(orpFDR)isaratioofexpectationsE(V) E(R).MoredetailsaboutpFDRcanbefoundinStorey'spaper.InChapter2,wediscusstherelationshipbetweenBFDRandourproposedcriterionbyshowingtheequivalenceundercertainconditions. 1.6AsymptoticBayesOptimalityunderSparsity[ 2 ]Bogdan,Chakrabarti,FrommletandGhosh(2011)introduceanotherimportantmeasurenamedAsymptoticBayesOptimalityunderSparsity(ABOS).SupposerandomvariablesXn1,...,Xnnareindependentidenticaldistributed.Foreachi=1,...,n,XnimarginallyhasamixturenormaldistributionXni(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)N(0,2n)+pnN(0,2n+2n)wherepn2(0,1),2n>0and2n>0.Thewordsparsityreferstothesituationwhenpn0.ConsiderthemultipletestingproblemofsimultaneouslytestingH0niversusHAnifori=1,...,n.Foreachi,twostatesH0niandHAnihappenwithprobabilities(1)]TJ /F3 11.955 Tf 11.84 0 Td[(pn)andpnrespectively.UnderH0ni,XniN(0,2n),andunderHAni,XniN(0,2n+2n).ThelossstructureformakingadecisionintheithtestisindicatedinTable 1-2 .Underanadditivelossfunction,theBayesdecisionproblemleadstoaprocedurechoosingthealternativehypothesisHAniinthecasesuchthatfAn(Xni) f0n(Xni)>(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)0n pnAn=:1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn pnnwherefAnandf0narethedensitiesofXniunderalternativeandnullrespectively.Substitutingthecorrespondingnormaldensities,oneobtainsaformalprocedure: rejectH0niifX2ni 2n>c2n,i=1,...,n. 13

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wherec2n=2n+2n 2nlog2n+2n 2n+2log1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn pnn.ThisprocedureisdenedasBayesoracleduetotheuseofunknownparamters.BCFGintroducethefollowingassumptionunderwhichboththetypeIandtypeIIerrorsarenontrivial.TheyprovetwobasicresultsunderthisassumptionanddenetheAsymptoticallyBayesOptimalunderSparsitybasedontheseresults.Assumption()Asequenceofvectorsf(pn,2n,2n,0n,An)g1n=1satisesthisassumptionifthecorrespondingsequenceofparametervectors(pn,an,bn)fulllsthefollowingconditions:pn!0,an!1,bn!1andlogbn an!C2(0,1),asn!1,wherean=2n 2nandbn=an1)]TJ /F8 7.97 Tf 6.58 0 Td[(pn pnn2.Result1.1UnderAssumption(),theprobabilitiesoftypeIerrorandtypeIIerrorusingtheBayesoraclearerespectivelyt1n=exp)]TJ /F3 11.955 Tf 10.49 8.09 Td[(C 2r 2 anlogbn(1+on(1))andt2n=(2(p C))]TJ /F6 11.955 Tf 11.96 0 Td[(1)(1+on(1)).Result1.2Underanadditivelossfunction,theBayesriskforaxedthresholdmultipletestingprocedureisgivenbyR=nXi=1[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1ni0n+pnt2niAn]=n[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n+pnt2nAn].Therefore,underAssumption(),theBayesriskofusingBayesoracleisRoptimal=npnAn(2(p C))]TJ /F6 11.955 Tf 11.95 0 Td[(1)(1+on(1)).Denition1.1ForasequenceofparameterssatisfyingAssumption(),amultipletestingruleiscalledAsymptoticallyBayesOptimalunderSparsity(ABOS)ifitsriskRsatises 14

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R Roptimal!1asn!1whereRoptimalistheoptimalriskgiveninResult1.2.BCFGinvestigatedtherelationshipbetweentheFDR,theBFDRandtheABOS,andtheyhavealsoprovedthattheBenjamini-HochbergprocedureisABOSwithcertainconditions.[ 6 ]NeuvialandRoquain(2012)extendedBCFG'sworktotheexponentialpowerfamilyofdistributions,includingthedoubleexponentialdistributionasaspecialcase.Thisclassofdistributionsarisesascertainscalemixturesofthenormaldistribution. 1.7MultipleTestingforExponentialDistributions[ 3 ]DonohoandJin(2006)discussedaboutthemutipletestingproblemsforexponentialdistributions.Theymentionseveralareaswheretheexponentialdistributionapplies.Onepossibleapplicationisrelatedtomultiplelifetimeanalysis.Theyconsiderthefailuretimesofmanycomparableindependentsystems,whereasmallproportionofthesystemsmayhavesignicantlyhigherexpectedlifetimesthanthetypicalsystems.Thefailuretimeshaveexponentialdistributions,andweneedanapproachtodetectthesignicantones.ThereisanotherapplicationformultipletestingwithLehmannalternatives.Supposethatweconductmanyindependentstatisticaltests,eachprovidingap-value,saypi,andthatmostofthesetestsarethecaseswithtruenullswhileasmallproportioncorrespondtocaseswhereaLehmannalternativeistrue.Thenlog(1=pi)Exp(i),wheremostoftheiare1,correspondingtotruenullhypotheses,whiletherestaregreaterthan1,correspondingtoLehmannalternatives.Theyprovedasymptoticminimaxityoffalsediscoveryratethresholdingforexponentialdataundersparsity.Theirworkmotivatesustoconsiderthemultipletestingproblemsinthecontextofexponentialdistributions. 15

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1.8ProposedResearchThegenesisofmydissertationisbasedonthetwopapersby[ 2 ]Bogdanetal.(2011)and[ 3 ]DonohoandJin(2006).Whiletheformerconsideredatwocomponentmixturenormalmodel,weconsiderinsteadmixtureinverse-gammadistributionsforexponentialscaleparameters.Unlike[ 3 ]DonohoandJin(2006),ouraimisnottoproveasymptoticminimaxity,butestablishtheasymptoticBayesoptimalityundersparsity.TwogeneralresultsprovidesthresholdingconditionsunderwhichagiventestingprocedureachievestheBayesianoptimalityasymptotically.Inparticular,itisshownhowmultipletestingproceduresof[ 1 ]BenjaminiandHochberg(1995),[ 4 ]EfronandTibshirani(2002)and[ 5 ]GenoveseandWasserman(2002)canachievethisBayesianoptimalityasymptotically. 16

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CHAPTER2ASYMPTOTICBAYESOPTIMALITYUNDERSPARSITYFOREXPONENTIALDISTRIBUTIONFromthischapteron,thenotationsinthepreviouschapterarevoided,andsomeofthenotationsarereassignedtootherrepresentations.Suppose,foreachxedn,Xn1,...,Xnnareindependentexponentialswithrespectivepdfsfni(x)=)]TJ /F5 7.97 Tf 6.59 0 Td[(1niexp()]TJ /F3 11.955 Tf 9.3 0 Td[(x=ni),x>0,ni>0;i=1,...,n.WewilldenotethesepdfsbyExp(ni),i=1,...,n.OurobjectiveistotestsimultaneouslyH0niversusH1ni,whereunderHjni,nihasInverse-Gammapdfnj()=[exp()]TJ /F9 11.955 Tf 9.3 0 Td[(jn=ni)jn]=[+1ni\()],j=0,1,1n>0n>0,>1.Then,marginallyunderHjni,Xnihaspdffjn(x)=jn=(x+jn)+1,j=0,1.where1n>0nbothunknownand>1known.WeassumethatH0niandH1nioccurwithprobabilitiespnand1)]TJ /F3 11.955 Tf 12.55 0 Td[(pnrespectively.Thusmarginally,Xnihasthetwo-groupmixtureParetopdf(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)0n=(x+0n)+1+pn1n=(x+1n)+1.Thewordsparsityreferstothesituationpn0.FollowingBCFG,considertheadditivelossPni=1L(Xni),whereL(Xni)=0n(Xni)+1n(1)]TJ /F9 11.955 Tf 12.09 0 Td[((Xni).Here(Xni)equals1or0accordingtotherejectionoracceptanceofH0.Table 2-1 describesthislossstructure. 17

