<%BANNER%>

Estimator Benchmark and Optimal Test for Location to Fechfner-Asymmetry

Permanent Link: http://ufdc.ufl.edu/UFE0043580/00001

Material Information

Title: Estimator Benchmark and Optimal Test for Location to Fechfner-Asymmetry
Physical Description: 1 online resource (116 p.)
Language: english
Creator: Athienitis, Demetris J
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: 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: Robustness measures and procedures have traditionally been developed and implemented through the contamination model. A model that partially samples from a contaminant distribution as presented by Hampel et al. (1986). We present a distortion model that takes a symmetric probability density function and skews it infinitesimally in a specific direction thereby distorting every realization of the distribution, a model based upon the work of Fechner (1897). Robustness to distortion of a symmetric signal is determined as the rate of change of the functional of an affine equivariant location estimator under the Fechner model. The mean, median and Hodges-Lehman estimators are compared under this model. Cassart et al. (2008) propose an optimal test for testing asymmetry and compare it to several traditional tests including the triples test of Randles et al. (1980). We derive the efficacy of the triples test and compare it to their signed-rank version. Using the Fechner model comparative to Cassart et al. (2008), locally optimal parametric and semiparametric signed-rank tests for location that are robust to asymmetry are created for both specified and unspecified asymmetry. The efficient score component of location is regressed onto the asymmetry component to create the residuals, that under regularity conditions, follow a normal distribution. The observations are replaced by a vector of signs and ranks to construct the sign-rank version test, and an adaptive testing procedure is used for choosing an adequate score density function.
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 Demetris J Athienitis.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Randles, Ronald H.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043580:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043580/00001

Material Information

Title: Estimator Benchmark and Optimal Test for Location to Fechfner-Asymmetry
Physical Description: 1 online resource (116 p.)
Language: english
Creator: Athienitis, Demetris J
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: 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: Robustness measures and procedures have traditionally been developed and implemented through the contamination model. A model that partially samples from a contaminant distribution as presented by Hampel et al. (1986). We present a distortion model that takes a symmetric probability density function and skews it infinitesimally in a specific direction thereby distorting every realization of the distribution, a model based upon the work of Fechner (1897). Robustness to distortion of a symmetric signal is determined as the rate of change of the functional of an affine equivariant location estimator under the Fechner model. The mean, median and Hodges-Lehman estimators are compared under this model. Cassart et al. (2008) propose an optimal test for testing asymmetry and compare it to several traditional tests including the triples test of Randles et al. (1980). We derive the efficacy of the triples test and compare it to their signed-rank version. Using the Fechner model comparative to Cassart et al. (2008), locally optimal parametric and semiparametric signed-rank tests for location that are robust to asymmetry are created for both specified and unspecified asymmetry. The efficient score component of location is regressed onto the asymmetry component to create the residuals, that under regularity conditions, follow a normal distribution. The observations are replaced by a vector of signs and ranks to construct the sign-rank version test, and an adaptive testing procedure is used for choosing an adequate score density function.
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 Demetris J Athienitis.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Randles, Ronald H.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-12-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043580:00001


This item has the following downloads:


Full Text

PAGE 1

ESTIMATORBENCHMARKANDOPTIMALTESTFORLOCATIONTOFECHNER-ASYMMETRYByDEMETRISATHIENITISADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

PAGE 2

c2011DemetrisAthienitis 2

PAGE 3

Tomylovingparents,PhidiasandJenniferAthienitisforalltheirsupport,understandingandpatience,andtomywonderfulwife,Dr.JenniferGoldman,forallherloveandencouragement 3

PAGE 4

ACKNOWLEDGMENTS First,andforemost,IwouldliketothankmyadvisorDr.RonaldRandlesfortheopportunitytoworkwithhimforthepurposeofthisdissertation;andforhiswealthofknowledgeandinvaluableguidancethroughthisprocess.Withouthimthiswouldhaveneverbeenpossible.IwouldalsoliketoacknowledgeandgivespecialthankstoDr.MarcHallinfortheinspiration,insightandaidofcriticalpartsofthisdissertationandtoDr.BrettPresnellforhisassistancewithspecictopicsthatwereconsidered. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 10 CHAPTER 1REVIEWOFMEASURESOFROBUSTNESSANDASYMMETRYMODELS 11 1.1IntroductionToRobustness .......................... 11 1.2RobustnessOfEstimators ........................... 11 1.2.1Huber'sMinimaxApproach ...................... 13 1.2.2TheInuenceFunction ......................... 13 1.2.2.1Measuresofrobustness ................... 17 1.2.2.2Globalreliability:breakdownpoint ............. 18 1.3Asymmetry ................................... 19 1.3.1FechnerAsymmetry .......................... 20 1.3.1.1GeneralFechnerasymmetry ................ 21 1.3.1.2Inferentialaspects ...................... 23 1.3.2AClassOfSkewDistributions ..................... 24 1.3.3ScalingTheTails ............................ 25 1.3.4AsymmetricDensitiesBasedOnEdgeworthApproximations .... 26 2DISTORTIONSENSITIVITYOFESTIMATORSOFLOCATION ......... 30 2.1Afne-Equivariance .............................. 30 2.2DistortionAndContaminationModels .................... 31 2.3ComparisonOfEstimatorsOfLocation .................... 32 2.3.1Mean,MedianandHodges-Lehmann ................. 33 2.3.2M-Estimators .............................. 42 2.3.3L-Estimators ............................... 43 2.3.4R-Estimators .............................. 44 2.3.5P-Estimators .............................. 45 2.4MultivariateEstimatorsOfLocation ...................... 45 2.4.1MultivariateFechnerModel ...................... 47 2.4.2MeanandMedian ........................... 48 3REVIEWOFOPTIMALDETECTIONOFFECHNER-ASYMMETRY ...... 54 3.1UniformLocalAsymptoticNormality ..................... 54 3.2ParametricOptimalTestsForSymmetry ................... 56 3.2.1SpeciedDensity,SpeciedLocation ................. 57 5

PAGE 6

3.2.2SpeciedDensity,UnspeciedLocation ............... 58 3.2.3Pseudo-GaussianTests ........................ 59 3.3RankBasedTestsForSymmetry ....................... 59 3.3.1Signed-RankVersionsoftheCentralSequence ........... 59 3.3.2TestWithUnspeciedLocation .................... 60 3.4AsymptoticRelativeEfciencies ....................... 62 4REVISITOFTHETRIPLESTESTOFASYMMETRY ............... 64 4.1FunctionalRepresentation ........................... 65 4.2AsymptoticRelativeEfciency ........................ 66 5OPTIMALTESTINGOFLOCATIONROBUSTTOFECHNER-ASYMMETRY 71 5.1ULANAndParametricallyOptimalTestForLocation ............ 71 5.1.1ULAN .................................. 71 5.1.2OptimalTesting:SpeciedDensity,SpeciedAsymmetry ..... 73 5.1.3OptimalTesting:SpeciedDensity,UnspeciedAsymmetry .... 74 5.2OptimalSigned-RankBasedTests ...................... 76 5.2.1OptimalSigned-RankTestsForLocation:SpeciedAsymmetry .. 78 5.2.2OptimalSigned-RankTestsForLocation:UnspeciedAsymmetry 80 5.3AsymptoticRelativeEfciencies ....................... 82 5.3.1SpeciedAsymmetry .......................... 82 5.3.2UnspeciedAsymmetry ........................ 86 5.4ConsistentEstimationOfParameters ..................... 86 5.5PracticalImplementationAndSimulationResults .............. 87 5.5.1SpeciedAsymmetry .......................... 87 5.5.2UnspeciedAsymmetry ........................ 88 6SUMMARYANDCONCLUSIONS ......................... 97 APPENDIX ADISTORTIONSENSITIVITYOFHODGES-LEHMANN .............. 99 BDETAILSOFEXAMPLE2.1 ............................. 102 CDISTORTIONSENSITIVITYOFM-ESTIMATORS ................ 104 DDISTORTIONSENSITIVITYOFL-ESTIMATORS ................. 105 EDISTORTIONSENSITIVITYOFR-ESTIMATORS ................ 107 FDISTORTIONSENSITIVITYOFP-ESTIMATORS ................ 108 GPROOFOFPROPOSITION3.2 .......................... 109 HCONSISTENTESTIMATIONOFCROSS-INFORMATIONQUANTITIES .... 111 REFERENCES ....................................... 113 6

PAGE 7

BIOGRAPHICALSKETCH ................................ 116 7

PAGE 8

LISTOFTABLES Table page 2-1Distributionalsensitivityratiosforcommondistributions ............. 51 2-2DistributionalSensitivityRatiounderPowerExponentialFamily ......... 51 2-3DistributionalSensitivityRatiounderStudent'stFamily ............. 53 4-1AREsundervariousdensitiesofthesigned-rankversiontestwithrespecttothetriplestest ..................................... 70 5-1AREsundervariousdensitiesofthesigned-rankversiontestwithrespecttothesigntestforspecied ............................. 90 5-2AREsundervariousdensitiesofthesigned-rankversiontestwithrespecttothet-testforspecied=0 ............................. 91 5-3AREsundervariousdensitiesofthesigned-rankversiontestwithrespecttotheWilcoxonsigned-rankforspecied=0 .................... 91 5-4Efcaciesundervariousdensitiesofthesigned-rankversiontestforunspecied,computedwhen=0.6 .............................. 92 5-5Rejectionfrequencies,withspeciedasymmetry=0,0.3and0.6(forrst,secondandthirdrowrespectively),ofthesigned-rankversiontestwithxedscoredensityfunctionft20 .............................. 92 5-6Rejectionfrequencies,withspeciedasymmetry=0,ofthesigned-rankversion,signtestandWilcoxonsigned-ranktest ................. 93 5-7Rejectionfrequencies,withspeciedasymmetry=0.3,ofthesigned-rankversionandsigntest ................................. 93 5-8Rejectionfrequencies,withspeciedasymmetry=0.6,ofthesigned-rankversionandsigntest ................................. 94 5-9Rejectionfrequencies,withunspeciedasymmetry=0,0.3and0.6(forrst,secondandthirdrowrespectively),ofthesigned-rankversiontestwithxedscoredensityfunctionft20 .............................. 94 5-10Rejectionfrequencies,withunspeciedasymmetry,ofthesigned-rankversiontest.Simulatedwith=0 .............................. 95 5-11Rejectionfrequencies,withunspeciedasymmetry,ofthesigned-rankversiontest.Simulatedwith=0.3 ............................. 95 5-12Rejectionfrequencies,withunspeciedasymmetry,ofthesigned-rankversiontest.Simulatedwith=0.6 ............................. 95 8

PAGE 9

LISTOFFIGURES Figure page 1-1GaussianFechnerclassfora()=andb()=)]TJ /F9 7.97 Tf 6.59 0 Td[(1 ............... 27 1-2GaussianFechnerclassfora()=1+andb()=1)]TJ /F4 11.955 Tf 11.96 0 Td[( ............ 28 1-3Skew-normalclass .................................. 28 1-4Separatelyscaledtailswithunderlyingstandardnormaldistribution ....... 29 1-5GaussianEdgeworthfamily ............................. 29 2-1Halfre-scaledmixturedensityfunctiongs ..................... 52 2-2ComplementarycumulativedistributionfunctionofUniform(0,1)andGs .... 52 2-3Unitcircleshiftedd0fromtheorigin ........................ 53 5-1Densityfunctionofsigned-rankversiontestforunspeciedasymmetrywith=0,=1and=0.6forthepowerexponentialdistributionwith=4 .... 96 B-1Plotoffunction,P(X>a))]TJ /F5 11.955 Tf 11.96 0 Td[(1+2afora2(0,1). ................. 103 9

PAGE 10

AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyESTIMATORBENCHMARKANDOPTIMALTESTFORLOCATIONTOFECHNER-ASYMMETRYByDemetrisAthienitisDecember2011Chair:RonaldRandlesMajor:Statistics Robustnessmeasuresandprocedureshavetraditionallybeendevelopedandimplementedthroughthecontaminationmodel.Amodelthatpartiallysamplesfromacontaminantdistributionaspresentedby Hampeletal. ( 1986 ).Wepresentadistortionmodelthattakesasymmetricprobabilitydensityfunctionandskewsitinnitesimallyinaspecicdirectiontherebydistortingeveryrealizationofthedistribution,amodelbasedupontheworkof Fechner ( 1897 ).RobustnesstodistortionofasymmetricsignalisdeterminedastherateofchangeofthefunctionalofanafneequivariantlocationestimatorundertheFechnermodel.Themean,medianandHodges-Lehmanestimatorsarecomparedunderthismodel. Cassartetal. ( 2008 )proposeanoptimaltestfortestingasymmetryandcompareittoseveraltraditionaltestsincludingthetriplestestof Randlesetal. ( 1980 ).Wederivetheefcacyofthetriplestestandcompareittotheirsigned-rankversion. UsingtheFechnermodelcomparativeto Cassartetal. ( 2008 ),locallyoptimalparametricandsemiparametricsigned-ranktestsforlocationthatarerobusttoasymmetryarecreatedforbothspeciedandunspeciedasymmetry.Theefcientscorecomponentoflocationisregressedontotheasymmetrycomponenttocreatetheresiduals,thatunderregularityconditions,followanormaldistribution.Theobservationsarereplacedbyavectorofsignsandrankstoconstructthesign-rankversiontest,andanadaptivetestingprocedureisusedforchoosinganadequatescoredensityfunction. 10

PAGE 11

CHAPTER1REVIEWOFMEASURESOFROBUSTNESSANDASYMMETRYMODELS 1.1IntroductionToRobustness Instatisticalinference,explicitand/orimplicitassumptionsaremadeinordertojustifystatisticalmethodsandtostudytheirproperties.Distributionalassumptions,aswellasassumptionsconcerningrandomness,independenceandpriordistributionsinaBayesianframeworkareamongthosemostcommonlyused. Certainpopularstatisticalproceduresaresensitivetosmalldeparturesfromtheseassumptionswhichhasgivenrisetothedevelopmentofrobuststatisticalprocedures,i.e.thoselesssensitivetoassumptions.Someprocedurescanimproveefciencywhentherearedeviations(ofanymagnitude)fromtheunderlyingassumptionsandcanevenmatchtheasymptoticefciencyofclassicalprocedures.Forexample,atwo-sampleranktestforlocationshifttestwithVanderWaerdennormalscorescanmatchtheefciencyofaclassicaltwo-samplet-testevenwhenthenormaldistributionassumptionsholdandhavebetterefciencywithallotherdistributions. Wewillbereferringtorobustnessinthesamesensethat Huber&Ronchetti ( 2009 ,chap.1)refersto,thatofinsensitivitytosmalldeviationsfromtheassumptions.Arobustprocedureshouldalsobeabletosafeguardfromacompletedisasterwhentherearelargerdeviations.Inparticular,wewillconcernourselveswithrobustnesstodistributionalassumptions.Theseassumptionsincertainstatisticalproceduresmaybeanoversimplicationorpossiblyamisidenticationoftheobserveddistribution.Whentheassumptionsareincorrect,itmayinvalidatethestatisticalprocedureordiminishitsperformanceincomparisontootherprocedures. 1.2RobustnessOfEstimators Theearlyworkonrobustprocedureswasfocusedontesting. Tukey ( 1960 )wasoneofthepioneerstoventureintorobustestimationafterdemonstratingtheseverelackofrobustnessofthemeantoextremedatapoints. 11

PAGE 12

Inthisworkwedescribeestimatorsandtheirparametersasfunctionals.ThefunctionalT(F)isdenedfordistributionfunctionsF(orsomeappropriatesubsetofthem).IfXFthenwewriteeitherT(F)orT(X).InexpressingthefunctionalwewilluseT(F)andT(X)interchangeablydependingonthesetting.TheestimatorcorrespondingtoT()willbedenotedbyTn=T(Fn)=Tn(X1,...,Xn) whereFnistheempiricaldistributionfunctioncorrespondingtoX1,...,Xni.i.d.F.WewillconsiderfunctionalsthatareFisherconsistent,thatis,oneswhichsatisfyTnp!T(F). Inaddition,asymptoticnormalityisalsooftenestablishedsincetheconditionsnecessaryforthisresultwillfrequentlybemet,thatis p n(Tn)]TJ /F3 11.955 Tf 11.95 0 Td[(T(F))d!N(0,V(T,F)),(1) whereV(T,F)iscalledtheasymptoticvarianceofTnatF. InquantitativerobustnessaneighborhoodoftheunderlyingdistributionFisintroducedsoastodescribehowsmallchangesinFaffectTn.ThecontaminationmodelforadistributionfunctionFisdenedas C=f(1)]TJ /F4 11.955 Tf 11.96 0 Td[()F+xg,2(0,1).(1) Thefunctionxisthedistributionfunctionofapointmassatx,alsoknownastheDiracdeltafunction. Huber ( 1964 )introducedageneralneighborhood,thegrosserrormodel P=f(1)]TJ /F4 11.955 Tf 11.95 0 Td[()F+H,H2Mg,(1) 12

PAGE 13

whereMisthespaceofalldistributionfunctions.ThismodelhassincebeingextensivelyusedintherobustnessliteratureasaneighborhoodofaparametricmodelF. 1.2.1Huber'sMinimaxApproach OneofthecharacteristicsthatHuberusesfromthegrosserrormodelisthemaximumbiassupG2PjT(G))]TJ /F3 11.955 Tf 11.96 0 Td[(T(F)j, whereheconcludesthatforsymmetricunimodaldistributionsthesamplemedianistheestimatorthatminimizesthemaximumbiasandhencethemaximumasymptoticbiaswhenPisaLevyneighborhood,i.e.fGjF(t)]TJ /F4 11.955 Tf 12.73 0 Td[())]TJ /F4 11.955 Tf 12.74 0 Td[(GF(t+)+,8tg,asthemaximumbiasandmaximumasymptoticbiasareequivalent.Thesamplemedianisthereforetheestimateofchoiceforsampleswherethestandarddeviationoftheestimateislessthanorequaltothebias. MoreinterestingaretheconclusionsderivedfromminimizingthemaximalasymptoticvariancelimnsupG2PQt(G,Tn)2,whereQt(G,Tn)isthenormalizedt-quantilerangeoftheasymptoticdistributionofp nTnandisdenotedasQt(G,Tn)=G)]TJ /F9 7.97 Tf 6.59 0 Td[(1(1)]TJ /F3 11.955 Tf 11.95 0 Td[(t))]TJ /F3 11.955 Tf 11.95 0 Td[(G)]TJ /F9 7.97 Tf 6.59 0 Td[(1(t) )]TJ /F9 7.97 Tf 6.59 0 Td[(1(1)]TJ /F3 11.955 Tf 11.95 0 Td[(t))]TJ /F5 11.955 Tf 11.95 0 Td[()]TJ /F9 7.97 Tf 6.58 0 Td[(1(t). Firstthemaximalvariance,thesupremumofsupV(G,T)forG2Pisminimizedandtheobservationsareassumedtobei.i.d.fromG(x)]TJ /F4 11.955 Tf 11.96 0 Td[(),withGbeingfromthesetP. LettingG02PdenotethedistributionwiththesmallestFisherinformation,denotedasI(G0)=Zg00 g02g0dx. Then,undercertainregularityconditionsitisshown( Huber&Ronchetti 2009 ,chap.4)thatG0istheuniquememberofthesetPthatminimizestheFisherinformation,hencesolvingtheminimaxproblem. 1.2.2TheInuenceFunction 13

PAGE 14

Hampel ( 1968 1974 )developedaheuristictool,theinuencefunction(IF)torepresenttherateofchange,inasymptoticbiasofthefunctionalTcausedbyaddinganinnitesimalpointatx.Theneighborhoodusedisaspecialcaseof( 1 )wherethecontaminantdistributionisthepointmassatx.TheinuencefunctionofTforthedistributionfunctionFisgivenby IF(x;T,F)=limt#0T((1)]TJ /F3 11.955 Tf 11.95 0 Td[(t)F+tx))]TJ /F3 11.955 Tf 11.96 0 Td[(T(F) t(1) forvaluesofxwherethelimitexists.ThepointmasscanbereplacedbyamoregeneraldistributionfunctionH,andTissaidtobeGateauxdifferentiableatF,ifthereexistsarealfunctionaFsuchthatlimt#0T((1)]TJ /F3 11.955 Tf 11.96 0 Td[(t)F+tH))]TJ /F3 11.955 Tf 11.96 0 Td[(T(F) t=ZaF(x)dH(x), butaF(x)lackspracticalinterpretationandsoinpracticeapointmassisusedasthecontaminant.ThefunctionaisassumedtobeconvenientlystandardizedsuchthatRaF(x)dF(x)=0. TheIFofafunctionalprovidesaconvenienttoolforexplicitlyformulatingthevarianceterminequation( 1 )undercertainregularityconditions.IfTisweaklycontinuousinaneighborhoodofF,theIFcanbeusedintheTaylorexpansionofTas T(G)=T(F)+ZIF(x;T,F)d(G)]TJ /F3 11.955 Tf 11.96 0 Td[(F)(x)+remainder(1) foradistributionfunctionGnearF.Forasampleofi.i.d.observationswehavebytheGlivenko-CantellitheoremthatFn!Fandhenceinequation( 1 )thedistributionfunctionGcanbereplacedbyFn.AssumingthatRIF(x;T,F)dF(x)=0,thenequation( 1 )canberewrittenasT(Fn)=T(F)+ZIF(x;T,F)dFn(x)+remainder, 14

PAGE 15

andwithsomesimplecalculusmanipulationasp n(Tn)]TJ /F3 11.955 Tf 11.96 0 Td[(T(F))=1 p nnXi=1IF(Xi;T,F)+remainder. Bythecentrallimittheoremthesummationtermisasymptoticallynormalasn!1anditisoftentruethattheremainderbecomesnegligible,sowehavethat p n(Tn)]TJ /F3 11.955 Tf 11.96 0 Td[(T(F))d!n!1N0,ZIF(x;T,F)2dF(x).(1) Arigorousproofofthisstatementcanbefoundin Reeds ( 1976 ), Boos&Sering ( 1980 ),and Fernholz ( 1983 ).Insteadoftryingtoverifysomeofthediverseandrigorousregularityconditionsthereisaneasierwaytoassurethatasymptoticnormalityisobtainedforasampleofi.i.d.observationswithdistributionF.Assume,ZIF(x;T,F)dF(x)=0and0
PAGE 16

@ @ZIF(x;T,F0)dF(x)0=@ @ZT(F))]TJ /F3 11.955 Tf 11.95 0 Td[(T(F0)dF(x)0=@ @[T(F)]0=@ @0=1, andbychangingtheorderofintegrationanddifferentiationitresultsin1=ZIF(x;T,F0)@ @[f(x)]0d(x)=ZIF(x;T,F0)@ @[lnf(x)]0dF0(x). ThenapplyingtheCauchy-SchwarzinequalityyieldstheCramer-RaoinequalityZIF(x;T,F0)2dF0(x)1 I(F0). Hencetheestimatorisasymptoticallyefcient(i.e.equalityholds)ifandonlyifIF(x;T,F0)=I(F0))]TJ /F9 7.97 Tf 6.58 0 Td[(1@ @[lnf(x)]0. WenowdescribetheIFforsomecommonlocationestimators.LetandMdenotethepopulationmeanandmedianofFrespectivelyandsupposethattheprobabilitydensityfunction(pdf)ofthedistributionfunctionFexists.TheIFforthesamplemeanisIF(x)=x)]TJ /F4 11.955 Tf 11.96 0 Td[(, andforthesamplemedianitisIF(x)=sgn(x)]TJ /F3 11.955 Tf 11.95 0 Td[(M) 2f(M). 16

PAGE 17

TheHodges-Lehmannestimator( HodgesJr&Lehmann 1963 )fori.i.d.observationsisdenedby medianijXi+Xj 2.(1) IthasIF(x)=1)]TJ /F5 11.955 Tf 11.95 0 Td[(2F(2T(F))]TJ /F3 11.955 Tf 11.96 0 Td[(x) 2Zf(2T(F))]TJ /F3 11.955 Tf 11.96 0 Td[(t)dF(t)=1)]TJ /F5 11.955 Tf 11.95 0 Td[(2F()]TJ /F3 11.955 Tf 9.3 0 Td[(x) 2Zf()]TJ /F3 11.955 Tf 9.3 0 Td[(t)dF(t),assumingT(F)=0=2F(x))]TJ /F5 11.955 Tf 11.95 0 Td[(1 2Zf2(t)dt,assumingfissymmetric. 1.2.2.1Measuresofrobustness AstheIFrepresentstherateofchangeofthefunctionalTwhencontaminationisadded,anaturalrobustnessqualitydesiredisaboundedIF.Thiswouldimplythatatinyamountofbaddatacanhaveatmost,alimitedinuenceonthebehaviorofthefunctional.Thegross-errorsensitivityisameasurethatquantiesthesupremumoftheabsolutevalueofIFforaxedFoverallpossiblelocationsoftheplacementofthepointmasscontaminationandisdenedby(T,F)=supx2ST,FjIF(x;T,F)j, whereST,F=fx2RjIF(x;T,F)existsg. Thelocalshiftsensitivitymeasuresthestandardizedeffectofslightmovementsinthepointx.Itisdenotedas(T,F)=supx,y2ST,Fx6=yjIF(y;T,F))]TJ /F3 11.955 Tf 11.95 0 Td[(IF(x;T,F)j jy)]TJ /F3 11.955 Tf 11.95 0 Td[(xj. Itmeasuresthepotentialeffectsofsmalluctuationsintheobservations(e.g.rounding).Thelocalshiftsensitivitycanpotentiallybeinnitelylargeduetoaninnitesimallysmalldifferencejy)]TJ /F3 11.955 Tf 11.96 0 Td[(xjevenifjIF(y;T,F))]TJ /F3 11.955 Tf 11.95 0 Td[(IF(x;T,F)jisnite. 17

PAGE 18

Therejectionpointisarobustnessmeasurethatprovidesaboundaryvalueontheregionwherecontaminationdoesnothaveanyeffect,forsymmetricdistributions,andisdenedtobe(T,F)=inffr>0jIF(x;T,F)=0whenjxj>rg. Hence,adesirablepropertyofestimatorsis<1. 1.2.2.2Globalreliability:breakdownpoint TheIFasdenedinequation( 1 )andthemeasuresofrobustnessderivedfromitareinherentlymeasuresoflocalreliability.TheydonotgiveanyinformationaboutthemaximumdistancethatthecontaminationcanbelocatedfromthemodeldistributionFwhilestillobtainingsomerelevantinformationaboutFfromtheestimator. Thegross-errorbreakdownpointisameasureofglobalreliabilitythatrepresentsthesmallestfractionofgrosserrorsthatcancarrythefunctionalT()innitelyfarawayfromthetargetT(F).Forthegrosserrormodelinequation( 1 ),itis=infsupH2M[jT(G,H))]TJ /F3 11.955 Tf 11.95 0 Td[(T(F)j]=+1forG2P. Hencealargerbreakdownpointisdesirableandanestimatorwithahighbreakdownpointisoftencalledaresistantstatistic.Althoughtherangeofis[0,1],thelargestvaluethatanestimatorcanattainis0.5,becauseifmorethanhalfoftheobservationsarefromthecontaminatingdistribution(H),theestimatorcannotdistinguishbetweenthetargetdistributionFandthecontaminantH. Themedianisanestimatorthatattainsthemaximumvaluewith=0.5.Themedianalsohasotherdesirableproperties.Itisanafne-equivariantandmonotonicestimator.Letx=(x1,...,xn)0.AnestimateT(x)isafne-equivariantifT(ax+b1)=aT(x)+b1,forallbanda6=0andmonotonicifT(x)T(x0)forxix0i,i=1,...,n. Bassett ( 1991 )showedthatthemedianistheonlyafne-equivariant,monotonicestimatorwith50%breakdownpoint.Asacorollary,heshowedthatanyafne-equivariant,monotonicestimatorwitha100r%breakdownmustliebetweenthe 18

PAGE 19

rand(1)]TJ /F3 11.955 Tf 12.35 0 Td[(r)samplequantiles.Estimatorswithtwoofthethreepropertiesincludethemean,whichisafne-equivariantandmonotonicbuthasa0%breakdown,andtheleastmedianofsquares( Rousseeuw 1984 )whichisafne-equivariantandhas50%breakdownbutisnotmonotonic. Themean,with=0,hasthesmallestpossiblebreakdownpointandcannottolerateanygrosslybaddata.TheHodges-Lehmannestimatorforexample,fallssomewhereinthemiddlewith=1)]TJ /F5 11.955 Tf 11.95 0 Td[(1=p (2)=0.293. Thebreakdownpointasdescribedsofarisstrictlyanasymptoticmeasureofglobalreliabilityandrobustness.Usuallyforlargesamples,amixturemodelmoreadequatelydescribesahighdegreeofcontaminationratherthanthegross-errormodelinequation( 1 ).Hencethebreakdownmaybemorevaluableinthenitesamplesetup. Donoho&Huber ( 1982 )distinguishthreewaystocorruptthedata.LetX=(x1,...,xn)beanitesample,thethreewaysare: 1.-contamination:addmarbitraryvaluesY=(y1,...,ym)tothesamplewhichwouldmakethefractionofbaddatainsamplebe=m=(n+m). 2.-replacement:replaceanarbitrarysubsetofsizemofthesamplebyarbitraryvaluesy1,...,ym.,then=m=n. 3.-modication:letbeanarbitrarydistancefunctionontheempiricalmeasurespace.LetFnandGnbetheempiricaldistributionfunctionsofthesampleXandanyothersampleX0respectivelysuchthat(Fn,Gn). Inthedenitionofthebreakdownpoint,thesupremumoverallpossiblecontaminatingdistributionsofthedifferencebetweenthecontaminatedandtargetfunctionals,providesaprobabilisticfreedomwhichmakesthebreakdownpointausefultoolbutremainsanunsuitablemeasureforoptimizingrobustnessduetotheexcessivefreedom. 1.3Asymmetry Inthisdissertation,robustnessisquantiedintermsofadeparturefromtheassumptionofsymmetryintheunderlyingdistribution.Itwillrequireastructurethatcantakeanarbitrarysymmetricdistributionand,bymeansofasingleparameter,convert 19

PAGE 20

thesymmetricdistributionintoanasymmetricone,i.e.anysymmetricdistributionbecomespartofalargerclassofdistributionsthatincludesboththesymmetricandasymmetricdistributions.Onesuchstructurewasproposedby Fechner ( 1897 )andfurtherdevelopedby Arellano-Valle&Genton ( 2005 ), Cassartetal. ( 2008 )and Mudholkar&Hutson ( 2000 ).Ageneralizationoftheclassisprovidedandtwospecialcasesarediscussed.Adifferentstructurewasproposedby O'Hagan&Leonard ( 1976 )whodevelopedafamilyofasymmetricdistributionsthatwasfurtherresearchedby Azzalini ( 1985 )and Liseo ( 1990 ).Anotherintuitiveprocessofcreatinganasymmetricdensitybysimplyscalingeachtailseparately,isalsodescribed.Inaddition,astructureforcreatingasymmetricdistributionsbasedonEdgeworthapproximationsthatwasproposedby Cassartetal. ( 2010 )isincluded. 1.3.1FechnerAsymmetry Aclassofdistributionfunctionsthathasonlyrecentlyreceivedsomeattentionhasaformthatcanbetracedbackto Fechner ( 1897 ).HenceforththisclassshallbereferredtoastheFechnerclassofdensities.AtrstitwaspresentedasafamilythatgeneralizestheGaussianfamilybyallowingforskewnessandkurtosis.ThestandardformofthedensityofY=(X)]TJ /F4 11.955 Tf 11.95 0 Td[()=ish(y)=Kexp)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 2M(y), whereMandKaredenedby M(y)=8><>:yify0;()]TJ /F3 11.955 Tf 9.29 0 Td[(My)ify<0; and1 K=1+1 M21=\(1 ) 20

PAGE 21

for00isgivenasfollows, f(xj,,)=2 (a()+b())fx)]TJ /F4 11.955 Tf 11.96 0 Td[( a()Ifxg+fx)]TJ /F4 11.955 Tf 11.96 0 Td[( b()Ifx
PAGE 22

Theexpressionofequation( 1 )istoogeneralforpracticalapplicationsandexpressionsfora()andb()mustusuallybespecied.Intheirresearchonasymmetry, Fernandez&Steel ( 1998 )deneasubclassofdistributionsbasedontheFechnerclassforspeciedaandbfunctionswiththeadditionalassumptionassumptionthatfisunimodalandthatf(jxj)isdecreasinginjxj.Theydenea()=andb()=)]TJ /F9 7.97 Tf 6.59 0 Td[(1for>0.Theclassisgivenby f(xj0,,)=2 (+)]TJ /F9 7.97 Tf 6.58 0 Td[(1)fx Ifx0g+fx Ifx<0g,(1) forf2Fandsymmetricaround0.Forunimodaldistributions,thisclassretainsthemodeattheoriginbutbecomesasymmetricwhen6=1.For2(0,1)itbecomesrightskewedandleftskewedfor>1.AnillustrationofthissubclasswithanisshowninFigure 1-1 Anothersubclassismotivatedbythefactthatifa()andb()aresmoothnear=0thenfora>0andb>0theyhavetheform,a()=1+a+o(2)andb()=1)]TJ /F3 11.955 Tf 11.96 0 Td[(b+o(2). Theconditionthata()]TJ /F4 11.955 Tf 9.3 0 Td[()=b()isnecessaryforsymmetriceffectsaboutthecenter,i.e.leftandrightasymmetrytreatedthesameway.Thus,ifsufcientlysmoothfunctionsarecontemplated,anaturalconsiderationwouldbeofa()=1+andb()=1)]TJ /F4 11.955 Tf 12.55 0 Td[(for2()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1).Withoutlossofgenerality,wewillassumethatthecenter(pointofsymmetry)is0whichsimpliesequation( 1 )to f(xj0,,)=1 fx (1+sgn(x)),(1) wheresgn(x)=8>>>><>>>>:)]TJ /F5 11.955 Tf 9.3 0 Td[(1ifx<0,0ifx=0,1ifx>0. 22

PAGE 23

ThedistributionfunctionofskewrandomvariableXSf(0,,)is F=F(xj0,,)=8>>><>>>:(1)]TJ /F4 11.955 Tf 11.96 0 Td[()Fx (1)]TJ /F4 11.955 Tf 11.95 0 Td[()ifx<0,)]TJ /F4 11.955 Tf 9.3 0 Td[(+(1+)Fx (1+)ifx0.(1) PlotsaredisplayedinFigure 1-2 foraFechnerasymmetricdensityfunctionwithanunderlyingstandardnormaldensityfunction. Themodelsofequations( 1 )and( 1 )differthroughthefunctionsaandbwhichatrstglancemayappeartobesmall,butthelargestdifferenceappearswhentheasymmetryparameterreachestheextremes.Underequation( 1 )lim!1f(xj)=0andlim!0+f(xj)=0. Astheasymmetryparameterapproachestheboundarypoints,themassofthedistributionisspreadevenlyacrossthereallinetotheextentthatatthelimitthefunctionisidentically0.ThisisillustratedinFigure 1-1 foranunderlyingnormaldistribution.Onthehandunderequation( 1 )lim!)]TJ /F9 7.97 Tf 15.06 0 Td[(1+f(xj)=f(x=2)Ifx0gandlim!1)]TJ /F3 11.955 Tf 8.25 5.81 Td[(h(xj)=f(x=2)Ifx<0g. Forthismodel,theconsequentdistributionisthehalfdistributionofthetargetwherethemassofthedensityisplacedonlyononesideof0(targetpointofcenter)dependingonwhichboundarypointtheasymmetryparameterisheadingto. 1.3.1.2Inferentialaspects StatisticalinferenceontheFechnerclassofdistributionswithanunderlyingstandardnormaldensityfunctionisdiscussedin Mudholkar&Hutson ( 2000 ).Theyprovideestimates(^,^,^)bythemethodofmomentsandbymaximumlikelihoodwhichhavedesirableasymptoticproperties.Theinclusionoftheasymmetryparameterandthesimultaneousestimationinthecaseofmaximumlikelihoodprovidemoreaccurateestimatesthantheclassicalprocedures. 23

PAGE 24

Parametricandnon-parametric,rankbased,testsforsymmetryarediscussedin Cassartetal. ( 2008 )whichareshowntobeoptimal(intheLeCamsense).AfunctionofobservationsfromtheFechnerclassofdistributionsisshowntobeuniformlyasymptoticallynormal(ULAN)withrespectto:=(,,)0,at(,,0),fromwhichateststatisticcanbeconstructed.RegularityconditionsareposedtoapplyLeCam'sthirdlemma( Hajeketal. 1999 )increatingteststatistics,totestforasymmetry,thatarelocallyasymptoticallyuniformlymostpowerful,locallyasymptoticallymostpowerfulandlocallyasymptoticallyoptimal(moststringent)dependingwhetherthedensityand/orlocationarespecied. 1.3.2AClassOfSkewDistributions Azzalini ( 1985 )and Liseo ( 1990 )developaclassofskew-normaldistributionsthatapproachtheGaussianfamilyastheincorporatedasymmetryparameterapproaches0.Furtherdevelopmentwaslaterdoneby Azzalini&Valle ( 1996 )and Branco&Dey ( 2001 )whoformallyextendtheclassbeyondthenormaldistributionframework.(Formoredetailsreferto Gentonetal. ( 2004 )).Inadditiontoasymmetry,thisclassalsoallowsforheaviertails. Azzaliniassignsthethefollowingdenitiontotheskew-normaldistribution. Denition2. Let2R.TherandomvariableZisaskew-normalrandomvariablewithparameterifithasadensityfunctionoftheform(z;)=2(z)(z),
PAGE 25