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Table2-1. MatrixofLosses chooseH0nichooseHAni H0nitrue00nHAnitrue1n0 Underthislossstructure,theBayesdecisionproblemleadstoaprocedurechoosingthealternativehypothesisHAniinthecasef1n(Xni) f0n(Xni)>(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)0n pn1nwheref1nandf0narethedensitiesofXniunderalternativeandnullrespectively.Thissimpliesto rejectH0niifXni>Cn,i=1,...,n.whereCn=Kn1n)]TJ /F9 11.955 Tf 11.96 0 Td[(0n 1)]TJ /F3 11.955 Tf 11.96 0 Td[(KnandKn=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)0n pn1n0n 1n1 +1.ThisprocedureisdenedasBayesoracleduetotheuseofunknownparamters.[ 2 ]Bogdanetal.(2011)denedasimilarBayesoraclepropertyforthemixturenormalmodel.InaBayesianframework,theBayesriskisanimportantquantitythatcanbeusedtomeasuretheperformanceofarule.InordertoderivetheBayesriskoftheBayesrule(theBayesoracle),werstlyneedtheprobabilitiesoftypeIandtypeIIerrors.WedenotetheprobabilitiesoftypeIandtypeIIerrorsrespectively,foralli,ast1n=PH0ni(H0niisrejected)=PH0ni(Xni>Cn)andt2n=PHAni(H0niisaccepted)=PHAni(Xni
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andunderHAni,Y1:=)]TJ /F3 11.955 Tf 9.3 0 Td[(log1n Xni+1nExp(1),theprobabilitiesofatypeIandtypeIIerrorsusingtheBayesoraclearegivenbyt1n=PY0>)]TJ /F3 11.955 Tf 9.3 0 Td[(log0n 1n)]TJ /F9 11.955 Tf 11.95 0 Td[(0n1)]TJ /F3 11.955 Tf 11.96 0 Td[(Kn Kn=0n 1n)]TJ /F9 11.955 Tf 11.96 0 Td[(0n1)]TJ /F3 11.955 Tf 11.95 0 Td[(Kn Kn =0n Cn+0n(2)andt2n=PY1<)]TJ /F3 11.955 Tf 9.3 0 Td[(log1n 1n)]TJ /F9 11.955 Tf 11.95 0 Td[(0n(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Kn)=1)]TJ /F7 11.955 Tf 11.95 16.86 Td[(1n 1n)]TJ /F9 11.955 Tf 11.96 0 Td[(0n(1)]TJ /F3 11.955 Tf 11.96 0 Td[(Kn) =1)]TJ /F7 11.955 Tf 11.96 16.85 Td[(1n Cn+1n.(2)InordertohavenontrivialprobabilitiesoftypeIIerrors,i.e.limn!1t2n2(0,1),weneedanassumption.Assumption()Asequenceofvectorsf(pn,0n,1n,0n,1n)g1n=1satisesthefollowingconditions,asn!1, pn!0,1n!1,0n!02(0,1),0n 1n!2(0,1),0n 1n
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Corollary2.1UnderAssumption(),pn1n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)0n1 A(1)]TJ /F3 11.955 Tf 11.96 0 Td[(A)t1n.Proof:FollowsimmediatelyfromTheorem2.2. ThenwecandenetheBayesriskoftheBayesrule(Bayesoracle),whichhastheminimalBayesriskamongallrules,inthefollowingway.Denition2.1Underanadditivelossfunction,theBayesriskforaxedthresholdmultipletestingprocedurewithrespectiveprobabilitiest01nandt02noftypeIandtypeIIerrorsisgivenbyR0=nXi=1[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t01n0n+pnt02n1n]=n[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t01n0n+pnt02n1n].Thus,theBayesriskcorrespondingtoBayesoraclewhichwewillrefertoastheoptimalriskis Roptimal=n[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n0n+pnt2n1n].(2)Inaddition,followingBCFG,wehaveDenition2.2whichformsthebasisforjudgingwhetheranarbitrarymultipletestingprocedureisgoodenoughornotbycomparingtheBayesriskoftheprocedurewiththeoptimalone.Denition2.2ForasequenceofparameterssatisfyingAssumption(),amultipletestingruleiscalledasymptoticallyBayesoptimalundersparsity(ABOS)ifitsriskRsatisesR Roptimal!1,asn!1,whereRoptimalistheoptimalriskgivenby( 2 ).Intherestofthispaper,ABOSwillbereferredtoasymptoticallyBayesoptimal(oroptimality)undersparsityuptothecontext.BasedonDenition2.2,wehavethefollowingthreetheoremswhichshowtheABOSoftwogeneralclassesofmutipletestingprocedures.Therstclassconsistsofthexedthresholdmultipletestingprocedures. 20

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Theorem2.3DeneaxedthresholdmutipletestingprocedurefFTMgby rejectH0niifXni>Cn,i=1,...,n,whereCn>0isaconstant.ThecorrespondingprobabilitiesoftypeIandtypeIIerrorsarerespectivelygivenby t1n=0n Cn+0nandt2n=1)]TJ /F7 11.955 Tf 11.96 13.27 Td[(1n Cn+1n.Theorem2.4indicatestheconditionunderwhichaxedthresholdmutipletestingprocedureisABOS.Theorem2.4AxedthresholdmutipletestingprocedurefFTMgdenedinTheorem2.3isABOSifandonlyifCnCn.Proof:TheriskofthemutipletestingprocedurefFTMgwithxedthresholdCnisR:=n[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n0n+pnt2n1n].Then,fFTMgisABOSifandonlyifR Roptimal!1,R)]TJ /F3 11.955 Tf 11.96 0 Td[(Roptimal Roptimal!0,(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)0n(t1n)]TJ /F3 11.955 Tf 11.96 0 Td[(t1n)+pn1n(t2n)]TJ /F3 11.955 Tf 11.96 0 Td[(t2n) (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)0nt1n+pn1nt2n!0,(t1n)]TJ /F3 11.955 Tf 11.95 0 Td[(t1n)+pn1n (1)]TJ /F8 7.97 Tf 6.58 0 Td[(pn)0n(t2n)]TJ /F3 11.955 Tf 11.95 0 Td[(t2n) t1n+pn1n (1)]TJ /F8 7.97 Tf 6.58 0 Td[(pn)0nt2n!0,(t1n)]TJ /F3 11.955 Tf 11.96 0 Td[(t1n)+t1n A(1)]TJ /F8 7.97 Tf 6.59 0 Td[(A)(t2n)]TJ /F3 11.955 Tf 11.95 0 Td[(t2n) t1n+1)]TJ /F5 7.97 Tf 6.59 0 Td[((1)]TJ /F8 7.97 Tf 6.59 0 Td[(A) A(1)]TJ /F8 7.97 Tf 6.59 0 Td[(A)t1n!0,A(1)]TJ /F3 11.955 Tf 11.96 0 Td[(A) 1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(A)+1t1n)]TJ /F3 11.955 Tf 11.96 0 Td[(t1n t1n+1 1)]TJ /F6 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(A)+1(t2n)]TJ /F3 11.955 Tf 11.96 0 Td[(t2n)!0.DeneL=A(1)]TJ /F8 7.97 Tf 6.59 0 Td[(A) 1)]TJ /F5 7.97 Tf 6.59 0 Td[((1)]TJ /F8 7.97 Tf 6.58 0 Td[(A)+1t1n)]TJ /F8 7.97 Tf 6.59 0 Td[(t1n t1n+1 1)]TJ /F5 7.97 Tf 6.58 0 Td[((1)]TJ /F8 7.97 Tf 6.58 0 Td[(A)+1(t2n)]TJ /F3 11.955 Tf 11.95 0 Td[(t2n).Therearethreepossiblecases.(1)IfCn=o(Cn),thent1n)]TJ /F8 7.97 Tf 6.58 0 Td[(t1n t1n!1andjt2n)]TJ /F3 11.955 Tf 11.95 0 Td[(t2nj62.ThenL!16=0.(2)IfCn=o(Cn),thent1n)]TJ /F8 7.97 Tf 6.58 0 Td[(t1n t1n!)]TJ /F6 11.955 Tf 24.57 0 Td[(1andt2n!1.ThenL!(1)]TJ /F8 7.97 Tf 6.58 0 Td[(A)+1 1)]TJ /F5 7.97 Tf 6.59 0 Td[((1)]TJ /F8 7.97 Tf 6.58 0 Td[(A)+16=0. 21

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(3)IfCnCnwhere2(0,1),thent1n)]TJ /F14 7.97 Tf 6.58 0 Td[(t1nandt2n!1)]TJ /F7 11.955 Tf 11.29 13.28 Td[(1)]TJ /F8 7.97 Tf 6.59 0 Td[(A 1+()]TJ /F5 7.97 Tf 6.58 0 Td[(1)A.Thust1n)]TJ /F8 7.97 Tf 6.58 0 Td[(t1n t1n!)]TJ /F14 7.97 Tf 6.58 0 Td[()]TJ /F6 11.955 Tf 11.95 0 Td[(1andt2n)]TJ /F3 11.955 Tf 11.96 0 Td[(t2n!(1)]TJ /F3 11.955 Tf 11.96 0 Td[(A))]TJ /F7 11.955 Tf 11.96 13.27 Td[(1)]TJ /F8 7.97 Tf 6.58 0 Td[(A 1+()]TJ /F5 7.97 Tf 6.58 0 Td[(1)A.Therefore,L!A(1)]TJ /F3 11.955 Tf 11.96 0 Td[(A) 1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(A)+1()]TJ /F14 7.97 Tf 6.59 0 Td[()]TJ /F6 11.955 Tf 11.96 0 Td[(1)+1 1)]TJ /F6 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(A)+1(1)]TJ /F3 11.955 Tf 11.96 0 Td[(A))]TJ /F7 11.955 Tf 11.96 16.86 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(A 1+()]TJ /F6 11.955 Tf 11.96 0 Td[(1)A=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(A) 1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(A)+1A)]TJ /F14 7.97 Tf 6.59 0 Td[(+(1)]TJ /F3 11.955 Tf 11.96 0 Td[(A))]TJ /F6 11.955 Tf 11.96 0 Td[([A+(1)]TJ /F3 11.955 Tf 11.95 0 Td[(A)])]TJ /F14 7.97 Tf 6.58 0 Td[(.Since>1andA2(0,1),byJensen'sinequalityA)]TJ /F14 7.97 Tf 6.59 0 Td[(+(1)]TJ /F3 11.955 Tf 11.17 0 Td[(A)>[A+(1)]TJ /F3 11.955 Tf 11.17 0 Td[(A)])]TJ /F14 7.97 Tf 6.58 0 Td[(withequalityholdingifandonlyif=1.ThusL!0ifandonlyif=1.Hence,fFTMgisABOSifandonlyifCnCn. Remark:Theorem2.4givesanecessaryandsufcientconditionforachievingABOSofxedthresholdmultipletestingprocedure,inasimpleformwhichinvolvestheratiooftwoxedthresholds.Unliketheprevioustheorem,thefollowingTheorem2.5givesasufcientconditionofABOSforagivenrandomthresholdmultipletestingprocedurerequiringacertainconvergencerateforthedifferenceofthetwothreshholdstozero.Theorem2.5DenearandomthresholdmutipletestingprocedurefRTMgbyrejectH0niifXni>^Cn,i=1,...,n,where^Cnisafunctionofrandomvariables.Ifforall>0, Pj^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj>=o(M)]TJ /F14 7.97 Tf 6.59 0 Td[(n),(2)thenfRTMgisABOS.Proof:Notethatbythemodelassumption,Pj^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)PH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>+pnPHAnij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>.From( 2 )andTheorem2.1,thisimplies PH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>=o(M)]TJ /F14 7.97 Tf 6.58 0 Td[(n)(2) 22

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and PHAnij^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj>=o(1).(2)DenetheprobabilitiesoftypeIandtypeIIerrorsrespectivelyby^t1ni=PH0niXni>^Cnand^t2ni=PHAniXni<^Cn.Wehave,forall>0,^t1ni=PH0niXni>^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cn+Cn=PH0ni[Xni>^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>]+PH0ni[Xni>^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj<]>PH0ni[Xni>Cn+]\[j^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj<]>PH0ni(Xni>Cn+))]TJ /F3 11.955 Tf 11.95 0 Td[(PH0nij^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj>=PH0n1(Xn1>Cn+))]TJ /F3 11.955 Tf 11.95 0 Td[(PH0nij^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj>and^t1ni=PH0niXni>^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn=PH0ni[Xni>^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>]+PH0ni[Xni>^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj<]6PH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>+PH0ni[Xni>Cn)]TJ /F9 11.955 Tf 11.96 0 Td[(]\[j^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj<]6PH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>+PH0ni(Xni>Cn)]TJ /F9 11.955 Tf 11.95 0 Td[()=PH0n1(Xn1>Cn)]TJ /F9 11.955 Tf 11.95 0 Td[()+PH0nij^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj> 23

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Similarly,^t2ni=PHAniXni<^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn=PHAni[Xni<^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>]+PHAni[Xni<^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj<]>PHAni[XniPHAni(Xni=PHAn1(Xn1and^t2ni=PHAniXni<^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn=PHAni[Xni<^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj>]+PHAni[Xni<^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj<]6PHAnij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>+PHAni[Xni+PHAni(Xni 24

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Therefore,asn!1,theratiooftheBayesrisk^RnoffRTMgtotheBayesoptimalrisk^Rn Roptimal=Pni=1[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)^t1ni0n+pn^t2ni1n] n[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n+pnt2n1n]=1 nP^t1ni t1n(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n (1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n0n+pnt2n1n+1 nP^t2ni t2npnt2n1n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n+pnt2n1n>PH0n1(Xn1>Cn+))]TJ /F5 7.97 Tf 13.22 4.71 Td[(1 nPPH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj> PH0n1(Xn1>Cn)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n (1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n0n+pnt2n1n+PHAn1(Xn1 PHAn1(Xn1Cn)]TJ /F9 11.955 Tf 11.96 0 Td[()+1 nPPH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj> PH0n1(Xn1>Cn)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n (1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n0n+pnt2n1n+PHAn1(Xn1 PHAn1(Xn1
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Hence,fRTMgisABOS. Theorem2.4and2.5provideusefultoolswhichcanbeappliedtoprovetheABOSofanyspecicmultipletestingprocedurewithineitherofthesetwoclasses.Inthefollowingsection,weinvestigatethreeexamplesgivenrespectivelyby[ 4 ]EfronandTibshirani(2002),[ 5 ]GenoveseandWasserman(2002)and[ 1 ]BenjaminiandHochberg(1995).WeprovethattheyareallABOSundercertainconditions. 26

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CHAPTER3ASYMPTOTICBAYESOPTIMALITYUNDERSPARSITYOFSOMEPROCEDURES 3.1ConnectingtheFalseDiscoveryRate,theBayesianFalceDiscoveryRateandAsymptoticBayesOptimalityunderSparsityForaxedthresholdingprocedure,[ 4 ]EfronandTibshirani(2002)denedanothermeasureoftheaccuracyofmultipletestingprocedure,theBayesianFalseDiscoveryRate(BFDR)asBFDR=P(H0niistruejH0niwasrejected)=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n (1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n+pn(1)]TJ /F3 11.955 Tf 11.95 0 Td[(t2n).ThefollowingtheoremshowsthattheBayesoracleprocedureisasymptoticallycontrollingBFDRatacertainlevel.Theorem3.1.1UnderAssumption(),theBayesoracleprocedureiscontrollingBFDRatthelevelBOnwhichsatises,asn!1,BOn!A A+.Proof:BythedenitionofBFDR,BOn=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n+pn(1)]TJ /F3 11.955 Tf 11.95 0 Td[(t2n).FromTheorem2.1andTheorem2.2,wehaveBOn!A A+. Ontheotherhand,thefollowingTheorem3.1.2indicatesthat,foraxedthresholdingprocedure,undercertainconditions,controllingtheBFDRatacertainlevelisasymp-toticallyequivalenttoachievingtheABOS,anditspeciesonenecessaryandsufcientcondition.ThistheoremoffersforthisanopportunitytocomparemethodsbasedonBFDRwiththosebasedonABOS. 27