TheclassofdistributionscanbeexplicatedbyconditioningthesymmetricrandomvariableX,withmeanandvariance(>0)ontheeventthattheindependentsymmetricrandomvariableX0,withmean0andunitvariance,isgreaterthan0.Theskew-symmetricrandomvariableYisdenedasY=[XjX0>0]andisdenotedasYSS(,,),where2[)]TJ /F5 11.955 Tf 9.3 0 Td[(1,1]istheasymmetryparameter.TheprobabilitydensityfunctionhastheformfY(y)=2 p Z(y)]TJ /F13 7.97 Tf 6.59 0 Td[()0g(2)r2+(y)]TJ /F4 11.955 Tf 11.95 0 Td[()2 dr. whereg(2)isasphericaltwodimensionaldensityand=)]TJ /F7 5.978 Tf 5.75 0 Td[(1 p 1)]TJ /F13 7.97 Tf 6.58 0 Td[(2)]TJ /F7 5.978 Tf 5.75 0 Td[(1.Thesamedistributionclasscanbeobtainedbyatransformationmethod,wherebythetherandomvariableYisdenedasalinearcombinationofXandX0.Y=jX0j+p 1)]TJ /F4 11.955 Tf 11.96 0 Td[(2X. Thisclassofdistributionsdoesposecertainrestrictions. Branco&Dey ( 2001 )notethattheclassofgeneratorfunctionsissmallersincetheprobabilitydensityisobtainedbymarginalizingatwo-dimensionaldensity.IncomparisontotheFechnerclasswherethemodeofthe(unimodal)distributionsremainsxed,fortheskew-normalclassthemodechanges(seeFigure 1-3 )aschanges. 1.3.3ScalingTheTails Asimpleandstraightforwardmethodofcreatinganasymmetricdensityfromasymmetriconeistosimplyscaleeachside(fromthecenter)differently.Letfdenoteasymmetriccontinuousdensityfunctionwithmean0andunitvariance.Wecanobtainanasymmetricdensityfunctiongbyintroducingtheasymmetryparameterinthefollowingway,g(x)=ef(xe)Ifx0g+e)]TJ /F13 7.97 Tf 6.59 0 Td[(f(xe)]TJ /F13 7.97 Tf 6.59 0 Td[()Ifx<0g. Thismethodthoughcreatesajumpdiscontinuityatthecenter,asshowninFigure 1-4 ,sincelimx!0)]TJ /F3 11.955 Tf 8.74 -.29 Td[(g(x)6=limx!0+g(x)for6=0. 25

PAGE 26

1.3.4AsymmetricDensitiesBasedOnEdgeworthApproximations Cassartetal. ( 2010 )withasimilarapproachtotheir2008paperconstructparametricandsignedrankoptimaltestsfortestingtheexistenceofasymmetryusingrstorderEdgeworthapproximationsofsymmetricdensitiesindexedbyanasymmetryparameter.Arstorder(Gaussian)Edgeworthapproximationisoftheform(x)]TJ /F4 11.955 Tf 11.95 0 Td[()+p n(x)]TJ /F4 11.955 Tf 11.95 0 Td[()(x)]TJ /F4 11.955 Tf 11.95 0 Td[()((x)]TJ /F4 11.955 Tf 11.96 0 Td[()2)]TJ /F4 11.955 Tf 11.95 0 Td[(), whereisthestandardnormaldensityfunction,isthelocationparameter,istheGaussiankurtosiscoefcientandisthemeasureofasymmetry.Forthenon-Gaussiangeneralization,considertheclassofdensitiesfsatisfying 1.f2F(asdenedin( 1 )) 2.Thereexists_fsuchthat,8t2R,f(t)=Rt_f(z)dz>0; 3.z7!f(z):=)]TJ /F5 11.955 Tf 10.8 2.66 Td[(_f(z)=f(z)ismonotoneincreasing; 4.I(f):=R12f(z)f(z)dz,J(f):=R1z22f(z)f(z)dz,K(f):=R1z42f(z)f(z)dzarenite; 5.Thereexists>0suchthatR1af(z)dz=O(jaj)]TJ /F13 7.97 Tf 6.59 0 Td[()andf(z)=o(jzj=2)]TJ /F9 7.97 Tf 6.59 0 Td[(2)asz!1. Theasymmetricdensitywithlocationparameter2R,scaleparameter2R+0andasymmetryparameter2Risthendenedtobeh(x)=1 fx)]TJ /F4 11.955 Tf 11.95 0 Td[( )]TJ /F4 11.955 Tf 11.95 0 Td[(1 _fx)]TJ /F4 11.955 Tf 11.95 0 Td[( x)]TJ /F4 11.955 Tf 11.96 0 Td[( 2)]TJ /F4 11.955 Tf 11.95 0 Td[((f)!Ifjx)]TJ /F4 11.955 Tf 11.96 0 Td[(jjzjg+sgn()1 fx)]TJ /F4 11.955 Tf 11.95 0 Td[( (Ifx)]TJ /F4 11.955 Tf 11.96 0 Td[(>sgn()]TJ /F4 11.955 Tf 9.3 0 Td[()zg)]TJ /F3 11.955 Tf 20.59 0 Td[(Ifx)]TJ /F4 11.955 Tf 11.96 0 Td[(
PAGE 27

Innextchapterwepresentadistortionmodelthattakesasymmetricprobabilitydensityfunctionandskewsitinnitesimallyinaspecicdirectiontherebydistortingeveryrealizationofthedistribution.Themodelweuseisbasedupontheworkof Fechner ( 1897 )andisusedforitsintrinsicsimplicity.Robustnesstodistortionofasymmetricsignalisdeterminedastherateofchangeofthefunctionalofanafneequivariantlocationestimator,similartothemethodologyoftheinuencefunction.Themean,medianandHodges-Lehmanestimatorsarecompared. Figure1-1. GaussianFechnerclassfora()=andb()=)]TJ /F9 7.97 Tf 6.58 0 Td[(1 27

PAGE 28

Figure1-2. GaussianFechnerclassfora()=1+andb()=1)]TJ /F4 11.955 Tf 11.96 0 Td[( Figure1-3. Skew-normalclass 28

PAGE 29

Figure1-4. Separatelyscaledtailswithunderlyingstandardnormaldistribution Figure1-5. GaussianEdgeworthfamily 29

PAGE 30

CHAPTER2DISTORTIONSENSITIVITYOFESTIMATORSOFLOCATION Inthischapterwewishtoestablishcertainrobustnesspropertiesofseveralclassesoflocationestimatorstotheideaofasymmetry,inthesensethateveryobservationisacquiredfromaspecicdistributionfunctionthathasbeenalteredfromthesymmetricideal. 2.1Afne-Equivariance AfunctionalT(X)ofap-dimensionalrandomvariableXissaidtobeafne-equivariantifT(AX+v)=AT(X)+v, whereAisanarbitrarynonsingularppmatrixandvanarbitraryp1vector.Wheneverthedataisrotatedorthemeasurementscaleofoneormorecomponentsislinearlytransformed,thispropertyallowstheestimatortomovecorrespondinglyinthesamedirection.Inaddition,itensurestheconsistentperformanceofthelocationestimatoroverchangesinthevariance-covariancestructureofthepopulation. LetXdenotearandomvariablethatissymmetricallydistributedaroundthep1locationparameterinthesensethatX)]TJ /F19 11.955 Tf 12.02 0 Td[(d=)]TJ /F10 11.955 Tf 12.03 0 Td[(X.Anafne-equivariantfunctionalTappliedtoXwillalwaysbethepointofsymmetry,,becauseT(X)]TJ /F19 11.955 Tf 11.96 0 Td[()=T()]TJ /F10 11.955 Tf 11.95 0 Td[(X),T(X))]TJ /F19 11.955 Tf 11.96 0 Td[(=)]TJ /F3 11.955 Tf 11.95 0 Td[(T(X),2T(X)=2,T(X)=. Thisshowsthatwhentheunderlyingdistributionissymmetric,allestimatorsbasedonafneequivariantfunctionalswillestimatethesameparameter,namelythepointofsymmetry. 30

PAGE 31

2.2DistortionAndContaminationModels InSection 2.1 ,weshowedthatallafne-equivariantestimatorsareaimingatthesametargetforsymmetricdistributions.Inapplicationsthedataarerarelyperfectlysymmetricthough.Consequently,thequestionofwhichlocationestimatoristhrownfurthestofftargetbytheimperfectiontothetotallysymmetricmodelisofimportanceandthatisthepurposeofthischapter. AswiththeIFwestudythederivativeofthefunctional(rateofchange)whenaninnitesimalamountofasymmetryisintroducedintoasymmetricdistributionF.InsteadofacontaminationmodelwewillfocusonthedistortionmodelwherebythesymmetrictargetdistributionFisgeneralizedtoabroaderclassofasymmetricdistributionsthroughtheintroductionofanasymmetryparameter.Underthismodelaconsistentforceisperceivedaspushingeachobservationinaxeddirection.Unlikethecontaminationmodelwheresomecontaminantsareaddedfromanotherdistributionfunction(xintheIFframework),hereeveryobservationisfromadistorteddistribution.Thedistortionmodelallowsforadirectionalderivativetobecalculatedfordatathatareconsistentlyandsystematicallydistortedawayfromthesymmetricframework. DuetoitsintrinsicsimplicitytheFechnerclassofdistributionsisselectedasthemodelframework,wherebyanarbitrarysymmetricdistributionisembeddedwithinanasymmetricfamily.SpecicallyletF()denotethedistributionfunctionforthedistributionwithdensity, f(xj0,,)=1 fx (1+sgn(x)),2()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1),x2R(2) foraxedbutarbitrarysymmetricdensityf2Fwhere F=ff()]TJ /F3 11.955 Tf 9.3 0 Td[(x)=f(x),f(x)08x2R,fisaboslutelyconituousandZ1f(x)dx=1(2) Thisclasstreatsleftandrightasymmetrysimilarlythroughtheasymmetryparameter2()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)asthefunctionsa()andb()inequation( 1 )aretakentobesymmetric, 31

PAGE 32

unlikethesubclassofequationsillustratedinequation( 1 ). Arellano-Valleetal. ( 2005 )illustratethatthelimitbehaviorofthedensityfunctionisvitallydifferent.Forthesubclassinequation( 1 )lim!1f(xj)=0andlim!0+f(xj)=0, whilefortheFechnersubclasslim!1)]TJ /F3 11.955 Tf 8.24 5.81 Td[(f(xj)=fx 2Ifx0gandlim!)]TJ /F9 7.97 Tf 15.06 0 Td[(1+f(xj)=fx 2Ifx<0g. ComparedtootherdistortionmodelsinSection 1.3 themodedoesnotshiftforsymmetricunimodaldistributionsastheasymmetryparameterdeviatesfrom0.ButtheFechnerfamilyremainscontinuouswhentheunderlyingdensityf()iscontinuous. SimulationusingthisclassisrelativelyeasyduetothestochasticrepresentationshowninthepropositioninSection 1.3.1.1 .TogenerateanobservationXfromtheFechnerclasssimplygenerateYF,andUwithP(U=1+)=(1+)=2=1)]TJ /F3 11.955 Tf 11.96 0 Td[(P(U=)]TJ /F5 11.955 Tf 11.95 0 Td[(1)independentofY,andsetX=UjYj. ComparedtothedensitiesthatarebasedonEdgeworthapproximation,theFechnerclassisamongthosewiththeleaststringentassumptions.Nowwehaveamodeltorepresentacollectionofdatapointswherebyeachandeveryoneisderivedfromanasymmetricdistribution.Hence,wecanproceedtodescriberobustnessproperties(withintheasymmetrysetup)ofestimatorsoflocation. 2.3ComparisonOfEstimatorsOfLocation Wewishtocompareafne-equivariantestimatorsoflocationwithrespecttotheirrobustnesstodistortion,i.e.asymmetryasdescribedinthedistortionmodel.ProceedingwithamethodologysimilartotheIF,andassumingnecessaryregularityconditions,wetaketherstpartialderivativewithrespecttotheasymmetryparameterofthefunctionalrepresentationofthelocationestimatorforaFechnerasymmetricdensity. 32

PAGE 33

Denition3. DenoteT=T(F).Theterm,DS=@T @=0 isdenedasthedistortionsensitivityandisameasureoftheratebywhichtheestima-tormovesawayfromthecenter,asshiftsawayfromthesymmetriccase,i.e.=0.Thisquantity,DSwilldependonTandthesymmetricdistributionF. Withoutlossofgeneralitywewillassumethatthecenter(parameteroflocation)forthesymmetricdensitiesis0. 2.3.1Mean,MedianandHodges-Lehmann Themeanisthemostcommonlyusedmeasureoflocationandit'sfunctionalrepresentationisT=ZxdF(x), whereFbelongstotheFechnerclassofdistributions.ThisrepresentationcanbeexpandedtoT=Z0x1 fx (1)]TJ /F4 11.955 Tf 11.96 0 Td[()dx+Z10x1 fx (1+)dx=(1)]TJ /F4 11.955 Tf 11.96 0 Td[()2Z0yf(y)dy+(1+)2Z10yf(y)dy. Toobtainitsdistortionsensitivitywetaketherstorderpartialderivativeofthefunctionalwithrespectandevaluateitat=0.@T @=)]TJ /F5 11.955 Tf 9.3 0 Td[(2(1)]TJ /F4 11.955 Tf 11.96 0 Td[()Z0yf(y)dy+2(1+)Z10yf(y)dy andhence @T @=0=)]TJ /F5 11.955 Tf 9.3 0 Td[(2Z0yf(y)dy+2Z10yf(y)dy=4Z10ydF(y). (2) 33

PAGE 34

Equation( 2 )canbere-expressedintermsofadistributionfunctionF+:R+![0,1]thatisuniquelydeterminedfromF,thatisF+=2F)]TJ /F5 11.955 Tf 12.39 0 Td[(1thedistributionfunctionofjXjwithdensityfunction f+(x)=8><>:2f(x)ifx0,0ifx<0.(2) Thisarticulatesequation 2 intheformofanexpectation,@T @=0=2Z10xdF+(x)=2EF+(X), asF+isuniquelydenedbyFandviseversa.ThisgivesadistortionsensitivityofDSmean=2EF+(X). Themedianisanothercommonlyusedestimatoroflocationthatisgenerallyspeakingmorerobustthanthemean.ThemedianTofFcanbeimplicitlyrepresentedfunctionallyasthesolutionof1 2=F(T). Withoutlossofgenerality,assumethat>0andhencethatT>0,sothefunctionalrepresentationcanbere-expressedas1 2=1 2(1+)+ZT01 fx (1+)dx=1 2(1+)+(1+)ZT (1+)0f(y)dy. Thus,takingthederivativewithrespecttoandapplyingtheLeibnizrulefordifferentiationundertheintegralsignforcontinuousf,wehave0=)]TJ /F5 11.955 Tf 10.49 8.08 Td[(1 2+ZT (1+)0f(y)dy+(1+)fT (1+)@T @(1+))]TJ /F3 11.955 Tf 11.96 0 Td[(T (1+)22. Evaluatingtheexpressionat=0yields@T @=0= 2f(0)= f+(0) 34

PAGE 35

producing DSmedian= f+(0).(2) Anotherlocationestimatorthatwewilllookinto,encompassesboththeideasofthemeanandthemedian.Forcontinuousdistributions,thefunctionalderivativefortheHodges-Lehmannestimatorinequation( 1 )is@T @=0=1=2+Z10xf+(x)dF+(x) Z10f+(x)dF+(x)=1=2+EF+(Xf+(X)) EF+(f+(X)). DetailsofthederivationarefoundinAppendix A .ItfollowsthatDSH-L=1=2+Z10xf+(x)dF+(x) Z10f+(x)dF+(x)=1=2+EF+(Xf+(X)) EF+(f+(X)). Forthelocationestimatorsconsidered,thedistortionsensitivityisamultipleofthescaleparameter.Takingtheratiosofthesederivativesprovidesamethodforcomparingthedistortionsensitivityoftheseafne-equivariantestimatorsoflocation,eliminatingthenuisanceparameter.Weseethatthedistortionsensitivityratioofthemeantothemedianis DSRmeanmedian=2f+(0)EF+(X),(2) forthemediantotheHodges-LehmannitisDSRmeanH-L=2EF+(X)EF+(f+(X)) 1=2+EF+(Xf+(X)), andoftheHodges-LehmannestimatortothemedianisDSRH-Lmedian=f+(0)1=2+EF+(Xf+(X)) EF+(f+(X)). LikeanAsymptoticRelativeEfciency(ARE),theseratiosarefreeofanylocationandscaleparametersintheunderlyingdistributionF.(MoredetailsaboutAREcanbefoundin Randles&Wolfe ( 1979 )).However,thedistortionsensitivityratioslackthe 35

PAGE 36

interpretationofpowerasARE'sdoandconsequentlyanyattempttostandardizethemhasnopracticalinterpretationastheyaremeasuringtherateofchangeofthemeasuretolocationandnotatest. Table 2-1 providesnumericalvaluesfortheseratiosforselectedsymmetricdistributionsF.Wenotethatforunimodaldistributionssuchasthenormal,Laplace,Student'standthetriangulardistributionthatthemedianexhibitsthehighestlevelofrobustnessrelativetothemeanandHodges-Lehmann,whilethemeanistheleastrobust.ForthecustomU-shapeddistributionwithdensityproportionalto(x2+1=2)forx2[)]TJ /F5 11.955 Tf 9.29 0 Td[(1,1]theHodges-Lehmannismostrobustcomparedtothemeanandmedian,whilethenowthemeanismorerobustthanthemedian.TheshiftedBeta(1 2,1 2)alsoillustratesthis.Fortheuniformdistributiontheestimatorsperformequivalently. WhenthetransformationusedintheFechnerdistortionmodelisappliedtoauniform()]TJ /F3 11.955 Tf 9.3 0 Td[(a,a)distribution,itseffectistoshiftthelocationofthedistribution.Theamountoftheshiftdependsonthemagnitudeoftheasymmetryparameter,whilethedirectioniscontingentonitssign.Thereisashifttotherightfor>0andtotheleftfor<0.LetUbeauniformrandomvariable.WecanperceivetheFechnertransformationasU+,whererepresentstheshift.Thenforanafne-equivariantestimatorT,wehavethatT(U+)=T(U)+.Therefore,theestimatordependsononlythroughandnotT.SotherstorderderivativewithrespecttooftheestimatorisidenticalforallafneequivariantestimatorsT. Proposition2.1. Letf()beasymmetriccontinuousunimodalprobabilitydensityfunctionsuchthatsupx2Rf(x)=f(0)andx=0istheuniquepointthatascertainsthesupremum.ThemedianisthemostrobustcomparedtothemeanortheHodges-Lehmannwithrespecttodistortionsensitivity,i.e.DSRmeanmedian>1andDSRH-Lmedian>1. 36