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WerstprovidesthespecicformofthethresholdofaxedthresholdprocedurecontrollingBFDRatagivenlevelinLemma3.1.1.ThenweproveTheorem3.1.2basedontheresultofLemma3.1.1.Lemma3.1.1UnderAssumption(),foraxedthresholdrulecontrollingBFDRatthelevelBn2(0,1),thethresholdCBnhastheformCBn=(1n)]TJ /F9 11.955 Tf 11.95 0 Td[(0n)"1)]TJ /F9 11.955 Tf 13.15 8.09 Td[(0n 1n1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn pn1)]TJ /F9 11.955 Tf 11.96 0 Td[(Bn Bn1 #)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 11.96 0 Td[(1n.Proof:BythedenitionofBFDR,Theorem2.1andTheorem2.2,wecanobtainCBnbysolvingBn=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)tB1n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)tB1n+pn(1)]TJ /F3 11.955 Tf 11.95 0 Td[(tB2n)=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)0n CBn+0n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)0n CBn+0n+pn1n CBn+1n.DirectcalculationgivesCBn=(1n)]TJ /F9 11.955 Tf 11.96 0 Td[(0n)"1)]TJ /F9 11.955 Tf 13.15 8.09 Td[(0n 1n(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)(1)]TJ /F9 11.955 Tf 11.96 0 Td[(Bn) pnBn1 #)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 11.95 0 Td[(1n. Theorem3.1.2UnderAssumption(),axedthresholdingrulewiththethresholdCBncontrollingBFDRatthelevelBn2(0,1)isABOSifandonlyifBn!A A+.Proof:TheprooffollowsfromTheorem2.4andLemma3.1.1. Inanearlierwork,[ 1 ]BenjaminiandHochberg(1995)introducedtheFalseDiscoveryRate(FDR),asFDR=EV RwhereRisthetotalnumberofnullhypothesesrejected,VisthenumberoffalserejectionsanditisassumedthatV R=0whenR=0.[ 5 ]GenoveseandWasserman(2002)provedthatwhilethenumberofteststendstoinnityandthefractionoftruealternativesremainsxed,axedthresholdmultiple 28

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testingprocedurewiththresholdCGWncanapproximatelycontrolFDRatleveln,ifitscorrespondingtypeIerrortGW1nandtypeIIerrortGW2nsatisfyn=tGW1n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)tGW1n+pn(1)]TJ /F3 11.955 Tf 11.95 0 Td[(tGW2n).WemaynotethatnissimilartoBnwiththeexceptionofthefactor1)]TJ /F3 11.955 Tf 11.96 0 Td[(pnappearingthenumeratorofthelatter.ThussimilartoLemma3.1.1,Lemma3.1.2givesanexpressionforthexedthresholdofGW.ThenwehavethefollowingTheorem3.1.3fortherelationshipbetweencontrollingFDRandachievingABOS.Lemma3.1.2UnderAssumption(),foraxedthresholdingrulecontrollingFDRattheleveln2(0,1),theGenovese-WassermanapproximationthresholdCGWnhastheformCGWn=(1n)]TJ /F9 11.955 Tf 11.96 0 Td[(0n)"1)]TJ /F9 11.955 Tf 13.15 8.08 Td[(0n 1n1)]TJ /F6 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)n pnn1 #)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 11.96 0 Td[(1n.Proof:InvokingTheorem2.3andTheorem2.2,oneobtainsCGWnbysolvingn=tGW1n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)tGW1n+pn(1)]TJ /F3 11.955 Tf 11.95 0 Td[(tGW2n)=0n CGWn+0n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)0n CGWn+0n+pn1n CGWn+1n.DirectcalculationgivesCGWn=(1n)]TJ /F9 11.955 Tf 11.96 0 Td[(0n)"1)]TJ /F9 11.955 Tf 13.15 8.09 Td[(0n 1n1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)n pnn1 #)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F9 11.955 Tf 11.95 0 Td[(1n. Theorem3.1.3UnderAssumption(),axedthresholdingrulewiththethresholdCGWncontrollingFDRattheleveln2(0,1)isABOSifandonlyifn!A A+.Proof:TheresultfollowsfromTheorem2.4andLemma3.1.2.Remark:Foraxedthresholdingprocedure,Theorems3.1.2and3.1.3demonstratethatundercontrollingeitherBFDRorFDR,ABOSholdsifandonlyifthereisone-to-oneasymptoticcorrespondingbetweenthelossratioandthelevelwhereBFDRorFDRis 29

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heldxed.Asaresultthereof,itisthelossratiowhichasymptoticallydictatesthelevelatwhichwewanttocontrolFDRorBFDR.Wemayconsiderthelossratioasatuningparamterinthiscase.Inaddtion,hereweproveTheorem3.1.4whichgivesaresultofindependentinterestandwillbeneededinSection3.2.Theorem3.1.4UnderAssumption(),asn!1,anecessaryandsufcientconditionsuchthatCGWn=Cn+o(1)isn=A A++o(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1n).Proof:ByLemma3.1.2,wehaveCGWn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cn=o(1),ifandonlyif(1n)]TJ /F9 11.955 Tf 11.95 0 Td[(0n)"1)]TJ /F9 11.955 Tf 13.15 8.09 Td[(0n 1n1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)n pnn1 #)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F9 11.955 Tf 11.96 0 Td[(1n)]TJ /F3 11.955 Tf 13.15 8.09 Td[(Kn1n)]TJ /F9 11.955 Tf 11.95 0 Td[(0n 1)]TJ /F3 11.955 Tf 11.96 0 Td[(Kn=o(1),(1n)]TJ /F9 11.955 Tf 11.95 0 Td[(0n)8<:"1)]TJ /F9 11.955 Tf 13.15 8.08 Td[(0n 1n1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)n pnn1 #)]TJ /F5 7.97 Tf 6.58 0 Td[(1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(Kn))]TJ /F5 7.97 Tf 6.59 0 Td[(19=;=o(1),(1n)]TJ /F9 11.955 Tf 11.96 0 Td[(0n)"0n 1n1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)n pnn1 )]TJ /F3 11.955 Tf 11.95 0 Td[(Kn#=o(1),1)]TJ /F6 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)n pnn1 )]TJ /F9 11.955 Tf 13.15 8.09 Td[(1n 0nKn=o(1),M)]TJ /F5 7.97 Tf 6.58 0 Td[(1nK)]TJ /F5 7.97 Tf 6.59 0 Td[(1n1)]TJ /F6 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)n pnn1 =1+o(M)]TJ /F5 7.97 Tf 6.58 0 Td[(1n).ByTaylor'sexpansion,thisholdsifandonlyif1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)n MnKnpnn=1+o(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1n)=1+o(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1n), 30

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ByTheorem2.1,thisisequivalentto1 n=A+ A+o(M)]TJ /F5 7.97 Tf 6.58 0 Td[(1n),orequivalentlyn=A A++o(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1n). 3.2AsymptoticBayesOptimalityunderSparsityoftheBenjamini-HochbergProcedureTheBenjamini-HochbergprocedurecontrolsFDRatleveln,bysortingp-valuesinanascendingorderp(1)6p(2)6...6p(n)andrejectingthenullhypothesesforwhichthecorrespondingp-valuesaresmallerthanorequaltop(t)wheret=argmaxip(i)6in n.Insteadofusingp-values,toconsideranequivalentprocedurewitharandomthresholdforobservationsXni,i=1,...,n,wedenote1)]TJ /F6 11.955 Tf 14.29 2.66 Td[(^Fn(y)=#fXni>yg n,and1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(y)=P(Xn1>y)=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)PH0n1(Xn1>y)+pnPHAn1(Xn1>y).Itisshown,e.g.[ 4 ]EfronandTibshirani(2002),thattheBHprocedurerejectsthenullhypothesisH0niwhenXni>^CBHnwheretherandomthresholdisgivenby^CBHn=infy:PH0n1(Xn1>y) 1)]TJ /F6 11.955 Tf 13.17 2.66 Td[(^Fn(y)6n.ToprovethattherandomthresholdingBHprocedureisABOS,weneedsomerestrictionsontheparameterspaceinadditiontoAssumption(*).ThefollowingTheorem3.2.1providesasufcientconditionunderwhichtherandomthresholdingBHprocedureisABOS.Theorem3.2.1UnderAssumption(),therandomthresholdingBHprocedurecontrollingFDRatlevelofnisABOSifn=A A++o(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1n), 31

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andM++2n[logn+log(Mn)]=o(n),where>0isarbitrary.Proof:Let^CBHnbetheBHrandomthresholdatleveln.Theproofofthetheoremisbasedonseveraltechnicallemmas.Lemma3.2.1isatechnicaldevicetoLemma3.2.2whichshowstheconnectionbetweentherandomthresholdofBHandamodiedversionofthexedthresholdofGW.Then,Lemma3.2.3andLemma3.2.4showthatPj^CBHn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>=o(M)]TJ /F14 7.97 Tf 6.59 0 Td[(n).HencetheresultofTheorem3.2.1followsfromTheorem2.5. Lemma3.2.1SupposeAssumption()holdsandn=A A++o(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1n).LetCGWnbetheGWthresholdatleveln.Foranyconstant2(0,1),wehaveP 1)]TJ /F6 11.955 Tf 13.18 2.65 Td[(^Fn(CGWn) 1)]TJ /F3 11.955 Tf 11.95 0 Td[(F(CGWn)>1+!6exp)]TJ /F6 11.955 Tf 10.5 8.09 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.95 0 Td[(A AnM)]TJ /F14 7.97 Tf 6.59 0 Td[(n2[1+o(1)],andP 1)]TJ /F6 11.955 Tf 13.18 2.66 Td[(^Fn(CGWn) 1)]TJ /F3 11.955 Tf 11.95 0 Td[(F(CGWn)<1)]TJ /F9 11.955 Tf 11.96 0 Td[(!6exp)]TJ /F6 11.955 Tf 10.5 8.09 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.95 0 Td[(A AnM)]TJ /F14 7.97 Tf 6.59 0 Td[(n2[1+o(1)].Proof:ByTheorem2.3andTheorem3.1.4,1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGWn)=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)PH0n1(Xn1>CGWn)+pnPHAn1(Xn1>CGWn)=A+ A1)]TJ /F3 11.955 Tf 11.95 0 Td[(A AM)]TJ /F14 7.97 Tf 6.59 0 Td[(n[1+o(1)]Notethat1)]TJ /F6 11.955 Tf 13.38 2.66 Td[(^Fn(CGWn)istheaverageofnBernoullirandomvariableswithsuccessprobability1)]TJ /F3 11.955 Tf 12.33 0 Td[(F(CGWn).Hence,byBernstein'sinequality(cf.[ 7 ]Sering,1980,p.96), 32

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onegetsP[(1)]TJ /F6 11.955 Tf 13.17 2.66 Td[(^Fn(CGWn)))]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGWn))]>(1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGWn))exp[)]TJ /F3 11.955 Tf 20.78 8.09 Td[(n2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGWn))2 2(1+)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGWn))]exp[)]TJ /F6 11.955 Tf 9.3 0 Td[((n2=4)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGWn))]=exp)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.96 0 Td[(A AnM)]TJ /F14 7.97 Tf 6.59 0 Td[(n2[1+o(1)]andsimilarlyP[(1)]TJ /F6 11.955 Tf 13.17 2.66 Td[(^Fn(CGWn)))]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGWn))]<)]TJ /F6 11.955 Tf 9.3 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGWn))6exp)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.96 0 Td[(A AnM)]TJ /F14 7.97 Tf 6.58 0 Td[(n2[1+o(1)].HenceP 1)]TJ /F6 11.955 Tf 13.18 2.66 Td[(^Fn(CGWn) 1)]TJ /F3 11.955 Tf 11.95 0 Td[(F(CGWn)>1+!6exp)]TJ /F6 11.955 Tf 10.5 8.09 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.95 0 Td[(A AnM)]TJ /F14 7.97 Tf 6.59 0 Td[(n2[1+o(1)],andP 1)]TJ /F6 11.955 Tf 13.18 2.66 Td[(^Fn(CGWn) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGWn)<1)]TJ /F9 11.955 Tf 11.96 0 Td[(!6exp)]TJ /F6 11.955 Tf 10.49 8.08 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.96 0 Td[(A AnM)]TJ /F14 7.97 Tf 6.59 0 Td[(n2[1+o(1)]. Lemma3.2.2SupposeAssumption()holdsandn=A A++o(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1n).Let^CBHnbetheBHrandomthresholdatlevelnandletCGW1nbetheGWthresholdatlevel1n=n(1)]TJ /F9 11.955 Tf 11.96 0 Td[(n)wheren=o(M)]TJ /F5 7.97 Tf 6.58 0 Td[(1n)asn!1.ItfollowsthatCGW1n=Cn+o(1),andP^CBHn>CGW1n6exp)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.96 0 Td[(A AnM)]TJ /F14 7.97 Tf 6.59 0 Td[(n2n[1+o(1)].Proof:Notethat1n=n(1)]TJ /F9 11.955 Tf 11.95 0 Td[(n)=A A++o(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1n). 33

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ByTheorem3.1.4,CGW1n=Cn+o(1).Observethatfromthedenitionof^CBHn, PH0n1(Xn1>CGW1n) 1)]TJ /F5 7.97 Tf 7.43 1.77 Td[(^Fn(CGW1n)6nimplies^CBHn6CGW1n.NotePH0n1(Xn1>CGW1n) 1)]TJ /F8 7.97 Tf 6.58 0 Td[(F(CGW1n)=1n=n(1)]TJ /F9 11.955 Tf 11.96 0 Td[(n).Therefore,P^CBHn6CGW1n>PPH0n1(Xn1>CGW1n) 1)]TJ /F6 11.955 Tf 13.17 2.66 Td[(^Fn(CGW1n)6n=PPH0n1(Xn1>CGW1n) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGW1n)1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGW1n) 1)]TJ /F6 11.955 Tf 13.17 2.66 Td[(^Fn(CGW1n)6n=P1n1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGW1n) 1)]TJ /F6 11.955 Tf 13.18 2.66 Td[(^Fn(CGW1n)6n=P 1)]TJ /F6 11.955 Tf 13.17 2.66 Td[(^Fn(CGW1n) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGW1n)>1)]TJ /F9 11.955 Tf 11.95 0 Td[(n!.Hence,P^CBHn>CGW1n6P 1)]TJ /F6 11.955 Tf 13.17 2.66 Td[(^Fn(CGW1n) 1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(CGW1n)<1)]TJ /F9 11.955 Tf 11.96 0 Td[(n!6exp)]TJ /F6 11.955 Tf 10.49 8.08 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.96 0 Td[(A AnM)]TJ /F14 7.97 Tf 6.58 0 Td[(n2n[1+o(1)]. Lemma3.2.3SupposeAssumption()holds,n=A A++o(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1n)andM++2nlog(Mn)=o(n).Let^CBHnbetheBHrandomthresholdatleveln.ItfollowsthatP(^CBHn>Cn+)=o(M)]TJ /F14 7.97 Tf 6.59 0 Td[(n).Proof:NotethatM++2n[logn+log(Mn)]=o(n),whichimpliesthatM++2nlog(Mn)=o(n).Choosingn=2M)]TJ /F11 5.978 Tf 5.75 0 Td[((1+=2)n q A+ A(1)]TJ /F12 5.978 Tf 5.76 0 Td[(A A),byLemma3.2.2,P(^CBHn>Cn+)6exp)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.95 0 Td[(A AnM)]TJ /F14 7.97 Tf 6.58 0 Td[(n2n[1+o(1)]=exp)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.95 0 Td[(A AnM)]TJ /F5 7.97 Tf 6.58 0 Td[((++2)n[1+o(1)]=o(M)]TJ /F14 7.97 Tf 6.59 0 Td[(n). 34