PAGE 37

Proof. Theserelationshipsarefreeofanyscaleparameter,soforconveniencewecanre-scalethefunctionf+,denotedgs,suchthatgs(0)=1,i.e. gs(x)=1 f+(0)f+x f+(0).(2) TheratioDSRT1T2comparesthedistortionsensitivityoftheestimatorT1toT2.IfDSRT1T2islessthat1itimpliesthatT1islesssensitivethanT2todistortioninthesenseoftheFechnermodelforthespecieddistributionfunctionF,andviceversaiftheratioisgreaterthan1.IfDSRT1T2=1itmeansthatthetwoestimatorsperformequivalently(inthecontextofthedesignatedFechnermodel). Applyingthere-scaleddistributioninequation( 2 )wecanre-expresstheratioofthedistortionsensitivityofthemeantothemedianasDSRmeanmedian=2Z10xdGs(x)=2EGs(X). Since,theidentityfunctionisanincreasingfunction,theexpectationobtainsitsminimumvaluewhenasmuchmassaspossible,ofthedensityofgs,isplacedascloseto0aspossible.Undertherestrictionthatsupx2Rf(x)=f(0)thisimpliesthattheminimumisobtainedwhengsisthedensityfunctionofaUniform(0,1)randomvariable,andexpectationwouldbeidentically1=2.WhentheuniquenessconditionisalsoimposedthenEGs(X)>1=2andhenceDSRmeanmedian>1. Similarly,wecanre-expressDSRH-LmedianasDSRH-Lmedian=1=2+EGs(Xf+(X)) EGs(f+(X)). WeneedtoshowthatDSRH-Lmedian>1orequivalentlythatR10(x)]TJ /F5 11.955 Tf 12.51 0 Td[(1)g2s(x)dx>)]TJ /F5 11.955 Tf 9.29 0 Td[(1=2.Sincethefunctionm(t)=t)]TJ /F5 11.955 Tf 12.56 0 Td[(1isanincreasingfunction,theintegrandisminimizedwhenthefunctiong2s,orequivalentlygs,placesmostofitsmassnear0.Asabove,this 37

PAGE 38

minimumoccurswhengsisUniform(0,1).Hence,Z10(x)]TJ /F5 11.955 Tf 11.96 0 Td[(1)g2s(x)dx=Z10(x)]TJ /F5 11.955 Tf 11.96 0 Td[(1)dx=)]TJ /F5 11.955 Tf 9.3 0 Td[(1=2, andisgreaterthat)]TJ /F5 11.955 Tf 9.3 0 Td[(1=2forallotherdistributions(undertheassumptionsoftheproposition). InProposition 2.1 theratioconcerningthemeanandmedianalsoholdsundertheweakerassumptionofstrictsecond-orderstochasticdominationforacontinuousf( Davidson&Duclos 2000 ).Thisconditionallowsfortheexistenceofasetofpoints,besides0,forwhichthefunctionfevaluatedatthosepointsisgreaterorequaltof(0),i.e.fx2R)]TJ /F5 11.955 Tf 12.18 0 Td[(0jf(x)f(0)g,whilestillrestrictingonhowfarabovethef(0)thresholdthefunctionfisallowedtogo. Denition4. SupposetherandomvariablesXandYhavesupporton[l,u].ThenXstrictlysecond-orderstochasticallydominatesYifZalP(X>t)dt>ZalP(Y>t)dt8a>l. Proposition2.2. LetXGs.IfXstrictlydominatesaUniform(0,1)randomvariableinsecond-orderstochasticsense,thenDSRmeanmedian>1. Proof. LetYUwhereUistheUniform(0,1)distributionfunction.TheexpectationofXcanbeexpressedasEGs(X)=Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Gs(t))dt, which,byDenition 4 isstrictlygreaterthanEU[Y]=Z10[1)]TJ /F3 11.955 Tf 11.96 0 Td[(U(t)]Ift2(0,1)gdt=Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(t)dt=1 2. Therefore,wehavethatDSRmeanmedian>1. Tobetterillustratetheconditionofsecondorderstochasticdominance,weintroducethefollowingexample. 38

PAGE 39

Example2.1. Letfbethedensityfunctionofamixtureofthreenormaldistributions,f(x)=1 41 0.1x+0.2 0.1+1 21 2x 2+1 41 0.1x)]TJ /F5 11.955 Tf 11.96 0 Td[(0.2 0.1. Thendenegstobethehalfre-scaleddensityoff,derivedbyequations( 2 )and( 2 ).InFigure 2-1 ,thefunctiongsisplottedwhereitcanbeseenthatthereexistsasetofpointsdistinctlydifferentfrom0wheregs(x)>f+(0). Furthermore,arandomvariableXwithdensityfunctiongsdoesnotrstorderstochasticallydominateaUniform(0,1)randomvariableU,seeingasP(X>t)P(U>t)forallt(showninFigure 2-2 ).ItdoeshoweversecondorderstochasticallydominateUwithanexpectedvaluethatisgreaterthan1=2.(ThedetailsarefoundinAppendix B ). FortheratioofdistortionsensitivityofthemeantotheHodges-Lehmannestimatorwerequireanadditionalcondition,tothoseofProposition 2.1 ,thatensuresthatDSRmeanH-L>1. Proposition2.3. LetXGsandc=(2R10xgs(x)dx))]TJ /F9 7.97 Tf 6.59 0 Td[(1.IfXstrictlydominatesaUniform(0,1)randomvariableinsecond-orderstochasticsense,andthereexistsauniqueinterval[tL,tU]withtL>0,tU<1(withoutexcludingthepossibilitythattL=tU),forwhichgs(t)=c,8t2[tL,tU], thenDSRmeanH-L>1. Proof. Applyingthere-scalingmethodofequation( 2 ),theexpressionDSRmeanH-L>1isequivalentto2Z10g2s(x)dxZ10xgs(x)dx)]TJ /F15 11.955 Tf 11.96 16.27 Td[(Z10xg2s(x)dx>1 2. Denec=(2R10xgs(x)dx))]TJ /F9 7.97 Tf 6.59 0 Td[(1.Recallthattheexpressionisfreeofanyscalingparameterssowecanre-scalethedensityfunctiontogc(x)=(1=c)gs(x=c)suchthatR10xgc(x)dx=1=2.NowallthatisnecessaryistoshowthatR10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g2c(x)dx>1=2. 39

PAGE 40

First,wenotethatgc(0)>1.BysecondorderstochasticdominancewehavethatR10xgs(x)dx>1=2,whichconnotesthatc<1bydenitionofc.Therefore,gc(0)=1 cgs(0)=1 c>1. Next,weobservethatgc(x)>0forsomex>1andforallx>1,that0gc(x)<1.TheseconditionsholdbytheassumptionsandtheconstructofR10xgc(x)dx=1=2.Ifgc(x)=0forallx>1,itwouldimplythatR10xgc(x)dx=EGc(X)<1=2sincegc(0)>1and[tL,tU]istheuniquesetsuchthatgc(t)=1,8t2[tL,tU].Thisimpliesacontradictionandconsequently,gc(x)>0forsomex>1.Inaddition,gc(z)1forsomespecicz>1violatestheassumptionoftheset[tL,tU].Hence0f+(t)<1forallt>1. TakeAR10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g2c(x)dx.ExpandingAleadsto A=Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g2c(x)dx+Z11(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)g2c(x)dx>Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)g2c(x)dx+Z11(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)gc(x)dx=Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)fg2c(x))]TJ /F3 11.955 Tf 11.95 0 Td[(gc(x)gdx+Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)gc(x)dxB+1 2. AllthatisnecessarynowistoshowthatB0forthesetofdensityfunctionsthatsatisfytheassumptions.Letk(x)=gc(x)+(1)]TJ /F4 11.955 Tf 12.15 0 Td[()u(x)for2(0,1),whereuistheUniform(0,1)densityfunction.WenextshowthatbyexpressingBexplicitlyintermsofthedensityk,Battainsitsminimumvalue,0,at=0,for2[0,1],andisgreaterthan0forall2(0,1].Firstweneedtoshowthatksatisesthesameconditionsasgc. 40

PAGE 41

Theexpectationofarandomvariablewithprobabilitydensityfunctionkis1=2since Z10xk(x)dx=Z10xgc(x)dx+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()Z10xu(x)dx=1 2+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()1 2=1 2. Inaddition,k(0)=gc(0)+(1)]TJ /F4 11.955 Tf 12.07 0 Td[()u(0)>+(1)]TJ /F4 11.955 Tf 12.08 0 Td[()=1,sincegc(0)>1and,usingthesetwoconstructsasbeforewehavethat0k(x)1forall2(0,1]andx>1. ReplacinggcbykinBgives, 2Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)fg2c(x))]TJ /F5 11.955 Tf 11.95 0 Td[(2gc(x)gdx+1 2+Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)gc(x)dx)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2,(2) whichistrivially0for=0andhasroots,for,0and1 2)]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)gc(x)dx Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)f1)]TJ /F3 11.955 Tf 11.96 0 Td[(gc(x)g2dx. Thesecondrootisnegativeandnotwithintherangeof.ThedenominatorispositiveandthenumeratorisnegativebecausebyconstructionR10(1)]TJ /F3 11.955 Tf 12.62 0 Td[(x)gc(x)dx=1=2soR10(1)]TJ /F3 11.955 Tf 12.06 0 Td[(x)gc(x)dx>1=2.Therstandsecondorderpartialderivativesofequation( 5 )withrespecttoarerespectively,Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)f2g2c(x)+gc(x))]TJ /F5 11.955 Tf 11.95 0 Td[(4gc(x)gdx+)]TJ /F5 11.955 Tf 13.15 8.08 Td[(1 2, and2Z10(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)fgc(x))]TJ /F5 11.955 Tf 11.96 0 Td[(1g2dx. 41

PAGE 42

Notethatthesecondpartialderivativeisstrictlygreaterthan0andhence,equation( 5 )asafunctionofisconvexwithitsglobalminimaattainedat=1 2)]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z10(1)]TJ /F3 11.955 Tf 11.96 0 Td[(x)gc(x)dx Z10(1)]TJ /F3 11.955 Tf 11.95 0 Td[(x)f2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(gc(x))g2dx, whichisnegative.ThereforeB0. Furthermore,analogoustotheresultof Bassett ( 1991 ),undercertainconditionsthemedianistheuniquefunctionalthatattainstheminimumpossibledistortionsensitivity. Proposition2.4. Foraunimodaldensityfunctionf()thatyieldsgs()byequation( 2 ),if@T @=0isstochasticallyincreasingings(),thenDSRTmedian1 Proof. Thedistortionsensitivityratioisfreeofthescaleparameter,sowithoutlossofgeneralityassumethat=1.Byconstructionofgs(_)andequation( 2 )DSmedian=1 foralldistributions.FortheUniform(0,1)distributionwehaveshownthatthedistortionsensitivityforallafneequivariantestimatorsisidentically1.Byassumption,@T @=0isstochasticallyincreasingings(),andhenceforanydistributionthatisstochasticallygreaterthantheUniform(0,1)impliesthatDSRTmedian1. Wehaveseenthattheassumptionof@T @=0stochasticallyincreasingings()isquiteplausibleforthemeanandHodges-LehmannestimatorsthatareexpressedintermsofexpectationswithrespecttoGs 2.3.2M-Estimators EstimatorsoflocationthataredenedbyaminimizationproblemofthetypeXi(xi;Tn)=minTn! 42

PAGE 43

whereisanarbitrarydistancefunctionandTnisalocationstatistic,arecalledM-estimators.Althoughthederivativeofmaynotalwaysexist,ifweassumethat (x,)=(@=@)(x,)existsanimplicitequationfortheM-estimatorTnisXi (xi;Tn)=0. TheM-estimatorT=T(F)isimplicitlydenedinfunctionalformasthequantitywhichsatises:Z (x;T)dF(x)=0, whichfortheFechnerdistortionmodel(equation( 2 ))is,Z (x;T)dF(x)=0. Assumingthat isdifferentiablethenthedistortionsensitivityforTbelongingtotheclassofM-estimatorsisdenotedbyDSRM=Z[ (x)+x 0(x)]dG(x) Z 0(x)dG(x). DetailsofthederivationcanbefoundinAppendix C 2.3.3L-Estimators EstimatesthatarelinearcombinationsoforderstatisticsoftheformTn=PiaiX(i)aredenedtobeL-estimates.Accordingto Hampeletal. ( 1986 ),anaturalsequenceoflocationestimatorsisobtainedbylettingai=Z[i)]TJ /F7 5.978 Tf 5.76 0 Td[(1 n,i n]h()d Z[0,1]h()d, 43

PAGE 44

whereh:[0,1]7!RandR[0,1]h()d6=0.Forsymmetricdistributionsweassumethathissymmetricabout=1=2,andhencethefunctionalisT(F)=Zxh(F(x))dF(x) Zh(F(x))dF(x). ThedistortionsensitivityofanL-estimator,assumingthatisahdifferentiablefunctionisDSRL=2Z10x[2h(F(x)))]TJ /F3 11.955 Tf 11.96 0 Td[(h0(F(x))(1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(x))]dF(x) Z1h(F(x))dF(x). DetailsofthederivationcanbefoundinAppendix D 2.3.4R-Estimators TodeneanR-estimatorrstconsideratwo-samplelocationranktestbasedonthesamplesX1,...,XmandY1,...,YnwithdistributionfunctionsF(x)andF(x+)respectively,whereisthelocationshift.ArankteststatisticforthehypothesesHo:=0versusHa:>0isSN=1 mmXi=1aN(Ri), whereRiistherankofXiinthepooledsampleofsizeN=n+m.TheweightsaN(i)aregeneratedbyafunction:[0,1]7!RbymeansofaN(i)=NZi Ni)]TJ /F7 5.978 Tf 5.76 0 Td[(1 N(u)du, whereisskewsymmetric,i.e.(1)]TJ /F3 11.955 Tf 10.39 0 Td[(u)=)]TJ /F4 11.955 Tf 9.3 0 Td[((u),suchthatR(u)du=0.ThefunctionalrepresentationforTisthroughtheimplicitfunctionZ1 2F(x)+1 2(1)]TJ /F3 11.955 Tf 11.95 0 Td[(F(2T)]TJ /F3 11.955 Tf 11.95 0 Td[(x))dF(x)=0. 44

PAGE 45

Duetothesymmetrywemayonlyconsiderthecasewhere>0andassumingisdifferentiable,thedistortionsensitivitybecomesDSRR=Z102(F(x))+0(F(x))F(x))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2+20(F(x))xf(x)dF(x) Z100(F(x))f(x)dF(x). DetailsofthederivationcanbefoundinAppendix E 2.3.5P-Estimators GeneralizationsofPitmanestimatorsforlocationareTn=ZYi(xi)]TJ /F4 11.955 Tf 11.96 0 Td[()d ZYi(xi)]TJ /F4 11.955 Tf 11.95 0 Td[()d, wheredoesnotnecessarilyhavetocoincidewiththeprobabilitydensityfunctionf.ThefunctionalforTisgivenbyZ0(x)]TJ /F3 11.955 Tf 11.95 0 Td[(T) (x)]TJ /F3 11.955 Tf 11.95 0 Td[(T)dF(x)=0. If=fandfistwicedifferentiable,thedistortionsensitivityisDSRP=Z10xf00(x)dx Z10f00(x))]TJ /F5 11.955 Tf 13.15 8.09 Td[((f0(x))2 f(x)dx. DetailsofthederivationcanbefoundinAppendix F 2.4MultivariateEstimatorsOfLocation Firstly,weassumethatthep-dimensionaldistributionisellipticallysymmetricaround0andthelocationestimatorsareafne-equivariantasdescribedinSection 2.1 .NotateA0asthetransposeofthematrixA.SupposeXhasaellipticallysymmetricdistributionaround0withsymmetricnon-negativecovariancematrix.Foraonedimensionalreal 45

PAGE 46

valuedfunctiong()thatisindependentofp,thedensitytakestheform:f(x)=cp )]TJ /F9 7.97 Tf 6.59 0 Td[(1=2g(x0)]TJ /F9 7.97 Tf 6.59 0 Td[(1x), wherecpisascalarproportionalityconstant.ThedistanceofthemultivariatelocationfunctionalT(X)fromtheorigin0shouldbemeasuredvia [T(X)]0)]TJ /F9 7.97 Tf 6.59 0 Td[(1[T(X)]1=2.(2) AnyellipticalsymmetricrandomvariableXcanbeexpressedintermsofasphericallysymmetricrandomvariableYaround0withcovariancetheidentitymatrixI.ViavirtueoftheCholeskydecomposition,=PP0,wherePisalowertriangularppmatrixandhencewecanexpressX=P)]TJ /F9 7.97 Tf 6.58 0 Td[(1Y.Bytheafneequivariantproperty,expression( 2 )canbeviewedasf[T(Y)]0[T(Y)]g1=2. whichisthespatialsphericaldistanceofTfromtheorigincomputedusingthesphericaldistributionofY.WithoutlossofgeneralityassumeXhasasphericallysymmetricdistributionaround0withdensity:f(x)=cpg(x0x). ArandomvectorXissaidtohaveap-sphericallysymmetricdistributionifXcanbewrittenasaproductoftwoindependentrandomvariablesX=UR,whereUisuniformlydistributedonthep-sphereandR=kXkisanon-negativeunivariaterandomvariableoftheformh(r)=8><>:kprp)]TJ /F9 7.97 Tf 6.58 0 Td[(1g(r2)ifr>00otherwise. 46

PAGE 47

2.4.1MultivariateFechnerModel Letd0denoteanarbitraryp-dimensionalunitvector.Thenforanyunitvectorudener0(,u)=u0d0+)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(2(u0d0)2+1)]TJ /F4 11.955 Tf 11.95 0 Td[(21=2, whichisinterpretedasthedistancefromtheorigintoapointontheunitsphereinthedirectionofuwhentheunitsphereisshiftedfromtheoriginunitsinthedirectiond0(seeFigure 2-3 ).When=0,r0(,u)=1foreveryu. Assumingthatg()isabsolutelycontinuous,wecanrepresentarandomvectorfromtheFechnermodelwith2()]TJ /F5 11.955 Tf 9.29 0 Td[(1,1)ashavingthesamedistributionasUR,wherethedensityofUisproportionalto[r0(,u)]p,i.e.bp[r0(,u)]pwherebpisthenormalizingconstantandtheconditionaldensityofRgivenUis1 r0(,u)hr r0(,u). Therefore,thedensityofmultivariateFechnermodelhasthefollowingform f(u,r)=bpkp[r0(,u)]p1 r0(,u)hr r0(,u).(2) Thederivativewithrespecttoat=0foranafneequivariantT(F)takestheform@T(F) @=0=SC(T,p,g())d0 andwedenethedistortionsensitivitytobe DST=@T(F) @=0=SC(T,p,g())(2) In Cassartetal. ( 2008 )asimplermultivariateextensionisintroducedbutourproposedextensionoffersthesameresultasintheunivariatecasethatwhenfistheuniformdensityfunctionwithinaunitp-sphere,theaboveFechnerextensionshifts(translates)theunitp-sphereunitsinthedirectionoftheunitvectord0.Atthisuniformdistribution,everyafneequivariantTsatisestheconditionthatDST=1. 47

PAGE 48

2.4.2MeanandMedian ThemeanfunctionalusingtheFechnermultivariatemodelrepresentationofequation( 2 )isT=ZZbpkpurp0(,u)rp prp0(,u)gr2 2r20(,u)drdu=ZZbpkpurp+10(,u)spg(s2)dsdu=Zbpurp+10(,u)duZkpspg(s2)ds=Zbpurp+10(,u)duE(R). Asrp+10(0,u)=1,wenotethat@ @rp+10(,u)=0=(p+1)u0d0, andhence@T @=0=d0(p+1)Zbpuu0duE(R)=d0(p+1)E(uu0)E(R)=d0Ippp+1 pE(R). Thedistortionsensitivityforthemean,bydenitionofequation( 2 ),isthereforeDSmean=p+1 pE(R), whereisascaleparameterof)]TJ /F9 7.97 Tf 6.59 0 Td[(1f(x=)andR=kXk.ThusthemeanwillhaveaninnitedistortionsensitivityifE(R)canbeinnite(assuming<1). Thereisaplethoraofpossiblemultivariateextensionsintheliteratureforthemedianandweuseanafneequivariantmultivariatemedianproposedin Hettmansperger&Randles ( 2002 ).Herethelocationp-vectorT(X)ischosentosatisfy EA(X)]TJ /F3 11.955 Tf 11.96 0 Td[(T(X)) kA(X)]TJ /F3 11.955 Tf 11.96 0 Td[(T(X))k=0,(2) 48