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Lemma3.2.4SupposeAssumption()holds,n=A A++o(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1n)andM++2n[logn+log(Mn)]=o(n).Let^CBHnbetheBHrandomthresholdatleveln.ItfollowsthatP(^CBHny) 1)]TJ /F8 7.97 Tf 6.58 0 Td[(F(y)ismonotonicallydecreasinginy, ^CBHn^CBHn) 1)]TJ /F8 7.97 Tf 6.59 0 Td[(F(^CBHn)>PH0n1(Xn1>CGW2n) 1)]TJ /F8 7.97 Tf 6.59 0 Td[(F(CGW2n)=n(1+n).Bythedenitionof^CBHn,PH0n1(Xn1>^CBHn) 1)]TJ /F6 11.955 Tf 13.17 2.66 Td[(^Fn(^CBHn)6n.Thus,theevent^CBHn1+n.Therefore,P(^CBHn1+n!6P supc2[0,CGW2n)1)]TJ /F6 11.955 Tf 13.17 2.66 Td[(^Fn(c) 1)]TJ /F3 11.955 Tf 11.95 0 Td[(F(c)>1+n!.Lett=F(c),zn=F(CGW2n)and^Gn(t)=empiricalcdfofniidUniform(0,1).P(^CBHn1+n!.Notethatzn=1)]TJ /F7 11.955 Tf 9.3 9.68 Td[(A+ A)]TJ /F5 7.97 Tf 6.68 -4.97 Td[(1)]TJ /F8 7.97 Tf 6.59 0 Td[(A AM)]TJ /F14 7.97 Tf 6.59 0 Td[(n[1+o(1)].Letui=i nandkn=n)]TJ /F6 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F7 11.955 Tf 11.95 9.68 Td[(A+ A)]TJ /F5 7.97 Tf 6.68 -4.97 Td[(1)]TJ /F8 7.97 Tf 6.59 0 Td[(A AM)]TJ /F14 7.97 Tf 6.58 0 Td[(n.Fromthemonotonicityof^Gn(t)andt,itfollowsthatfort2[ui,ui+1 n),t)]TJ /F6 11.955 Tf 13.97 2.66 Td[(^Gn(t)1)]TJ /F3 11.955 Tf 13 0 Td[(ui)]TJ /F5 7.97 Tf 14.26 4.71 Td[(1 n.Thisimpliest)]TJ /F5 7.97 Tf 7.79 1.77 Td[(^Gn(t) 1)]TJ /F8 7.97 Tf 6.59 0 Td[(t
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Table3-1. SumLossComparisonbetweenBayesOracleandBHProcedure (pn,1n)(0.001,10)(0.01,10)(0.1,10)(0.001,100)(0.01,100)(0.1,100) LossofBO665679001LossofBH6676840165 ThereforeforsufcientlylargenP supt2[0,zn)1)]TJ /F6 11.955 Tf 13.71 2.66 Td[(^Gn(t) 1)]TJ /F3 11.955 Tf 11.95 0 Td[(t>1+n!6knXi=0P1)]TJ /F6 11.955 Tf 13.71 2.66 Td[(^Gn(ui)>(1)]TJ /F3 11.955 Tf 11.96 0 Td[(ui)]TJ /F6 11.955 Tf 13.25 8.09 Td[(1 n)(1+n).Nowobservethatforeveryi=0,...,kn,1)]TJ /F3 11.955 Tf 11.95 0 Td[(ui>1)]TJ /F3 11.955 Tf 13.15 8.08 Td[(kn n>A+ A1)]TJ /F3 11.955 Tf 11.96 0 Td[(A AM)]TJ /F14 7.97 Tf 6.58 0 Td[(n.Lettingni=1 n(1)]TJ /F8 7.97 Tf 6.59 0 Td[(ui),wehave1)]TJ /F3 11.955 Tf 13.47 0 Td[(ui)]TJ /F5 7.97 Tf 14.74 4.71 Td[(1 n=(1)]TJ /F3 11.955 Tf 13.48 0 Td[(ui)(1)]TJ /F9 11.955 Tf 13.47 0 Td[(ni),andni6A+ A)]TJ /F5 7.97 Tf 6.67 -4.98 Td[(1)]TJ /F8 7.97 Tf 6.58 0 Td[(A A)]TJ /F5 7.97 Tf 6.59 0 Td[(1Mn n=o(n).ThenbytheBernsteininequalityagain,foreveryi=0,...,kn,P1)]TJ /F6 11.955 Tf 13.71 2.66 Td[(^Gn(ui)>(1)]TJ /F3 11.955 Tf 11.95 0 Td[(ui)]TJ /F6 11.955 Tf 13.25 8.09 Td[(1 n)(1+n)=P1)]TJ /F6 11.955 Tf 13.71 2.66 Td[(^Gn(ui)>(1)]TJ /F3 11.955 Tf 11.96 0 Td[(ui)(1+n)(1)]TJ /F9 11.955 Tf 11.95 0 Td[(ni)6exp)]TJ /F6 11.955 Tf 10.49 8.08 Td[(1 4n(1)]TJ /F3 11.955 Tf 11.95 0 Td[(ui)2n[1+o(1)]6exp)]TJ /F6 11.955 Tf 10.49 8.09 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.96 0 Td[(A AnM)]TJ /F14 7.97 Tf 6.59 0 Td[(n2n[1+o(1)]Therefore,P(^CBHn1+n!6knXi=0P1)]TJ /F6 11.955 Tf 13.71 2.66 Td[(^Gn(ui)>(1)]TJ /F3 11.955 Tf 11.96 0 Td[(ui)]TJ /F6 11.955 Tf 13.25 8.09 Td[(1 n)(1+n)6nexp)]TJ /F6 11.955 Tf 10.5 8.08 Td[(1 4A+ A1)]TJ /F3 11.955 Tf 11.95 0 Td[(A AnM)]TJ /F14 7.97 Tf 6.59 0 Td[(n2n[1+o(1)]=o(M)]TJ /F14 7.97 Tf 6.58 0 Td[(n).Hence,P(^CBHn
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3.3SimulationStudyInthissection,weconductasimulationstudytoevaluatetheperformanceoftheprocedureusingBayesoracleandtheperformanceofBenjamin-Hochbergprocedureusingrandomthresholdofp-value's.WLOG,wechoosen=10000,=100,0n=1,and0n=1n=1,andconsiderf0.001,0.01,0.1gasthesetofcandidatepn'sandf10,100gasthesetofcandidate1n's.WegetthexedthresholdCnbyusingthetruevaluesoftheparameters.WecalculatetheFDRlevelnwhichtheBHprocedureiscontrollingatusingtherstequationfromTheorem3.2.1byreplacingAwithKnandignoringtheo(M)]TJ /F5 7.97 Tf 6.59 0 Td[(1n)term.Wegeneratedatafromsix(threepn'sbytwo1n's)differenttwo-groupmixtureParetodistribution.Thenusingthedatafromeachdifferentmodel,wecompareBayesoracleprocedureandBHprocedureinthesenseofcomparingthesumofloss.Table 3-1 showsthesumlosscomparisonbetweentheBayesoracleandtheBHprocedure.WhiletheratioMn=1n=0nis10,theBayesoracleandtheBHprocedurehavealmostthesamesumloss,andtheybothclaimsignicanceforabout65%ofthetruealternatives.WhiletheratioMn=1n=0nis100andpniseither0.001or0.01,boththeBayesoracleandtheBHproceduremakethecorrectdecisionsforalmostallthehypotheses.AlltheabovecasesshowthattheBHprocedurewellapproximatestheBayesruleinthesenseofminimizingtheBayesrisk.However,whiletheratioMn=1n=0nis100andpn=0.1whichmeansthesparsityismoderate,theBayesoracle'ssumlossis1,buttheBHprocedurehasasumlossof65.TheBayesoracleperformsmuchbetterthantheBHprocedureinthiscase,whichalsomeansthattheBHprocedurefailstoapproximatetheBayesruleinthesenseofminimizingtheBayesrisk.Notethatnoneofthesixcombinationsforpnand1nsatisesthesufcientconditioninTheorem3.2.1.However,onlythelastcombinationisanexampleof 37

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failureforapproximation.IttellsusthatthesufcientconditionsfortheBHproceduretoachieveABOScanberelaxedinadvance. 38

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CHAPTER4ALTERNATIVECHOICEFORTHEDENSITYOFTHESCALEPARAMETERFromthischapteron,thenotationsinthepreviouschapterarevoided,andsomeofthenotationsarereassignedtootherrepresentations.Suppose,foreachxedn,Xn1,...,Xnnareindependentexponentialswithrespectivepdfsfni(x)=)]TJ /F5 7.97 Tf 6.59 0 Td[(1niexp()]TJ /F3 11.955 Tf 9.3 0 Td[(x=ni),x>0,ni>0;i=1,...,n.WewilldenotethesepdfsbyExp(ni),i=1,...,n.OurobjectiveistotestsimultaneouslyH0niversusH1ni,whereunderHjni,nihasGammapdfnj()=[exp()]TJ /F9 11.955 Tf 9.3 0 Td[(ni=jn))]TJ /F5 7.97 Tf 6.58 0 Td[(1ni]=[jn\()],j=0,1,1n>0n>0,>0.Then,marginallyunderHjni,Xnihaspdffjn(x)=Z101 jn\())]TJ /F5 7.97 Tf 6.59 0 Td[(2niexp)]TJ /F9 11.955 Tf 10.49 8.09 Td[(ni jn)]TJ /F3 11.955 Tf 16.93 8.09 Td[(x nidni,j=0,1,where1n>0nbothunknownand>0known.Let=2xandjn=p xjn,thenforj=0,1,fjn(x)=1 jn\()2 1=2exp)]TJ /F9 11.955 Tf 14.21 8.08 Td[( jnZ10)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2ni2 )]TJ /F5 7.97 Tf 6.58 0 Td[(1=2)]TJ /F5 7.97 Tf 6.58 0 Td[(3=2niexp")]TJ /F9 11.955 Tf 17.32 8.09 Td[( 2nini jn)]TJ /F6 11.955 Tf 11.96 0 Td[(12#dni=1=2 \())]TJ /F14 7.97 Tf 6.59 0 Td[(jnx)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2exp)]TJ /F6 11.955 Tf 9.3 0 Td[(2r x jnEY)]TJ /F5 7.97 Tf 6.58 0 Td[(1=2jn,whereYjnInverseGaussian(jn,)=IG(p xjn,2x).For=1 2,3 2,5 2,...,theclosedformexpressionofEY)]TJ /F5 7.97 Tf 6.59 0 Td[(1=2jnmaybederived.Forsimplicity,wechoose=1 2.Inthiscase,EY)]TJ /F5 7.97 Tf 6.58 0 Td[(1=2jn=1. 39