PAGE 49

whereAisappuppertriangularmatrixwithaoneinitsupperleft-handentrysatisfyingEA(X)]TJ /F3 11.955 Tf 11.95 0 Td[(T(X)) kA(X)]TJ /F3 11.955 Tf 11.95 0 Td[(T(X))kA(X)]TJ /F3 11.955 Tf 11.96 0 Td[(T(X)) kA(X)]TJ /F3 11.955 Tf 11.96 0 Td[(T(X))k0=c)]TJ /F9 7.97 Tf 6.58 0 Td[(1Ipxp. Re-expressingequation( 2 )withs=r(r0(,u))]TJ /F9 7.97 Tf 6.59 0 Td[(1undertheFechnermodelwehaveZZA(r0(,u)su)]TJ /F3 11.955 Tf 11.96 0 Td[(T) kA(r0(,u)su)]TJ /F3 11.955 Tf 11.96 0 Td[(T)kbpkprp0(,u)sp)]TJ /F9 7.97 Tf 6.58 0 Td[(1g(s2)dsdu=0. Bynotingthat@ @A(r0(,u)su)]TJ /F3 11.955 Tf 11.95 0 Td[(T) kA(r0(,u)su)]TJ /F3 11.955 Tf 11.95 0 Td[(T)k=0=Au kAuku0d0)]TJ /F3 11.955 Tf 13.15 8.78 Td[(A@T @j=0 skAuk)]TJ /F3 11.955 Tf 19.13 8.08 Td[(Au kAuku0d0+u0A0A@T @j=0Au skAuk3 and@ @bpkprp0(,u)sp)]TJ /F9 7.97 Tf 6.59 0 Td[(1g(s2)=0=bpkppsp)]TJ /F9 7.97 Tf 6.59 0 Td[(1g(s2), weobtain0=ZZAu kAuku0d0bpkppsp)]TJ /F9 7.97 Tf 6.59 0 Td[(1g(s2)dsdu+ZZAu kAuku0d0bpkpsp)]TJ /F9 7.97 Tf 6.59 0 Td[(1g(s2)dsdu)]TJ /F4 11.955 Tf 9.3 0 Td[()]TJ /F9 7.97 Tf 6.58 0 Td[(1ZZA@T @j=0 kAuku0d0bpkpsp)]TJ /F9 7.97 Tf 6.59 0 Td[(2g(s2)dsdu)]TJ /F15 11.955 Tf 11.95 16.27 Td[(ZZAu kAuku0d0bpkpsp)]TJ /F9 7.97 Tf 6.59 0 Td[(1g(s2)dsdu+)]TJ /F9 7.97 Tf 6.58 0 Td[(1ZZu0A0A@T @j=0Au kAuk3u0d0bpkpsp)]TJ /F9 7.97 Tf 6.58 0 Td[(2g(s2)dsdu, orequivalently0=pZAu kAuku0d0bpdu)]TJ /F5 11.955 Tf 33.14 8.09 Td[(1 E(R)]TJ /F9 7.97 Tf 6.59 0 Td[(1)ZA@T @j=0 kAuku0d0bpdu+1 E(R)]TJ /F9 7.97 Tf 6.59 0 Td[(1)Zu0A0A@T @j=0Au kAuk3u0d0bpdu. ThedistortionsensitivitytothenormbasedmedianisthenDSmedian=p (p)]TJ /F5 11.955 Tf 11.96 0 Td[(1)E(R)]TJ /F9 7.97 Tf 6.59 0 Td[(1). SinceE(R)]TJ /F9 7.97 Tf 6.58 0 Td[(1)>0,estimatorsbasedonthespatialnormbasedmedianhaveanitedistortionsensitivitywhenp2. 49

PAGE 50

Tocomparethetwoestimatorsfurtherweformthedistortionsensitivityratioofthemeantothemedian,DSRmeanmedian=p2)]TJ /F5 11.955 Tf 11.95 0 Td[(1 p2E(R)E(R)]TJ /F9 7.97 Tf 6.58 0 Td[(1). Thenextstepistocomparethedistortionsensitivityratioofthemeantothemedianforaclassofdistributions.Weconsiderthemultivariatepowerexponentialfamilyormultivariategeneralizednormalfamilyofthefollowingformf(x)=kpexp()]TJ /F5 11.955 Tf 9.3 0 Td[([x0x]=c). When=1thisisthep-variatestandardnormaldistribution,when>1thedistributionislighter-tailedthanthemultivariatenormalandwhen<1itisheavier-tailedthanthenormal.TheresultsareshowninTable 2-2 WealsoconsiderStudent'stFamilyinTable 2-3 Itisnotedthatwithoutanyconditionsong()otherthanbothE(R)andE(R)]TJ /F9 7.97 Tf 6.59 0 Td[(1)beingbothnitethenDSRmeanmedianp2)]TJ /F5 11.955 Tf 11.96 0 Td[(1 p2, asJensen'sinequalitydeductsthatE(R)E(R)]TJ /F9 7.97 Tf 6.59 0 Td[(1)1.Furthermore,undersomeadditionalassumptions,itcanbeshownthatthemedianisthemostrobustestimatortothedistortionoftheFechnermodel. Proposition2.5. AssumeE(R)andE(R)]TJ /F9 7.97 Tf 6.58 0 Td[(1)arebothnite.Ifg()hasitsmodeat0andT()hasa@T(F) @=0E(R)]TJ /F9 7.97 Tf 6.59 0 Td[(1) thatisastochasticallyincreasinginG(),thenDSRTmedian1. ProofissimilartotheunivariateversionofProposition 2.4 50

PAGE 51

Table2-1. Distributionalsensitivityratiosforcommondistributions DistributionDSRmeanmedianDSRmeanH-LDSRH-Lmedian Normal1.27321.10021.1573Cauchy111.4053Laplace21.33331.5Logistic1.38631.16191.1931Uniform111Student'st51.44101.19981.2011Triangular(-1,1)1.33331.06671.25CenteredBeta(1 2,1 2)0.81061 11 13 5(x2+1 2)I[)]TJ /F9 7.97 Tf 6.59 0 Td[(1x1]0.721.05750.6809 Table2-2. DistributionalSensitivityRatiounderPowerExponentialFamily Dim.=0.1=0.25=0.5=1=4 p=211.922.51.51.181.02p=35.161.871.331.131.02p=43.431.611.251.101.02p=52.681.471.201.091.01p=101.641.221.101.041.01p=151.391.141.071.031.01 51

PAGE 52

Figure2-1. Halfre-scaledmixturedensityfunctiongs Figure2-2. ComplementarycumulativedistributionfunctionofUniform(0,1)andGs 52

PAGE 53

Figure2-3. Unitcircleshiftedd0fromtheorigin Table2-3. DistributionalSensitivityRatiounderStudent'stFamily Dim.df=3df=5df=10df=20 p=21.501.331.251.21p=31.441.281.201.16p=41.411.251.171.13p=51.381.231.151.12p=101.331.181.111.07p=151.311.171.091.06 53

PAGE 54

CHAPTER3REVIEWOFOPTIMALDETECTIONOFFECHNER-ASYMMETRY Symmetryisafundamentalconstituentofmanystatisticalassumptionsinaplethoraofelds.Thisconcepthasbeenwellstudiedandavarietyofparametric,nonparametricandasymptoticallynonparametrictestshavebeenproposed.Therearenonparametrictestsofsymmetrybasedonlinearcombinationsofranks,mostnotablythesigntestandWilcoxon'ssigned-ranktest. Cassartetal. ( 2008 )statethatthesetestsarenotoptimalinanysatisfactorysenseagainstasymmetryandthatinfactthesigntestisinsensitivetoasymmetricalternativesthatpreservethemedian. OthertestsincludetheFrasernormalscorestestandthetestofthevanderWaardentypereferredtoby Hajeketal. ( 1999 )thatareasymptoticallyoptimumtestsforthegeneralnormalfamilyofdensities.Kolmogorov-typetestsalsoexistthatrelyonameasureofdistanceoftheempiricaldistributionfunctiontothespaceofsymmetricdistributionfunctions.Suchatestwasproposedby Butler ( 1969 )butnooptimalityissuesareaddressed. Cassartetal. ( 2008 )considerapracticalconceptofoptimalityandbyadoptingthesimplegeneralclassofskeweddensitiesintroducedby Fechner ( 1897 ),derivelocallyandasymptoticallyoptimalparametricandsemiparametrictestingproceduresforthenullhypothesisofsymmetryforthecasesofspeciedorunspeciedlocationand/ordensity.TheFechnerfamilyofdistributionsisnotmeanttobearealisticrepresentationbutmoreofaconvenienttoolforcreatingthesetests. 3.1UniformLocalAsymptoticNormality Inthepaperof Cassartetal. ( 2008 )themaintechnicaltoolistheuniformasymptoticnormality(ULAN),withrespectto=(,,),at(,,0),oftheFechnerfamiliesP(n)f1=[>0fP(n),,;f1j2R,2()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)g, 54

PAGE 55

associatedwithf12F1.TheclassF1consistsofallstandardizedsymmetricdensities,i.e.f1(z)=f1()]TJ /F3 11.955 Tf 9.3 0 Td[(z)andR1f1(z)dz=0.75,thatareabsolutelycontinuous.TheprobabilitydensityfunctionofFechnerdistributionis 1 f1x)]TJ /F4 11.955 Tf 11.95 0 Td[( (1)]TJ /F5 11.955 Tf 11.96 0 Td[(sgn(x)]TJ /F4 11.955 Tf 11.96 0 Td[()),(3) whereitisimportanttonotethatrightskewnessisidentiedwhen<0. Forconveniencewestatetheirpropositionbyrstintroducingsomenotation.Let'f1=)]TJ /F5 11.955 Tf 10.79 2.65 Td[(_f1=f1,J(f1)=Z+1z2'2f1(z)f1(z)dz<1,I(f1)=Z+1'2f1(z)f1(z)dz<1,andM(f1)=Z+1jzj'2f1(z)f1(z)dz<1. Proposition3.1. Letf12F1.TheFechnerfamilyisULANatany=(,,0)with(writingZiforZ(n)i(,)=)]TJ /F9 7.97 Tf 6.59 0 Td[(1(X(n)i)]TJ /F4 11.955 Tf 11.96 0 Td[())centralsequence(n)f1()=0BBBBBBBBB@(n)f1;1()(n)f1;2()(n)f1;3()1CCCCCCCCCA=n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2nXi=10BBBBBBBBB@1 'f1(Zi)1 ('f1(Zi)Zi)]TJ /F5 11.955 Tf 11.96 0 Td[(1))]TJ /F4 11.955 Tf 9.29 0 Td[('f1(Zi)jZij1CCCCCCCCCA andfull-rankinformationmatrix )]TJ /F6 7.97 Tf 6.94 -1.8 Td[(f1()=0BB@)]TJ /F9 7.97 Tf 6.58 0 Td[(2I(f1)0)]TJ /F4 11.955 Tf 9.29 0 Td[()]TJ /F9 7.97 Tf 6.58 0 Td[(1M(f1)0)]TJ /F9 7.97 Tf 6.59 0 Td[(2(J(f1))]TJ /F5 11.955 Tf 11.96 0 Td[(1)0)]TJ /F4 11.955 Tf 9.3 0 Td[()]TJ /F9 7.97 Tf 6.59 0 Td[(1M(f1)0J(f1)1CCA.(3) Moreprecisely,forany(n)=((n),(n),0)suchthat(n))]TJ /F4 11.955 Tf 12.17 0 Td[(=O(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2)and(n))]TJ /F4 11.955 Tf 12.17 0 Td[(=O(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2),andforanyboundedsequence(n)=(t(n),s(n),(n))2R3,wehave,underP(n)(n);f1,asn!1,(n)+n)]TJ /F7 5.978 Tf 5.75 0 Td[(1=2(n)=(n);f1=(n)0(n)f1((n)))]TJ /F5 11.955 Tf 13.15 8.08 Td[(1 2(n)0)]TJ /F6 7.97 Tf 6.94 -1.8 Td[(f1()(n)+oP(1) 55

PAGE 56

and(n)f1((n))d!N(0,)]TJ /F6 7.97 Tf 6.94 -1.8 Td[(f1()). Remark3.1. InSection 1.3.4 anothergeneralclassofskeweddensities,basedonEdgeworthapproximations,wasreferenced.AlthoughadifferentmodelfromtheFechnerfamily,itisofconsiderablementionthatULANwasalsoestablishedby Cassartetal. ( 2010 )foritpostulatedadiagonalcovariancematrixconsequentlyreducingthelossofinformationcausedbyaco-dependencyoftheparameters. Considering=(,,)at(,,0),oftheparametricfamiliesP(n)f1=[>0fP(n),,;f1j2R,2Rg, yieldsthefollowingproposition Proposition3.2. Letf12F1.TheFechnerfamilyisULANatany=(,,0)with(writingZiforZ(n)i(,)=)]TJ /F9 7.97 Tf 6.59 0 Td[(1(X(n)i)]TJ /F4 11.955 Tf 11.96 0 Td[())centralsequence(n)f1()=0BBBBBBBBB@(n)f1;1()(n)f1;2()(n)f1;3()1CCCCCCCCCA=n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2nXi=10BBBBBBBBB@1 'f1(Zi)1 ('f1(Zi)Zi)]TJ /F5 11.955 Tf 11.96 0 Td[(1)'f1(Zi)(Z2i)-222(J(f1)=I(f1)1CCCCCCCCCA andfull-rankinformationmatrix)]TJ /F6 7.97 Tf 6.94 -1.79 Td[(f1()=0BB@)]TJ /F9 7.97 Tf 6.59 0 Td[(2I(f1)000)]TJ /F9 7.97 Tf 6.58 0 Td[(2(J(f1))]TJ /F5 11.955 Tf 11.95 0 Td[(1)000(f1)1CCA. whereI(f1)andJ(f1)aredenedasbefore,K(f1)=R+1z4'2f1(z)f1(z)dzand(f1)=K(f1))-222(J2(f1)=I(f1). 3.2ParametricOptimalTestsForSymmetry Inthissectionwereviewtheparametricallyoptimaltestsforsymmetryforthecasewherethedensityisspeciedbutthelocationisunspecied. 56

PAGE 57

3.2.1SpeciedDensity,SpeciedLocation AlocallyasymptoticallyuniformlymostpowerfultestofS>0fP(n),,0;f1giscreatedbysimplyusingthethirdcomponentofthecentralsequence(n)f1;3(,,0)inProposition 3.1 ,asthecovariancebetweenandiszero,implyingthatthereisnoimpact(asymptotically)on(n)f1;3fromthesubstitutionofaroot-nperturbationof.Thisisrequiredsothatthenumberofpossiblevaluesofanestimator,say^(n),inballswithO(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2)radiuscenteredattobeniteasn!1,whichimpliesthat(^)]TJ /F19 11.955 Tf 12.2 0 Td[()uniformly=OP(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2)intheasymptotictheory.Anestimator^(n)caneasilybediscretizedby ^(n)#=(cn1=2))]TJ /F9 7.97 Tf 6.58 0 Td[(1sgn(^(n))d(cn1=2j^(n)je).(3) ThetestthenbecomesTnf1(,^#)=1 p nJf1nXi=1'f1(Zi(,^#))jZi(,^#)j. Apossiblecontenderforaconsistentestimatorforistheempiricalmedian^=Med(jX(n)i)]TJ /F4 11.955 Tf 11.96 0 Td[(j),withnomomentassumptions. TheclassicalresultsonULANfamilies(seeChapter11of LeCam ( 1986 ))providethefollowingproposition. Proposition3.3. Letf2F1.Then, 1.T(n)f1(,^#)=T(n)f1(,)+oP(1)isasymptoticallynormal,withmeanzerounderP(n),,0;f1,mean(Jf1)1=2underP(n),,n)]TJ /F7 5.978 Tf 5.76 0 Td[(1=2;f1andvarianceoneunderboth; 2.thesequenceoftestsrejectingthenullhypothesisofsymmetry(withf1)wheneverT(n)f1(,^#)exceedsthe(1)]TJ /F4 11.955 Tf 12.52 0 Td[()standardnormalquantilezislocallyasymptot-icallyuniformlymostpowerful,atasymptoticlevel,for[2R+0fP(n),,0;f1gagainst[>0[2R+0fP(n),,;f1g. Twosidedtestscansimilarlyalsobederived. 57

PAGE 58

3.2.2SpeciedDensity,UnspeciedLocation HerethenullhypothesisisH(n)f1=S2RS2R+0fP(n),,0;f1g.Notethatfromequation( 3 )thelocationandskewnesscomponentsofthecentralsequenceinuenceeachother.TheauthorsstatethatLANandtheconvergenceoflocalexperimentstotheGaussianshiftexperiment( VanderVaart ( 2000 ))0BBBB@131CCCCAN0BBBB@0BBBB@)]TJ /F6 7.97 Tf 6.78 -1.8 Td[(f1,11())]TJ /F6 7.97 Tf 21.62 -1.8 Td[(f1,13())]TJ /F6 7.97 Tf 6.78 -1.8 Td[(f1,13())]TJ /F6 7.97 Tf 21.62 -1.8 Td[(f1,33()1CCCCA0BBBB@131CCCCA,0BBBB@)]TJ /F6 7.97 Tf 6.78 -1.8 Td[(f1,11())]TJ /F6 7.97 Tf 21.61 -1.8 Td[(f1,13())]TJ /F6 7.97 Tf 6.78 -1.8 Td[(f1,13())]TJ /F6 7.97 Tf 21.61 -1.8 Td[(f1,33()1CCCCA1CCCCA,(1,3)02R2 leadthemtocreatelocallyoptimaltestsonbyregressing3withrespectto1computedat(n)f1;3(,,0)and(n)f1;1(,,0).Theresidualisthus3)]TJ /F5 11.955 Tf 9.3 0 Td[(()]TJ /F6 7.97 Tf 11.66 -1.8 Td[(f1,11()))]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F6 7.97 Tf 6.77 -1.8 Td[(f1,13()1withresultingf1-efcientcentralsequenceforsymmetry(n)f1()=n)]TJ /F5 11.955 Tf 9.3 0 Td[(1=2nXi=1'f1(Zi(,))((f1))-221(jZi(,)j), where(f1)=Mf1=If1. FromtheclassicalresultsonULANfamiliesthetestis T(n)f1(^#,^#)=1 p nf1nXi=1'f1(Zi(^#,^#))M(f1) I(f1))-221(jZi(^#,^#)j,(3) where^#,^#arediscretizedroot-nconsistentestimators.Theestimate^istakentobeMed(X(n)i)andfor^,themedianabsolutedeviationestimate. TheclassicalresultsonULANfamilies(seeChapter11of LeCam ( 1986 ))providethefollowingproposition. Proposition3.4. Letf12F1.Then, 1.T(n)f1(^#,^#)=T(n)f1(,)+oP(1)isasymptoticallynormal,withmeanzerounderP(n),,0;f1,mean(f1)1=2underP(n),,n)]TJ /F7 5.978 Tf 5.75 0 Td[(1=2;f1andvarianceoneunderboth; 58

PAGE 59

2.thesequenceoftestsrejectingthenullhypothesisofsymmetry(withf1)wheneverT(n)f1(^#,^#)exceedsthe(1)]TJ /F4 11.955 Tf 12.57 0 Td[()standardnormalquantilezislocallyasymp-toticallyuniformlymostpowerful,atasymptoticlevel,for[2R+0fP(n),,0;f1gagainst[>0[2R[2R+0fP(n),,;f1g. Twosidedtestscansimilarlyalsobederived. 3.2.3Pseudo-GaussianTests Forcompletenessweprovidepseudo-GaussiantestsdevelopedbyextendingthetestsforunspecieddensitywithspeciedandunspeciedlocationofSections3.4and3.5of Cassartetal. ( 2008 ).Theteststatisticforsymmetrywithspeciedlocationis^T(n)'1()=1 q nm(n)4()nXi=1sgn(Xi)]TJ /F4 11.955 Tf 11.96 0 Td[()(Xi)]TJ /F4 11.955 Tf 11.96 0 Td[()2, wherem(n)k()=n)]TJ /F9 7.97 Tf 6.58 0 Td[(1Pni=1(Xi)]TJ /F4 11.955 Tf 12.35 0 Td[()k.Fortheunspeciedlocationcase,withm(n)k()=n)]TJ /F9 7.97 Tf 6.59 0 Td[(1Pni=1jXi)]TJ /F4 11.955 Tf 11.96 0 Td[(jk,thetestis^T(n)'1()=P(Xi)]TJ /F4 11.955 Tf 11.95 0 Td[()(2m(n)1())-221(jXi)]TJ /F4 11.955 Tf 11.95 0 Td[(j) q n(m(n)4())]TJ /F5 11.955 Tf 11.95 0 Td[(4m(n)1()m(n)3()+4(m(n)1())2m(n)2()). 3.3RankBasedTestsForSymmetry Inthissectionwereviewthetestforunspeciedlocationasitisofmorepracticalinterest. 3.3.1Signed-RankVersionsoftheCentralSequence Thegroupoftransformationsthatgeneratesthenullassumptionofsymmetrywithrespecttohasasamaximalinvariantthevectorofsigns(s1(),...,sn()),alongwiththevectorofranks(R(n)+,1(),...,R(n)+,n())wheresi()isthesignofXi)]TJ /F4 11.955 Tf 12.15 0 Td[(andR(n)+,i()therankofjXi)]TJ /F4 11.955 Tf 11.96 0 Td[(j. Conditioningthecentralsequence(n)f1()onthesignsandranksyieldsaccordingto Hallin ( 2003 )aversionofthesemiparametricallyefcient(atf1and)centralsequence.Thesigned-rankversionsoftherstandthirdcomponentsofthecentral 59

PAGE 60

sequencearee(n)f1;1(,)=1 p nnXi=1si()'f1 F)]TJ /F9 7.97 Tf 6.59 0 Td[(11+ R(n)+,i() n+1!!,e(n)f1;3()=)]TJ /F5 11.955 Tf 9.3 0 Td[(1 p nnXi=1si()'f1 F)]TJ /F9 7.97 Tf 6.59 0 Td[(11+ R(n)+,i() n+1!!F)]TJ /F9 7.97 Tf 6.59 0 Td[(11+ R(n)+,i() n+1!, whereF1+=2F1)]TJ /F5 11.955 Tf 11.95 0 Td[(1denotesthedistributionfunctiononjZij. InordertocreatetheoptimaltestwerequireanasymptoticrepresentationofthecentralsequencederivedfromclassicalHajektheoryforlinearsigned-rankstatisticsthatrequires'f1tobewrittenasthedifferenceoftwomonotoneincreasingfunctions.LetF1beasubsetofF1suchthatthisrequirementismet.Bytakingf12F1andg1tobeanystandardizedsymmetricdensityitwasshownthatthesigned-rankversionoftheasymmetrycomponentofthecentralsequenceisequalto(n)f1;3(,,0)+oP(1)underP(n),,0;f1withmeanzeroandavariancethatisasymptoticallyequivalenttoitsparametriccounterpart. 3.3.2TestWithUnspeciedLocation Forthecaseofunspeciedlocation,aconsistentestimatorhastobeusedinthecentralsequence,yieldingsignssi(^)andranksR(n)+,i(^)thataresubsequentlyalignedwithrespectto^.Thisistakencareofbyconditioningthe-componentofthecentralsequencetothe-componentunderunspecieddensityg1.Thedensityg1belongstothesetF1suchthatthecrossinformationquantitiesthatconcerntheandcomponentsarenite.Hence,underP(n),,0;g1, 0BBBB@e(n)f1;3()(n)g1;1(,,0)1CCCCA=1 p nnXi=10BBBB@)]TJ /F4 11.955 Tf 9.3 0 Td[('f1F)]TJ /F9 7.97 Tf 6.59 0 Td[(11(G1(Z(n)i(,)))jF)]TJ /F9 7.97 Tf 6.59 0 Td[(11(G1(Z(n)i(,)))j1 'g1(Zi)1CCCCA+oP(1)(3) 60

PAGE 61

isasymptoticallynormalwithmeanzeroandcrossinformationmatrix0BBBB@J(f1))]TJ /F4 11.955 Tf 9.3 0 Td[()]TJ /F9 7.97 Tf 6.59 0 Td[(1M(f1,g1))]TJ /F4 11.955 Tf 9.29 0 Td[()]TJ /F9 7.97 Tf 6.58 0 Td[(1M(f1,g1))]TJ /F9 7.97 Tf 6.59 0 Td[(2I(g1)1CCCCA, whereM(f1,g1)=Z10jF)]TJ /F9 7.97 Tf 6.59 0 Td[(11(u)j'f1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(F)]TJ /F9 7.97 Tf 6.59 0 Td[(11(u)'g1)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(G)]TJ /F9 7.97 Tf 6.58 0 Td[(11(u)du,I(f1,g1)=Z10'f1)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(F)]TJ /F9 7.97 Tf 6.59 0 Td[(11(u)'g1)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(G)]TJ /F9 7.97 Tf 6.58 0 Td[(11(u)du anddenoteFef1=fg12F1jI(f1,g1)<1,L(f1,g1)<1andM(f1,g1)<1g. Insimilarfashionto 3.2.2 byregressingtherstcomponenttothesecondcomponentofthecentralsequenceofequation( 3 )thesignrankequivalenttesttoequation( 3 )isTe(n)f1(^#)where Te(n)f1()=1 q ne(n)(f1)nXi=1si()'f1 F)]TJ /F9 7.97 Tf 6.59 0 Td[(11+ R(n)+,i() n+1!! e(n)(f1;))]TJ /F3 11.955 Tf 11.96 0 Td[(F)]TJ /F9 7.97 Tf 6.59 0 Td[(11+ R(n)+,i() n+1!!(3) andwheree(n)(f1)=1 nnXi=1'2f1F)]TJ /F9 7.97 Tf 6.58 0 Td[(11+r n+1e(n)(f1;))]TJ /F3 11.955 Tf 11.96 0 Td[(F)]TJ /F9 7.97 Tf 6.59 0 Td[(11+r n+12 Itshouldbenotedthate(n)(f1;)isaconsistentestimatorofe(n)(f1,g1)=M(f1,g1)=M(f1,g1). Thefollowingpropositionresultsbytheasymptoticequivalenceoftheefcientcentralsequencetoitsequivalentforwithsubstitutedbythediscretizedconsistentestimator^#. Proposition3.5. Letf12F1.Then 1.Te(n)f1(^#)isasymptoticallynormalwithmeanzerounderP(n),,0;g1,g12Fef1,meanL(f1,g1))-222(M(g1,f1)e(f1,g1) hJ(f1))]TJ /F5 11.955 Tf 11.96 0 Td[(2e(f1,g1)M(f1)+e2(f1,g1)I(f1)i1=2 61

PAGE 62

underP(n),,n)]TJ /F7 5.978 Tf 5.75 0 Td[(1=2;g1,g12Fef1,andvarianceoneunderboth; 2.thesequenceoftestsrejectingthenullhypothesis[g12Fef1[2R[2R+0fP(n),,0;g1gofsymmetrywithrespecttounspeciedwheneverTe(n)f1(^#)exceedthe(1)]TJ /F4 11.955 Tf 12.66 0 Td[()standardnormalquantilezislocallyasymptoticallyoptimal(moststringent),atasymptoticlevelforthenullhypothesisagainst[>0[2R[2R+0fP(n),,;f1g. 3.4AsymptoticRelativeEfciencies TherstcomponentofProposition 3.5 andtheprecedingpropositionsbasedontheparametrictests,provideatoolforcomputingtheefcaciesofthetests.Perturbationsinsymmetryoftheformn)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2facilitatethecomputationofefcacybytakingthederivativewithrespectto. Focusingontheunspeciedlocationcase,bytakingtheratioofthesquaredefcaciesofthesign-rankversiontesttothepseudo-Gaussian,theasymptoticrelativeefciencyishL(f1,g1))-221(M(g1,f1)e(f1,g1)i2=hJ(f1))]TJ /F5 11.955 Tf 11.95 0 Td[(2e(f1,g1)M(f1)+e2(f1,g1)I(f1)i [421)]TJ /F5 11.955 Tf 11.95 0 Td[(32]2=[4212)]TJ /F5 11.955 Tf 11.96 0 Td[(413+4], andfortheratiototheclassicaltesthL(f1,g1))-221(M(g1,f1)e(f1,g1)i2=hJ(f1))]TJ /F5 11.955 Tf 11.95 0 Td[(2e(f1,g1)M(f1)+e2(f1,g1)I(f1)i [)]TJ /F5 11.955 Tf 9.3 0 Td[(43+612]2=[6)]TJ /F5 11.955 Tf 11.95 0 Td[(624+932], wherek=Rzkg1(z)dzandk=Rjzjkg1(z)dz. Table1of Cassartetal. ( 2008 )displaysthenumericalasymptoticrelativeefciency(ARE)valuesforthet4.5,t6.5,t8,t10,t20,normal,exponentialpowerwithshapeparameter4,6and10,logistic,anddouble-exponentialdensities.Itcanbeseenthatthesign-rankversiontestsperformsespeciallywellwhenthetrueunderlyingdensityistheStudentdensitywith4.5and6.5degreesoffreedom. InthenextchapterwerevisitthetriplestestU-statisticproposedby Randlesetal. ( 1980 )andderiveitsasymptoticrelativeefciency,asitrelatestotheFechnerclassofdistributions. Cassartetal. ( 2008 )discusstheperformanceofthetriplestestasit 62

PAGE 63

relatestopowerandsignicancelevel,andourexaminationenablestheinclusionofthetriplestest. InChapter5wedevelopanoptimaltestforlocationthatisrobusttoasymmetrybyextendingthemethodologyofthischapterbutemphasizingourattentiontotherstcomponentofthecentralsequenceratherthanthethird. 63

PAGE 64

CHAPTER4REVISITOFTHETRIPLESTESTOFASYMMETRY Anasymptoticallydistribution-freeandlocationinvarianttestbasedonaU-statistic,fortestingsymmetryversusasymmetrywasproposedby Randlesetal. ( 1980 ).GivenarandomsampleX1,...,Xnthesmallestcardinalitysetrequiredtoidentifyasymmetryisatripleofobservation(Xi,Xj,Xk).Ifwetakethreeorderedobservations(X(i),X(j),X(k))fori0))]TJ /F3 11.955 Tf 11.96 0 Td[(P(X1+X2)]TJ /F5 11.955 Tf 11.96 0 Td[(2X3<0). TheasymptotictheoryfollowsalongthesamelinesofthetraditionalU-statistictheorytoexhibittheasymptoticnormalityofthestatisticanditstest. 64

PAGE 65

Cassartetal. ( 2008 )(and Cassartetal. ( 2010 ))refertothissignbasedtestanddeclarethesignsdonotfollowanyconceptofgroupinvarianceforsymmetry,unlikethesignsinSection 3.3.1 ,andhencearenotdistributionfree.Thetriplestest,however,foritssimplicity,appearstobeverycompetitiveintheirsimulationstudiesaswellastheonesillustratedin Randlesetal. ( 1980 ).Nextweintroduceamethodforthenumericalestimationoftheefcaciesforthetriplestest. 4.1FunctionalRepresentation InSection 1.2 thefunctionalrepresentationofestimatorsintermsoftheunderlyingdistributionfunctionwasintroduced.Thisnotationallowsforthedescriptionofcertaindesirablepropertiessuchasrobustnessandasymptoticnormality,throughtheuseoftheinuencefunction. LetFbetheunderlyingdistributionfunctionandX(i)denotetheithorderedrandomvariable.ThenanequivalentrepresentationoftheU-statistictothethetriplestestis T=3!ZZZx(1)>>>>>>>><>>>>>>>>>:1=3t3)]TJ /F3 11.955 Tf 11.96 0 Td[(t2>t2)]TJ /F3 11.955 Tf 11.96 0 Td[(t1)]TJ /F5 11.955 Tf 9.29 0 Td[(1=3t3)]TJ /F3 11.955 Tf 11.96 0 Td[(t2
PAGE 66

LetX=X=(1+sgn(X))and'f=)]TJ /F3 11.955 Tf 9.3 0 Td[(f0=f.Thederivativeofthefunctionalbecomes@ @T=3!ZZZx(1)
PAGE 67

3.d d[Un()]=0Un()andd d[Tn0()]=0Tn0() areassumedtoexistandbecontinuousinsomeclosedintervalabout=0with0Un()and0Tn0()beingnonzero. 4.limi!1Uni(i) Uni(0)=limi!1Tn0i(i) Uni(0)=1. 5.limn!1Uni(0) q n2Un(0)=KU>0andlimn0!1Tn0(0) q n02Tn0(0)=KT>0. Forthetestofsymmetry0=0.Forthesigned-rankversiontestforasymmetry,Proposition 3.5 easilyestablishestheconditions.Forthetriplestest,asymptoticnormalityofU-statisticsestablishestherstcondition.Next,wehavethatUn()=TandfromtheU-statistictheorythatUn()=n3)]TJ /F9 7.97 Tf 6.58 0 Td[(13Xk=13kn)]TJ /F5 11.955 Tf 11.95 0 Td[(33)]TJ /F3 11.955 Tf 11.96 0 Td[(kk, where1,=Eh(X(1),X(2),X(3))h(X(1),X(4),X(5)))]TJ /F3 11.955 Tf 11.96 0 Td[(T22,=Eh(X(1),X(2),X(3))h(X(1),X(2),X(4)))]TJ /F3 11.955 Tf 11.96 0 Td[(T23,=Eh(X(1),X(2),X(3))h(X(1),X(2),X(3)))]TJ /F3 11.955 Tf 11.96 0 Td[(T2=1=9)]TJ /F3 11.955 Tf 11.96 0 Td[(T2. Notingthatas!0yieldsthatF!Fandhence,functionally,T!Tandk,!k.Consequently,thelimitoftheratioofthevariancesofthestatisticunderthealternativeandnullhypothesesreachesoneandhenceveriesthesecondcondition.Thethirdcondition,concerningequation( 4 ),isestablishedbytheabsolutecontinuityofthesymmetricdensitiesthatgeneratetheFechnerfamily.Denote0Un(0)=@ @Tj=0,which 67

PAGE 68

gives 0Un(0)=E"h(X(1),X(2),X(3))3Xi=1(jX(i)j'f(X(i)))#.(4) Likewisetotheargumentforthesecondconditionwehavethatlimi!10Uni(i)=0Uni(0).Finally,U-statistictheorygivesusthat limn!1n2Un(0)=91=9Eh(X(1),X(2),X(3))h(X(1),X(4),X(5))(4) andtogetherwithequation( 4 )conrmsthefthcondition. Alternatively,replacingthedistributionfunctionFwithadistributionfunctionfromthecontaminationmodelinequation( 1 )andadoptingthemethodologyoftheinuencefunction(seeequation( 1 ))wecanobtainanalternativeformforq n2Un(0).LetIF(x)=@ @T(F)j=0,thenunderthenullassumptionof0=0andtheasymptoticnormalityresultofequation( 1 ) n2Un(0)=EIF2(X),(4) whereIF(t)=3264ZZx(1)
PAGE 69

ofthesigned-rankversiontestforasymmetrywithunspeciedlocationofequation( 3 )tothetriplestestviatheasymptoticrelativeefciency.TocompleteTable1in Cassartetal. ( 2008 ),theAREsofthesigned-rankversiontesttothetriplestestareillustratedinTable 4-1 ComparingtheAREvaluesofTable 4-1 toTable1in Cassartetal. ( 2008 )weimmediatelynotethatthetriplestestiscomparativelyequivalenttothesigned-rankversiontestundertheStudent-tdensities(forg1)andforscoref1.Inaddition,whentheactualdensities(i.e.g1)arethet4.5andt6.5thentriplestestoutperformsforthescoredensitiesofthepowerexponential,logisticandlaplace. Ontheotherhanditisimportanttonotethatthetriplesdoesnotperformaswellforthecorrectlyspeciednormaldensity'1withanasymptoticrelativeefciencyvalueof1.1254.Also,thetriplestestdoesnotachievehighefcacyvalues(andthereforelowAREs)underthelighter(thannormal)tailedpowerexponentialdensities.Thepseudo-Gaussianandclassicalprocedurearemorecompetitivethanthetriplestest.Howeverunderthelogisticandlaplacedensitiesthetriplesdoesappeartobesportingsimilarvaluesasthesigned-rankversiontestandforthepowerexponentialscoresattainlowerAREsthaneitherthepseudo-Gaussianorclassicalprocedure. Inconclusion,thetriplestestemergesasacontendingtestespeciallyforheaviertaileddensities. 69

PAGE 70

Table4-1. AREsundervariousdensitiesofthesigned-rankversiontestwithrespecttothetriplestest Scoref1g1 ft4.5ft6.5ft8ft10ft201fPE4fPE6fPE10fLogfL ft4.51.01031.01551.01071.01091.03981.03961.36361.88662.79891.01211.0571ft6.51.00451.02141.02181.02681.06631.07631.50452.15353.33291.02451.0395ft80.99801.02011.02301.03031.07481.08991.56722.27643.58601.02681.0291ft100.99011.01681.02201.03131.08041.10041.62292.38813.81981.02681.0184ft200.96651.00271.01271.02651.08541.11641.73682.62524.33021.02040.991010.92780.97340.98841.00691.07641.12541.84892.87724.89841.00000.6750fPE40.63160.68610.70910.73470.81610.88782.47585.094611.38480.72190.5641fPE60.46300.50720.52670.54830.61610.68002.28945.533614.67840.53930.3968fPE100.29970.33010.34380.35900.40680.45411.79334.964816.45570.35420.2483fLog0.99121.01521.01921.02751.07441.09261.54182.20843.41951.03071.0358fL0.76940.74080.72410.71320.71220.69320.57060.69310.95820.74291.2579 70