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Table4-1. MatrixofLosses chooseH0nichooseHAni H0nitrue00nHAnitrue1n0 WeassumethatH0niandH1nioccurwithprobabilitiespnand1)]TJ /F3 11.955 Tf 12.55 0 Td[(pnrespectively.Thusmarginally,Xnihasthetwo-groupmixturepdf(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)f0n(x)+pnf1n(x).Thewordsparsityreferstothesituationpn0.ConsidertheadditivelossPni=1L(Xni),whereL(Xni)=0n(Xni)+1n(1)]TJ /F9 11.955 Tf 12.3 0 Td[((Xni).Here(Xni)equals1or0accordingtotherejectionoracceptanceofH0.Table 4-1 describesthislossstructure.Underthislossstructure,theBayesdecisionproblemleadstoaprocedurewhichchoosesthealternativehypothesisHAniwhenf1n(Xni) f0n(Xni)>(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)0n pn1nwheref1nandf0narethedensitiesofXniunderalternativeandnullrespectively.ThissimpliestorejectH0niifs 0n 1nexp2p Xni1 p 0n)]TJ /F6 11.955 Tf 23.05 8.09 Td[(1 p 1n>(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)0n pn1n,orequivalently,rejectH0niif Xni>Cn,i=1,...,n.whereCn="1 2p 0n1n p 1n)]TJ 11.96 9.03 Td[(p 0nlog (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)0n pn1ns 1n 0n!#2.ThisprocedureisdenedasBayesoracleduetotheuseofunknownparamters.InaBayesianframework,theBayesriskisanimportantquantitythatcanbeusedtomeasuretheperformanceofarule.InordertoderivetheBayesriskofthe 40

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Bayesrule(theBayesoracle),werstlyneedtheprobabilitiesoftypeIandtypeIIerrors.WedenotetheprobabilitiesoftypeIandtypeIIerrorsrespectively,foralli,ast1n=PH0ni(H0niisrejected)=PH0ni(Xni>Cn)andt2n=PHAni(H0niisaccepted)=PHAni(Xnip Cn=exp )]TJ /F6 11.955 Tf 9.3 0 Td[(2s Cn 0n! =exp")]TJ 30.83 17.11 Td[(p 1n p 1n)]TJ 11.95 9.03 Td[(p 0nlog (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)0n pn1ns 1n 0n!#(4)andt2n=PH1niZni


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Denition4.1Underanadditivelossfunction,theBayesriskforaxedthresholdmultipletestingprocedurewithrespectiveprobabilitiest01nandt02noftypeIandtypeIIerrorsisgivenbyR0=nXi=1[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t01n0n+pnt02n1n]=n[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t01n0n+pnt02n1n].Thus,theBayesriskcorrespondingtoBayesoraclewhichwewillrefertoastheoptimalriskis Roptimal=n[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n0n+pnt2n1n].(4)Inaddition,wehaveDenition4.2togiveacriteriontojudgewhetheranarbitrarymultipletestingprocedureisgoodenoughornotbycomparingtheBayesriskoftheprocedurewiththeoptimalone.Denition4.2ForasequenceofparameterssatisfyingAssumption(),amultipletestingruleiscalledasymptoticallyBayesoptimalundersparsity(ABOS)ifitsriskRsatisesR Roptimal!1,asn!1,whereRoptimalistheoptimalriskgivenby( 4 ).Intherestofthispaper,ABOSwillbereferredtoasymptoticallyBayesoptimal(oroptimality)undersparsityuptothecontext.BasedonDenition4.2,wehavethefollowingthreetheoremswhichshowtheABOSoftwogeneralclassesofmutipletestingprocedures.Therstclassisofthexedthresholdmultipletestingprocedures.Theorem4.2ForaxedthresholdmutipletestingprocedurefFTMgdenedby rejectH0niifXni>Cn,i=1,...,n,whereCn>0.ThecorrespondingprobabilitiesoftypeIandtypeIIerrorsarerespectivelygivenby t1n=exp)]TJ /F6 11.955 Tf 9.3 0 Td[(2q Cn 0nandt2n=1)]TJ /F6 11.955 Tf 11.96 0 Td[(exp)]TJ /F6 11.955 Tf 9.3 0 Td[(2q Cn 1n.Theorem4.3providesnecessaryandsufcientconditionsunderwhichaxedthresholdmutipletestingprocedureisABOS. 42

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Theorem4.3AxedthresholdmutipletestingproceduredenedinTheorem4.2isasymptoticallyBayesoptimalundersparsity(ABOS)ifandonlyifoneofthefollowingconditionsissatised(a)CnCn;(b)Cn0nlog [1)]TJ /F5 7.97 Tf 6.58 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)]pn2.Proof:TheriskofthemutipletestingprocedurefFTMgwithxedthresholdCnisR:=n[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n0n+pnt2n1n].Then,byTheorem4.1,fFTMgisABOSifandonlyifR Roptimal!1,(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)0nt1n+pn1nt2n (1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)0nt1n+pn1nt2n!1,t1n+pn1n (1)]TJ /F8 7.97 Tf 6.59 0 Td[(pn)0nt2n t1n+pn1n (1)]TJ /F8 7.97 Tf 6.59 0 Td[(pn)0nt2n!1, 1)]TJ /F6 11.955 Tf 11.96 0 Td[(exp()]TJ /F3 11.955 Tf 9.29 0 Td[(A)t1n pn+t2n 1)]TJ /F6 11.955 Tf 11.96 0 Td[(exp()]TJ /F3 11.955 Tf 9.29 0 Td[(A)!1.DeneL= 1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)t1n pn+t2n 1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A).Therearevepossiblecases.[1]IfCn=o(Cn)andCn0nlog [1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)]pn2,then 1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)t1n pn=exp)]TJ /F7 11.955 Tf 9.29 13.6 Td[(q Cn 0n [1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)]pn!1andt2n 1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)!0.ThusL!1.[2]IfCn=o(Cn)andCn0nlog [1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)]pn2,then 1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)t1n pn=exp)]TJ /F7 11.955 Tf 9.29 13.6 Td[(q Cn 0n [1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)]pn91andt2n 1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)!0.ThusL91.[3]IfCn=o(Cn),then 1)]TJ /F5 7.97 Tf 6.58 0 Td[(exp()]TJ /F8 7.97 Tf 6.59 0 Td[(A)t1n pn!0andt2n 1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.59 0 Td[(A)!1 1)]TJ /F5 7.97 Tf 6.58 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A).ThusL!1 1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)6=1.[4]IfCnCn,then 1)]TJ /F5 7.97 Tf 6.58 0 Td[(exp()]TJ /F8 7.97 Tf 6.59 0 Td[(A)t1n pn!0andt2n 1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.59 0 Td[(A)!1.ThusL!1.[5]IfCnrCnwherer6=1,then 1)]TJ /F5 7.97 Tf 6.58 0 Td[(exp()]TJ /F8 7.97 Tf 6.59 0 Td[(A)t1n pn!0andt2n 1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.58 0 Td[(A)!1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ 6.59 5.91 Td[(p rA) 1)]TJ /F5 7.97 Tf 6.58 0 Td[(exp()]TJ /F8 7.97 Tf 6.59 0 Td[(A).ThusL!1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ 6.59 5.91 Td[(p rA) 1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.59 0 Td[(A)6=1.Hence,fFTMgisABOSifandonlyifeitherCn0nlog [1)]TJ /F5 7.97 Tf 6.59 0 Td[(exp()]TJ /F8 7.97 Tf 6.59 0 Td[(A)]pn2orCnCn. 43

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Remark:Theorem4.3givesanecessaryandsufcientconditionofachievingABOSforagivenxedthresholdmultipletestingprocedureinasimpleformwhichonlyinvolvesaratiooftwoxedthresholds.Unliketheabove,thefollowingTheorem4.4givesasufcientconditionofABOSforagivenrandomthresholdprocedurerequiringacertainconvergencerateforthedifferenceofthetwothreshholdstozero.Theorem4.4DenearandomthresholdmutipletestingprocedurefRTMgby rejectH0niifXni>^Cn,i=1,...,n,Ifforall>0, Pj^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj>=o(pn),(4)thenfRTMgisABOS.Proof:Notethatbythemodelassumption,Pj^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)PH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>+pnPHAnij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>.By( 4 ), PH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>=o(pn)(4)and PHAnij^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj>=o(1).(4)DenetheprobabilitiesoftypeIandtypeIIerrorsrespectivelyby^t1ni=PH0niXni>^Cnand^t2ni=PHAniXni<^Cn. 44

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Wehave,forall>0,^t1ni=PH0niXni>^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cn+Cn=PH0ni[Xni>^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>]+PH0ni[Xni>^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj<]>PH0ni[Xni>Cn+]\[j^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj<]>PH0ni(Xni>Cn+))]TJ /F3 11.955 Tf 11.95 0 Td[(PH0nij^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj>=PH0n1(Xn1>Cn+))]TJ /F3 11.955 Tf 11.95 0 Td[(PH0nij^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj>and^t1ni=PH0niXni>^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn=PH0ni[Xni>^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>]+PH0ni[Xni>^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj<]6PH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>+PH0ni[Xni>Cn)]TJ /F9 11.955 Tf 11.96 0 Td[(]\[j^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj<]6PH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>+PH0ni(Xni>Cn)]TJ /F9 11.955 Tf 11.95 0 Td[()=PH0n1(Xn1>Cn)]TJ /F9 11.955 Tf 11.95 0 Td[()+PH0nij^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj>Similarly,^t2ni=PHAniXni<^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn=PHAni[Xni<^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>]+PHAni[Xni<^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj<]>PHAni[XniPHAni(Xni=PHAn1(Xn1 45

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and^t2ni=PHAniXni<^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn=PHAni[Xni<^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.96 0 Td[(Cnj>]+PHAni[Xni<^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cn+Cn]\[j^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj<]6PHAnij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj>+PHAni[Xni+PHAni(XniTherefore,asn!1,theratiooftheBayesrisk^Rntotheoptimalrisk^Rn Roptimal=Pni=1[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)^t1ni0n+pn^t2ni1n] n[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n+pnt2n1n]=1 nP^t1ni t1n(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n (1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n0n+pnt2n1n+1 nP^t2ni t2npnt2n1n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n+pnt2n1n>PH0n1(Xn1>Cn+))]TJ /F5 7.97 Tf 13.22 4.7 Td[(1 nPPH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj> PH0n1(Xn1>Cn)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n (1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n0n+pnt2n1n+PHAn1(Xn1 PHAn1(Xn1
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whichfollowsfrom( 4 ),( 4 )andTheorem4.1.Similarly,^Rn Roptimal=Pni=1[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)^t1ni0n+pn^t2ni1n] n[(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n+pnt2n1n]=1 nP^t1ni t1n(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n (1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n0n+pnt2n1n+1 nP^t2ni t2npnt2n1n (1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n+pnt2n1n6PH0n1(Xn1>Cn)]TJ /F9 11.955 Tf 11.96 0 Td[()+1 nPPH0nij^Cn)]TJ /F3 11.955 Tf 11.95 0 Td[(Cnj> PH0n1(Xn1>Cn)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pn)t1n0n (1)]TJ /F3 11.955 Tf 11.96 0 Td[(pn)t1n0n+pnt2n1n+PHAn1(Xn1 PHAn1(Xn1
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CHAPTER5CONCLUSIONSMultipletestingproblemsoriginatedfrommultiplesourcesincludinggenomicsresearchandwitnessedadramaticdevelopmentduringthepasteighteenyears.WebrieydescribedtheimportanceanddevelopmentofresearchonthemultipletestingprobleminChapter1.Wereviewedtheliteratureonmultipletestingproblemsmostrelevanttoourresearch,includingpapersof[ 1 ]BenjaminiandHochberg(1995),Bogdan,Chakrabarti,FrommletandGhosh(2011),[ 4 ]EfronandTibshirani(2002),[ 5 ]GenoveseandWasserman(2002)and[ 8 ]Storey(2003).InChapter2,forastatisticalmodelofmixtureexponentialdistributions,wecontructedamultipletestingprocedurebasedonBayesiandecisionrulewithanassumptionoftheparameterspace.WederivedtheimportantquantitiessuchasprobabilitiesoftypeIandtypeIIerrorsandthecorrespondingoptimalBayesrisk.Then,weintroducedtheconceptoftheAsymptoticBayesOptimalityunderSparsitybasedontheoptimalBayesrisk.EventuallyweprovedtwogeneraltheoremsabouttheAsymptoticBayesOptimalityunderSparsityforthexedthresholdmultipletestingprocedureandtherandomthresholdmultipletestingprocedure.Thesetwotheoremsprovideusefultoolstoverifytheoptimalityofotherthresholdingprocedures.Next,inChapter3,weextendedtheaboveworktoinvestigatetherelationshipbetweentheAsymptoticBayesOptimalityunderSparsity,theFalseDiscoveryRateandtheBayesianFalseDiscoveryRate.Weprovedaseriesoftheoremstoprovidetheconditionsunderwhichtheabovethreemeasurescanbeequivalent.Weprovedasetoflemmasbyapplyingpreviousresults.TheselemmashelpedustoprovetheAsymptoticBayesOptimalityunderSparsityoftheBenjamini-Hochbergprocedureundercertainconditions,particularlyfortheexponentialdistribution.Wealsoranasimulationstudy. 48

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Finally,inChapter4,weconsideredaGammapriorasanalternativechoiceonthescaleparameters.WedenedtheAsymptoticBayesOptimalityunderSparsitybasedonthismodelandderivedaseriesofcorrespondingresults. 49

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REFERENCES [1] Benjamini,Y.andHochberg,Y.Controllingthefalsediscoveryrate:apracticalandpowerfulapproachtomultipletesting.J.R.Stat.Soc.Ser.B57(1995):289. [2] Bogdan,M.,Chakrabarti,A.,Frommlet,F.,andGhosh,J.K.AsymptoticBayes-optimalityundersparsityofsomemultipletestingprocedures.TheAn-nalsofStatistics39(2011):1551. [3] Donoho,D.andJin,J.Asymptoticminimaxityoffalsedicoveryratethresholdingforsparseexponentialdata.TheAnnalsofStatistics34(2006):2980. [4] Efron,B.andTibshirani,R.Empiricalbayesmethodsandfalsediscoveryratesformicroarrays.GeneticEpidemiology23(2002):70. [5] Genovese,C.andWasserman,L.Operatingcharacteristicsandextensionsofthefalsediscoveryrateprocedure.J.R.Stat.Soc.Ser.B64(2002):499. [6] Neuvial,P.andRoquain,E.Onfalsediscoveryratethresholdingforclassicationundersparsity.TheAnnalsofStatistics40(2012):2572. [7] Sering,R.J.ApproximationTheoremsofMathematicalStatistics.NewYork:Wiley,1980. [8] Storey,J.D.Thepositivefalsediscoveryrate:aBayesianinterpretationandtheq-value.TheAnnalsofStatistics31(2003):2013. [9] Storey,J.D.Theoptimaldiscoveryprocedure:anewapproachtosimultaneoussignicancetesting.J.R.Stat.Soc.Ser.B69(2007):347. [10] Sun,W.andCai,T.C.Oracleandadaptivecompounddecisionrulesforfalsediscoveryratecontrol.JASA102(2007):901. 50

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BIOGRAPHICALSKETCH KeListartedhisgraduatestudyatDepartmentofStatistics,UniversityofFloridainthefallof2008.HereceivedhisPh.D.fromtheUniversityofFloridainthesummerof2013. 51