PAGE 71

CHAPTER5OPTIMALTESTINGOFLOCATIONROBUSTTOFECHNER-ASYMMETRY Inthischapterweaimtoextendtheconceptsfoundin Cassartetal. ( 2008 )wherebylocallyandasymptoticallyoptimal(intheLeCamsense)testsarederivedforthenullhypothesisofknownpopulationlocationthatarerobusttodeparturesfromsymmetryintheFechnersense.WerstbeginbyprovidingtheULANmethodologythatwillpowertheparametricproceduresinSection 5.1 andthesigned-ranksemiparametricproceduresinSection 5.2 thatareasymptoticallyequivalentintermsofefciency. Thesigned-rankproceduresarenotnonparametricastheymakeuseofascorefunctionthatiscreatedfromaspecieddistributionfunction.Theconceptistodevelopaprocedurethatisrobusttomisspecieddistributionfunctions. Locationinthissettingreferstothemodeunderaparticularmodelforwhichthemodeisknownquantileofthedistribution.Obviously,inthecaseofsymmetrythemode,medianandmeanareallequalbutweaimtocreateatestthatisrobusttoFechner-asymmetry.Hence,themode,medianandmeanwillnot(usually)beequalbutintheFechnerset-upthemoderemainsunchanged,whentheunderlyingdensityisunimodeal. 5.1ULANAndParametricallyOptimalTestForLocation Firstly,theULANtheoryandmethodologyisdevelopedtocreateaparametricallyoptimaltestforlocationforthecaseofspecieddensityandspeciedasymmetryandforthecaseofspecieddensitybutunspeciedasymmetry.TheparametricallyoptimaltestforunspecieddensityandasymmetrycanthenbeobtainbyextendingthemethodologyofSection3.5in Cassartetal. ( 2008 ),makingitrelevantforatestoflocation. 5.1.1ULAN WewilluseULANwithrespectto=(,,),at(0,,),oftheFechnerfamilyP(n)f=[>0fP(n),,;fj2R,2()]TJ /F5 11.955 Tf 9.3 0 Td[(1,1)g. 71

PAGE 72

Withoutlossofgeneralityassumethatthelocationparameter=0. Proposition5.1. Letf2Fasdenedinequation( 2 )and'f=)]TJ /F5 11.955 Tf 10.79 2.65 Td[(_f=f,TheFechnerfamilyisULANatany=(0,,)with(writingZiforZ(n)i(,)=X(n)i=((1+sgn(X(n)i))))centralsequence(n)f()=0BBBBBBBBB@(n)f;1()(n)f;2()(n)f;3()1CCCCCCCCCA=1 p nnXi=10BBBBBBBBB@1 1 1+sgn(Zi)'f(Zi)1 ('f(Zi)Zi)]TJ /F5 11.955 Tf 11.95 0 Td[(1)1 1+sgn(Zi)'f(Zi)jZij1CCCCCCCCCA andfull-rankcovariance(information)matrix )]TJ /F6 7.97 Tf 6.94 -1.8 Td[(f()=0BB@)]TJ /F9 7.97 Tf 6.59 0 Td[(21 1)]TJ /F13 7.97 Tf 6.59 0 Td[(2I(f)0)]TJ /F9 7.97 Tf 6.58 0 Td[(11 1)]TJ /F13 7.97 Tf 6.59 0 Td[(2M(f)0)]TJ /F9 7.97 Tf 6.59 0 Td[(2(J(f))]TJ /F5 11.955 Tf 11.95 0 Td[(1)0)]TJ /F9 7.97 Tf 6.59 0 Td[(11 1)]TJ /F13 7.97 Tf 6.58 0 Td[(2M(f)01 1)]TJ /F13 7.97 Tf 6.59 0 Td[(2J(f)1CCA(5) whereI(f)=Z+1'2f(z)f(z)dz<1,J(f)=Z+1z2'2f(z)f(z)dz<1,M(f)=Z+1jzj'2f(z)f(z)dz<1. Moreprecisely,forany(n)=(0,(n),(n))suchthat(n))]TJ /F4 11.955 Tf 12.22 0 Td[(=O(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2)and(n))]TJ /F4 11.955 Tf 12.23 0 Td[(=O(n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2),andforanyboundedsequence(n)=(t(n),s(n),(n))2R3,wehave,underP(n)(n);f,asn!1,(n)+n)]TJ /F7 5.978 Tf 5.75 0 Td[(1=2(n)=(n);f=(n)0(n)f((n)))]TJ /F5 11.955 Tf 13.15 8.08 Td[(1 2(n)0)]TJ /F6 7.97 Tf 6.94 -1.8 Td[(f()(n)+oP(1) and(n)f((n))d!N(0,)]TJ /F6 7.97 Tf 6.94 -1.79 Td[(f()). SeeAppendix G fortheproof. 72

PAGE 73

Thecross-informationcoefcients)]TJ /F9 7.97 Tf 6.78 -1.79 Td[(12and)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(23arezeroasthecoefcientshavetheform)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(12=)]TJ /F9 7.97 Tf 6.59 0 Td[(2Z+1'f(z)(z'f(z))]TJ /F5 11.955 Tf 11.96 0 Td[(1)f(z)dz=)]TJ /F9 7.97 Tf 6.59 0 Td[(2Z+1'2f(z)zf(z)dz)]TJ /F15 11.955 Tf 11.96 16.27 Td[(Z+1'f(z)f(z)dz=0 and)]TJ /F9 7.97 Tf 6.77 -1.79 Td[(23=)]TJ /F9 7.97 Tf 6.58 0 Td[(1Z+1('f(z)z)]TJ /F5 11.955 Tf 11.96 0 Td[(1)'(z)jzjf(z)dz=)]TJ /F9 7.97 Tf 6.58 0 Td[(1Z+1'2f(z)zjzjf(z)dz)]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z+1'f(z)jzjf(z)dz=0 andarezeroastheintegrandsareoddfunctions. 5.1.2OptimalTesting:SpeciedDensity,SpeciedAsymmetry AstraightforwardexpansionofSection 3.2.1 yieldsalocallyasymptoticallyuniformlymostpowerfultestforlocationof[2R+0fP(n)0,,;fgbysimplyusingthe-componentofthecentralsequenceduetotheorthogonalitybetweentheand-components,evidentfromthecovariancematrix0BBBB@)]TJ /F9 7.97 Tf 6.58 0 Td[(21 1)]TJ /F13 7.97 Tf 6.59 0 Td[(2I(f)00)]TJ /F9 7.97 Tf 6.59 0 Td[(2(J(f))]TJ /F5 11.955 Tf 11.95 0 Td[(1).1CCCCA Thetestbasedon(n)f;1(0,^#,)denotedbyT(n)f;1(0,^#,)where^#isadiscretizedroot-nconsistentestimatorof,isT(n)f;1(0,,)=s 1)]TJ /F4 11.955 Tf 11.96 0 Td[(2 nI(f)nXi=i1 1+sgn(Zi)'f(Zi). 73

PAGE 74

TheclassicalresultsonULANfamiliesprovidethefollowingproposition.Werefertotheresultsof LeCam ( 1986 ,chap.11),andforanabridgedversionto LeCam&Yang ( 2000 ,chap.6). Proposition5.2. Letf2F.Then, 1.T(n)f(^#,)=T(n)f(,)+oP(1)isasymptoticallynormal,withmeanzerounderP(n)0,,;f,mean()]TJ /F9 7.97 Tf 6.59 0 Td[(2(1)]TJ /F4 11.955 Tf 11.95 0 Td[(2))]TJ /F9 7.97 Tf 6.59 0 Td[(1)1=2underP(n)n)]TJ /F7 5.978 Tf 5.76 0 Td[(1=2,,;fandvarianceoneunderboth; 2.thesequenceoftestsrejectingthenullhypothesisof=0(withf)wheneverT(n)f(^#,)exceedsthe(1)]TJ /F4 11.955 Tf 9.7 0 Td[()standardnormalquantilezislocallyasymptoticallyoptimal(moststringent),atasymptoticlevel,for[2R+0fP(n)0,,;fgagainst[>0[2R+0fP(n),,;fg. Locallyasymptoticallymaximumtwosidedtestsareeasilyderivedalongthesamelines. 5.1.3OptimalTesting:SpeciedDensity,UnspeciedAsymmetry Withtheasymmetryparameterunspeciedandandcomponentsofthecentralsequencenotbeingorthogonal,weuseULANandtheconvergenceoflocalexperimentstotheGaussianshiftexperiment0BBBB@131CCCCAN0BBBB@0BBBB@)]TJ /F6 7.97 Tf 6.77 -1.79 Td[(f,11())]TJ /F6 7.97 Tf 21.62 -1.79 Td[(f,13())]TJ /F6 7.97 Tf 6.77 -1.79 Td[(f,13())]TJ /F6 7.97 Tf 21.62 -1.79 Td[(f,33()1CCCCA0BBBB@131CCCCA,0BBBB@)]TJ /F6 7.97 Tf 6.78 -1.79 Td[(f,11())]TJ /F6 7.97 Tf 21.62 -1.79 Td[(f,13())]TJ /F6 7.97 Tf 6.78 -1.79 Td[(f,13())]TJ /F6 7.97 Tf 21.62 -1.79 Td[(f,33()1CCCCA1CCCCA,(1,3)02R2 tocreatelocallyoptimalinferenceonbasedontheresidualoftheregressionof1withrespectto3,computedat(n)f;1(0,,)and(n)f;3(0,,).Thisresidualtakestheform1)]TJ /F5 11.955 Tf 11.95 0 Td[(()]TJ /F6 7.97 Tf 11.66 -1.79 Td[(f,33))]TJ /F9 7.97 Tf 6.59 0 Td[(1)]TJ /F6 7.97 Tf 6.77 -1.79 Td[(f,133;thatbeing1)]TJ /F4 11.955 Tf 11.96 0 Td[()]TJ /F9 7.97 Tf 6.59 0 Td[(1M(f) J(f)3. Theresultingfefcientcentralsequenceforlocationisthus(letting(f)=M(f)=J(f))(n)f()=1 p nnXi=1'f(Zi) 1+sgn(Zi)(1)]TJ /F4 11.955 Tf 11.95 0 Td[((f)jZij). 74

PAGE 75

ThisefcientcentralsequenceunderP(n);fisasymptoticallynormalwithmeanzeroandvariance )]TJ /F9 7.97 Tf 6.59 0 Td[(2f=)]TJ /F9 7.97 Tf 6.58 0 Td[(21 1)]TJ /F4 11.955 Tf 11.96 0 Td[(2(I(f))]TJ /F4 11.955 Tf 11.96 0 Td[((f)M(f)).(5) Itfollowsthatlocallyasymptoticallymostpowerfultestsof[2R+0[2()]TJ /F9 7.97 Tf 6.59 0 Td[(1,1)fP(n)0,,;fgcanbebasedon(n)f(0,^#,^#),henceonT(n)f(^#,^#),whereT(n)f(,)=1 p n)]TJ /F9 7.97 Tf 6.59 0 Td[(2fnXi=1'f(Zi) 1+sgn(Zi)(1)]TJ /F4 11.955 Tf 11.96 0 Td[((f)jZij), andwhere^#and^#arediscretizedroot-nconsistentestimators.Consistentestimatorsusingthemethodofmomentsandmaximumlikelihoodbasedonthebroadfamilyofpowerexponentialdistributionsisdescribedin Arellano-Valleetal. ( 2005 ).Thereisalossofinformationforlocation(secondterminequation( 5 ),(f)M(f))duetothedependencyoftherstcomponentofcentralsequencetothethird. Randles ( 1982 )and Pierce ( 1982 )discusstheeffectofsubstitutingestimatorsfornon-orthogonalparametersbutbyLeCam'sthirdLemma,thecentralsequencecreatedbyregressingthecomponentofthesequencethatcorrespondstotheparameterofinteresttothenon-orthogonalcomponentsaddressedthisissue.Forconsistentestimationofthecross-informationquantitieswereferthereadertoSection4.5in Cassartetal. ( 2008 ). Proposition5.3. Letf2F.Then, 1.T(n)f(^#,^#)=T(n)f(,)+oP(1)isasymptoticallynormal,withmeanzerounderP(n)0,,;f,mean()]TJ /F9 7.97 Tf 6.59 0 Td[(2f)1=2underP(n)n)]TJ /F7 5.978 Tf 5.76 0 Td[(1=2,,;fandvarianceoneunderboth; 2.thesequenceoftestsrejectingthenullhypothesisof=0(withf)wheneverT(n)f(^#,^#)exceedsthe(1)]TJ /F4 11.955 Tf 12.59 0 Td[()standardnormalquantilezislocallyasymp-toticallyoptimal(moststringent),atasymptoticlevel,for[2R+0[2()]TJ /F9 7.97 Tf 6.58 0 Td[(1,1)fP(n)0,,;fgagainst[>0[2R+0[2()]TJ /F9 7.97 Tf 6.59 0 Td[(1,1)fP(n),,;fg. Theproofincontrivedbytheresultsof LeCam&Yang ( 2000 ,chap.6).Locallyasymptoticallymaximumtwosidedtestsareeasilyderivedalongthesamelines. 75

PAGE 76

5.2OptimalSigned-RankBasedTests Thesignsandranksthataretobeusedinthedevelopmentofthetestareappliedwithadifferentreasoningfromtheirtraditionalsense(suchastheWilcoxonRankSumandSigned-Rankedtests)andfromtheiruseinSection 3.3 .Asaforementioned,thesignsandranksareusedtocreateteststhatrobusttomisspecieddensities. MaximalinvarianttransformationsundertheassumptionofsymmetrywithrespecttoisthevectorofsignsalongwiththevectorofranksasdiscussedinSection 3.3 .Whentheasymmetryisspeciedthenullhypothesisofxedlocation,=0withrespectto(and)hascertaininvarianceproperties.Thevectorofsigns(si(),...,sn()),alongwiththevectorofranks(R(n)+,1(),...,R(n)+,n())aremaximalinvariant,wheresi()isthesignofXi=(1+sgn(Xi))andR(n)+,i()therankofjXi=(1+sgn(Xi))j.Howeverwhenisnotspecied,thevectorofranksremainsinvariantbutthevectorofsignsdoesnot.Thevectorofranksforasetofobservationsfromasymmetricdistribution,with=0,isequivalenttoasetwhere6=0bysimplyapplyingthetransformationX=(1+sgn(X)).Ontheotherhand,forthesymmetriccasewhen=0theprobabilityofapositivesignandthatofanegativeareequalat1/2,thoughthesamecannotbesaidfortheasymmetriccasewhen6=0.Thevectorofsignsandranksremainindependentbutusualteststatisticswouldnotbecenteredatzero.Bycreatingatestontheresidualsafterregressingtherstcomponentofthecentralsequencetotheothercomponents,iteffectivelyeliminatesissuesthatarisefromtheco-dependencyoftheparametersandfromthealignmentofthesignsandrankstoaconsistentestimatorofaswell.Italsoeliminatestheproblemofanon-centeredsigned-ranktest.Notethatsi()effectivelydoesnotdependonandhencewewillsimplyusesi. Hallin ( 2003 )showsthatconditioningthecentralsequence(n)f()onthesignsandranksyieldsaversionofthesemiparametricallyefcient(atfand)centralsequence.RecallthatZi=Z(n)i(,)=X(n)i (1+sgn(X(n)i)), 76

PAGE 77

andhencethedensityfunctionofZiis(1+sgn(z))f(z).Thedistributionfunctionistherefore FZ(z)=8>>>><>>>>:(1)]TJ /F4 11.955 Tf 11.96 0 Td[()F(z)ifz<0,(1+)F(z))]TJ /F4 11.955 Tf 11.95 0 Td[(ifz0,(5) andthequantilefunctionis F)]TJ /F9 7.97 Tf 6.59 0 Td[(1Z(u)=8>>>><>>>>:F)]TJ /F9 7.97 Tf 6.58 0 Td[(1u 1)]TJ /F4 11.955 Tf 11.95 0 Td[(if0
PAGE 78

byequations( 5 )and( 5 ).Thefollowingpropositionthenfollowsfrom Hajeketal. ( 1999 ,sec.3.3) Proposition5.4. Letf2Fandg2F,withdistributionfunctionsFandGrespectively.Then, 1.e(n)f;1(,)=E[(n)f;1(0,,)js1,...,sn,R(n)+,1(),...,R(n)+,n()]+oL2(1) =1 p nPni=11 1+si'f)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(F)]TJ /F9 7.97 Tf 6.59 0 Td[(1(G(Zi))+oL2(1) asn!1,underP(n)0,,;g,andhencee(n)f;1(,)=(n)f;1(0,,)+oP(1)underP(n)0,,;f; 2.ef;1hasmeanzeroandvariance)]TJ /F9 7.97 Tf 6.59 0 Td[(2e(n)(f)=)]TJ /F9 7.97 Tf 6.58 0 Td[(21 nnXi=1(1+si))]TJ /F9 7.97 Tf 6.59 0 Td[(2'2f F)]TJ /F9 7.97 Tf 6.59 0 Td[(1+ R(n)+,i() n+1!! underP(n)0,,;g.Moreovere(n)(f)=(1)]TJ /F4 11.955 Tf 11.95 0 Td[(2))]TJ /F9 7.97 Tf 6.59 0 Td[(1I(f)+o(1)asn!1. 5.2.1OptimalSigned-RankTestsForLocation:SpeciedAsymmetry Proposition 5.4 providestheinsightintoconstructingadistribution-freerank-basedtestoflocationwithrespecttospeciedasymmetry.ExtendingtheideabehindSection 3.2.1 welet Te(n)f(,)=e(n)f;1(,) 1 q e(n)(f)=1 q ne(n)(f)nXi=1si 1+si'f F)]TJ /F9 7.97 Tf 6.58 0 Td[(1+ R(n)+,i() n+1!!(5) wherethetestuses^#asadiscretizedroot-nconsistentestimatorofanddenethecross-informationcoefcient I(f,g)=Z10'f(F)]TJ /F9 7.97 Tf 6.59 0 Td[(1(u))'g(G)]TJ /F9 7.97 Tf 6.58 0 Td[(1(u))du.(5) DenotetheclassofdensitiesforwhichtheintegralexistsandisnitebyFe. Proposition5.5. Letf2F.Then 1.Te(n)f(^#,)isasymptoticallynormalwithmeanzerounderP(n)0,,;g,g2Ff,mean1 p (1)]TJ /F4 11.955 Tf 11.95 0 Td[(2)I(f,g) p I(f) 78

PAGE 79

underP(n)n)]TJ /F7 5.978 Tf 5.76 0 Td[(1=2,,;g,g2Fef,andvarianceoneunderboth; 2.thesequenceoftestsrejectingthenullhypothesis[g2Fef[2R+0fP(n)0,,;ggof=0withrespecttospeciedwheneverTe(n)f(^#,)exceedthe(1)]TJ /F4 11.955 Tf 11.44 0 Td[()standardnormalquantilezislocallyasymptoticallyoptimal(moststringent),atasymptoticlevelforthenullhypothesisagainst[>0[2R+0fP(n),,;fg. Theproofincontrivedinthesamemannerastheparametrictestsbytheresultsof LeCam&Yang ( 2000 ,chap.6).Thetwo-sidedversioneasilyensues.Thetestisonlyvalidforxed-alternatives. Asmentionedearlier,thevectorofsignsandranksaremaximalinvariantforthegroupgeneratedbythenullhypothesisandconsequentlythetestisalsodistributionfreeinthesensethatithasanulldistributionthatdoesnotdependonthedensityfunctiong(),butonlyonthespecied. Alogicalcompetitortothesigned-rankversiontestistheclassicalsigntestanditismeaningfultomentiontheirequivalencefortheLaplacescoredensityfunction.Notethat,'fL()=sgn(),andtherefore'fL F)]TJ /F9 7.97 Tf 6.58 0 Td[(1+ R(n)+,i() n+1!!1,8j. Let,Bbedenedas B=jfXijXi>0gj,(5) whichdenotesthecountofthenumberofsignsthataregreaterthan0.Ignoringthevariancestandardizationterm,theteststatisticinequation( 5 ),becomesnXi=1si 1+si=B 1+)]TJ /F3 11.955 Tf 13.15 8.08 Td[(n)]TJ /F3 11.955 Tf 11.95 0 Td[(B 1)]TJ /F4 11.955 Tf 11.95 0 Td[(=1 1)]TJ /F4 11.955 Tf 11.95 0 Td[(2(2B)]TJ /F3 11.955 Tf 11.95 0 Td[(n(1+)), Usingthebinomialdistribution,themeanvalueofBisn(1+)1=2,sothesigned-rankversionteststatisticisequivalenttothesigntest. Itisimportanttonotethatforthespecialcasewhen=0,thesigned-rankversionteststatisticisequivalenttothelinear-scoresigned-rankteststatisticasdenedin 79

PAGE 80

Hajeketal. ( 1999 ,pg.74)thatusestheasymptoticallyoptimalscorefunctionandproducesalocationtestwithmaximumAREforthatF.Specically,forthelogisticscoredensityfunctionthetestisequivalenttotheWilcoxonsigned-ranktestdenedasW+=nXi=11[X(n)i>0]R(n)+,i. Assuming=1withoutlossofgenerality,thethelogisticscoredensityfunctionisdenotedby'fLog(t)=1)]TJ /F5 11.955 Tf 11.95 0 Td[((2e)]TJ /F6 7.97 Tf 6.58 0 Td[(t)=(1+e)]TJ /F6 7.97 Tf 6.58 0 Td[(t)andF+(t)=2=(1+e)]TJ /F6 7.97 Tf 6.58 0 Td[(t))]TJ /F5 11.955 Tf 11.96 0 Td[(1.Therefore,'fLog F)]TJ /F9 7.97 Tf 6.59 0 Td[(1+ R(n)+,i(0) n+1!!=R(n)+,i(0) n+1,8j, establishingtheequivalence. Inasensethesigned-rankversionteststatisticisageneralizationofthesigned-rankteststatisticsforthecasewhen6=0. 5.2.2OptimalSigned-RankTestsForLocation:UnspeciedAsymmetry InasimilarwaytothemethodologyofSection 5.1.3 ,weregresse(n)f;1withrespecttoe(n)g;3thatwilladdressthedependencyonthenuisanceparameter,aswellasthealignmentofsiandR(n)+,1toaconsistentestimatorofandthebiasofsito.Thismethodologydoescomeatacostasshowninequation( 5 ).Theasymptoticjointdistribution,byProposition 5.4 ,underP(n)0,,;gof0BBBBBBBBB@e(n)f;1(n)g;2(n)g;31CCCCCCCCCA=1 p nnXi=i0BBBBBBBBB@1 (1+sgn(F)]TJ /F9 7.97 Tf 6.58 0 Td[(1(G(Zi))))'f)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(F)]TJ /F9 7.97 Tf 6.59 0 Td[(1(G(Zi))1 ['g(Zi)Zi)]TJ /F5 11.955 Tf 11.95 0 Td[(1]1 1+sgn(Zi)'g(Zi)jZij1CCCCCCCCCA+oP(1) isasymptoticallynormalwithmeanzeroandcovariance0BB@)]TJ /F9 7.97 Tf 6.58 0 Td[(21 1)]TJ /F13 7.97 Tf 6.59 0 Td[(2I(f)0)]TJ /F9 7.97 Tf 6.59 0 Td[(11 1)]TJ /F13 7.97 Tf 6.58 0 Td[(2M(f,g)0)]TJ /F9 7.97 Tf 6.58 0 Td[(2(J(g))]TJ /F5 11.955 Tf 11.96 0 Td[(1)0)]TJ /F9 7.97 Tf 6.59 0 Td[(11 1)]TJ /F13 7.97 Tf 6.59 0 Td[(2M(f,g)01 1)]TJ /F13 7.97 Tf 6.58 0 Td[(2J(g)1CCA. 80

PAGE 81

Theadditionalcrossinformationquantitiesaredenedby,J(f,g)=Z10jF)]TJ /F9 7.97 Tf 6.59 0 Td[(1(u)jjG)]TJ /F9 7.97 Tf 6.59 0 Td[(1(u)j'f(F)]TJ /F9 7.97 Tf 6.59 0 Td[(1(u))'g(G)]TJ /F9 7.97 Tf 6.59 0 Td[(1(u))du andM(f,g)=Z10jG)]TJ /F9 7.97 Tf 6.59 0 Td[(1(u)j'f(F)]TJ /F9 7.97 Tf 6.59 0 Td[(1(u))'g(G)]TJ /F9 7.97 Tf 6.59 0 Td[(1(u))du. Oncemore,thecross-informationcoefcientcorrespondingtoandandalsoandarezeroasinSection 5.1 .Denotetheclassofdensitiesforwhichtheintegralsexistsandarenite,inadditiontoequation( 5 ),byFe. LetFefbethesetofallg2Fsuchthatallthecrossinformationtermsarenite.Also,writingefore(f,g)=M(f,g)=J(f,g)andproceedinginasimilarfashiontoSection 5.1.3 ,theefcientcentralsequencee(n)f(e;)=1 p nnXi=1si 1+si'f F)]TJ /F9 7.97 Tf 6.59 0 Td[(1+ R(n)+,i() n+1!!"1)]TJ /F4 11.955 Tf 11.95 0 Td[(eF)]TJ /F9 7.97 Tf 6.59 0 Td[(1+ R(n)+,i() n+1!# andalsoe(n)f(e(n)(f;);,)are,byLeCamm'sthirdLemma,asymptoticallyinsensitivetoroot-nperturbationsofand.Theterme(n)(f;)isanyconsistentestimatorofe(n)(f,g)underP(n)0,,;g.Then,applyingLemma3.1from Cassartetal. ( 2008 )e(n)f(e(n)(f;^#);^#,^#))]TJ /F5 11.955 Tf 11.95 0 Td[(e(n)f(e(n)(f;);,)=oP(1) underP(n)0,,;gforanydiscretizedroot-nconsistentestimatorsforandandanyg2FfforFfFwherethecrossinformationtermsarenite.Estimatinge(andallcross-informationquantities)isdescribedinSection4.5in Cassartetal. ( 2008 ),usinge(f,f)=(f). TheteststatisticisthenbasedonTe(n)f(^#,^#),whereTe(n)f(,)=1 q n)]TJ /F9 7.97 Tf 6.59 0 Td[(2(n)fnXi=1si 1+si'f F)]TJ /F9 7.97 Tf 6.58 0 Td[(1+ R(n)+,i() n+1!!"1)]TJ /F4 11.955 Tf 11.95 0 Td[(e(n)(f;)F)]TJ /F9 7.97 Tf 6.58 0 Td[(1+ R(n)+,i() n+1!# 81

PAGE 82

andwheree(n)f=1 nnXi=1(1+si))]TJ /F9 7.97 Tf 6.59 0 Td[(2'2f F)]TJ /F9 7.97 Tf 6.59 0 Td[(1+ R(n)+,i() n+1!!"1)]TJ /F4 11.955 Tf 11.96 0 Td[(e(n)(f;)F)]TJ /F9 7.97 Tf 6.59 0 Td[(1+ R(n)+,i() n+1!#2 and^#and^#arediscretizedroot-nconsistentestimatorsofand. Proposition5.6. Letf2F.Then 1.Te(n)f(^#,^#)isasymptoticallynormalwithmeanzerounderP(n)0,,;g,g2Ff,mean1 p (1)]TJ /F4 11.955 Tf 11.96 0 Td[(2)I(f,g))]TJ /F4 11.955 Tf 11.96 0 Td[(eM(g,f) hI(f))]TJ /F5 11.955 Tf 11.96 0 Td[(2eM(f)+e2J(f)i1=2 underP(n)n)]TJ /F7 5.978 Tf 5.76 0 Td[(1=2,,;g,g2Fef,andvarianceoneunderboth; 2.thesequenceoftestsrejectingthenullhypothesis[g2Fef[2R+0[2()]TJ /F9 7.97 Tf 6.58 0 Td[(1,1)fP(n)0,,;ggof=0withrespecttounspeciedwheneverTe(n)f(^#,^#)exceedthe(1)]TJ /F4 11.955 Tf 12.41 0 Td[()standardnormalquantilezislocallyasymptoticallyoptimal(moststringent),atasymptoticlevelforthenullhypothesisagainst[>0[2R+0[2()]TJ /F9 7.97 Tf 6.59 0 Td[(1,1)fP(n),,;fg. Theproofincontrivedinthesamemannerastheparametrictestsbytheresultsof LeCam&Yang ( 2000 ,chap.6).Thetwo-sidedversioneasilyensues. 5.3AsymptoticRelativeEfciencies Weemphasizeourattentiontothesigned-rankversiontestsofSection 5.2 forthecomparisonoftheseteststotraditionaltestsoflocation.Proposition 5.5 inthecontextofspeciedasymmetryallowsforthecomparisonofTe(n)f(^#,)withthesign,theWilcoxonsigned-rankandttestsoflocation.Unfortunately,inthecontextofunspeciedasymmetryProposition 5.6 doesnoteasilyallowacomparisonofTe(n)f(^#,^#)toanothertest. 5.3.1SpeciedAsymmetry WhentheasymmetryparameterisspeciedthesigntestcanbeadaptedtoincorporatethefactthattheprobabilityoftheeventthatX<0isnolonger1=2but(1)]TJ /F4 11.955 Tf 12.93 0 Td[()=2forXGwhereGistheFechnerdistributionfunctionofequation 82

PAGE 83

( 1 ).Totestthenullhypothesisof[2R+0fP(n)0,,;ggagainsttheonesidedalternative[>0[2R+0fP(n),,;ggforg2F. Usingthesamemethodologyof( 4 ),thenforanywehavethatn()=E,(B)=nP,fX>0g=n[1)]TJ /F3 11.955 Tf 11.95 0 Td[(G()]TJ /F4 11.955 Tf 9.3 0 Td[()], whereBisdenedasinequation( 5 ).Therefore,thepartialderivativeofthemeanofBatthenullis@n() @=0=)]TJ /F9 7.97 Tf 6.59 0 Td[(1ng(0).ThevarianceofBatthenullhypothesisisn)]TJ /F9 7.97 Tf 6.68 -4.98 Td[(1)]TJ /F13 7.97 Tf 6.59 0 Td[( 2)]TJ /F9 7.97 Tf 14.14 -4.98 Td[(1+ 2=n1)]TJ /F13 7.97 Tf 6.59 0 Td[(2 2andconsequentlyK2sign=4 2(1)]TJ /F4 11.955 Tf 11.95 0 Td[(2)g2(0). ThetermK2signisthenusedinProposition 5.7 toderivetheasymptoticrelativeefciencies.Noteworthy,isthefactthatundertheLaplacescorefunctionthesigned-rankversiontestisequivalenttothesigntest.Viaformulation,theAREsinequation( 5 )arefreeofthescaleparameter,soforthesakeofargumentwetake=1.Consequently,as'fL()=sgn(),equation( 5 )canbeexpressedbyI(fL,g)=Z0)]TJ /F4 11.955 Tf 9.3 0 Td[('g(x)g(x)dx+Z10'g(x)g(x)dx=Z0_g(x)dx)]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z10_g(x)dx=2g(0), andasI(fL)=1,weprocurethatAREg(Te(n)fL(^#,)=T(n)sign())=1. AsmentionedinSection 5.2.1 thesigned-rankversiontestwithaspeciedasymmetryof=0isequivalenttoasigned-rankteststatisticandspecicallyforthelogisticscoredensityfunctiontothetheWilcoxonsigned-rankteststatistic.ThesquaredefcacyoftheWilcoxonsigned-rankteststatisticisknownthewellknow 83

PAGE 84

quantityK2W+=12)]TJ /F9 7.97 Tf 6.59 0 Td[(2Zg2(s)ds2. Thet-testisasymptoticallyequivalenttothestandardnormalz-test,sowederivetheefcacyforthelatter.Byequation(10)of Arellano-Valleetal. ( 2005 ),E(X)=+21andVar(X)=2[(1+32)2)]TJ /F5 11.955 Tf 12.42 0 Td[(4(1)2]2,wherek=Rjxjkg(x)dxaretheabsolutemomentswithrespecttodensityfunctiong.Subsequently,n()=+21 p n[(1+32)2)]TJ /F5 11.955 Tf 11.96 0 Td[(4(1)2]1=2, andinasmuch@n() @=0=p n [(1+32)2)]TJ /F5 11.955 Tf 11.96 0 Td[(4(1)2]1=2. Thevarianceofthez-teststatisticisoneunderthenullhypothesisandthereforeK2t=)]TJ /F9 7.97 Tf 6.59 0 Td[(2[(1+32)2)]TJ /F5 11.955 Tf 11.95 0 Td[(4(1)2])]TJ /F9 7.97 Tf 6.59 0 Td[(1. Thez-test(ort-test)onlybecomesrelevantfor=0asitisonlythenthatthetwotests,testthesameparameter.Thisallowsfortheformulationofthefollowingproposition. Proposition5.7. Letf2F.Then,theasymptoticrelativeefciencies,underg2Fefofthesigned-rankversiontestbasedonTe(n)f(^#,)withrespecttothesigntest,andfor=0totheWilcoxonsigned-ranktestandthet-testare AREg(Te(n)f(^#,)=T(n)sign())=1 2(1)]TJ /F4 11.955 Tf 11.96 0 Td[(2)I2(f,g) I(f) 4 2(1)]TJ /F4 11.955 Tf 11.96 0 Td[(2)g2(0)=I2(f,g) 4g2(0)I(f), (5) 84

PAGE 85

AREg(Te(n)f(^#,0)=T(n)W+(0))=)]TJ /F9 7.97 Tf 6.58 0 Td[(2I2(f,g) I(f) )]TJ /F9 7.97 Tf 6.59 0 Td[(212Zg2(s)ds2=I2(f,g) 12I(f)Rg2(s)ds2. andAREg(Te(n)f(^#,0)=T(n)t(0))=)]TJ /F9 7.97 Tf 6.59 0 Td[(2I2(f,g) I(f) )]TJ /F9 7.97 Tf 6.58 0 Td[(2[(1+3)2])]TJ /F9 7.97 Tf 6.58 0 Td[(1=I2(f,g)42 I(f), NotethattheAREfunctionforthesigned-rankversiontestcomparedtothesigntestisfreeof.NumericalvaluesofthoseAREsundert4.5,t6.5,t8,t10,t20,normal,power-exponential,logisticandLaplacedistributionsaredisplayedinTable 5-1 .Thesigned-rankversionoutperformsthesigntestacrosswhenthescoredensityfunctionisheaviertailedorapproximatelyequivalenttothetruedensityfunction.Itdoesnotfairaswellwhenthescoredensityfunctionislightertailedthanthetruedensityfunction. ThenumericalvaluesoftheAREsforcomparisonofthesigned-rankversiontestwithrespecttotheWilcoxonsigned-ranktestarepresentedonceagainonlyfor=0inTable 5-3 .Mostnotably,thesigned-rankversiontestperformswell,mostnotablyforthescoredensityfunctionoft10andalsowhenthetruedensityisalightertailedpowerexponentialtotheaheaviertailedpowerexponentialscoredensityfunction.Howeverwhenalightertailedpowerexponentialscoredensityfunctionisimplementedwheninfactthetruedensityisaheaviertdistributionthesigned-rankversionlackstoattainthesamepowerastheWilcoxonsignedranktestandperformstheworst.Also,itappearsthatthefortheLaplacescoredensityfunctionthesigned-rankversiontestlacksperformanceexceptwhenitcorrectlyspeciestheLaplacedistribution. 85

PAGE 86

Finally,thenumericalvaluesoftheAREsforcomparisonofthesigned-rankversiontestwithrespecttothet-testarepresentedonlyfor=0.Thesigned-rankversiontestmarginallyoutperformsthet-testwhenboththescoreandtruedensityfunctionsbelongtothet-family.Thesigned-rankversiontestisalsoprominentwhenboththedensityfunctionsareinthelighterthannormal,powerexponentialfamily.UnlikeTable 5-1 ,itdoesnotperformwellwhenat-densityfunctionisusedforthescoreandapowerexponentialwithshapegreaterthat2isthetruedensity.Itisnoteworthythough,thatwhenthenormaldensityisusedasascorefunctiontheAREsaregreaterorequalto1acrosstheboard.Thisisdirectresultfrom Chernoff&Savage ( 1958 )wherebytheefciencyofnormalscoresproceduresinneverlessthan1whensamplingfromasymmetricdistribution. 5.3.2UnspeciedAsymmetry FromProposition 5.6 theefcacyforthesigned-rankversiontestiseasilydeductedbutacomparisontootherstestsisnotincludedforlackofcompetingtests.Theefcacyforwhen=0.6isprovidedinTable 5-4 5.4ConsistentEstimationOfParameters Consistentestimationofandisrequiredforthesigned-rankversiontest. Mudholkar&Hutson ( 2000 )developconsistentestimatorsfor(,,)bymethodofmoments,maximumlikelihoodandbayesianinferencefortheGaussiandistribution. Arellano-Valleetal. ( 2005 )extendthemethodofmomentsforanyarbitrarysymmetricdensityinFwithnitefourthmomentandusethebroaderclassofthepowerexponentialfamilytodevelopmaximumlikelihoodestimatorsandtheaccompanyingasymptoticnormalitytheory,inclusiveofthecasewithunknownshapeparameter. Maximumlikelihoodestimationismoreefcientthanthemethodofmomentsbutbothrequiretheuseorspecicationofaparametricfamily.FromTable 5-1 andTable 5-2 ,wenoticethatwhenat-distributionisusedforthescorefunction,theAREsforthesigned-rankversiontestarecompetitiveacrosstheboard.Thismakes 86

PAGE 87

thet-distributionahardycandidateforthescorefunction. Lucas ( 1996 ,chap.3)derivescertainrobustnesspropertiesofthet-distributionbasedonpseudomaximumlikelihoodestimatorsforthet-distribution(MLT)thatareobtainedviaM-estimation,andconsequentlyillustratesthatasthedegreesoffreedomincrease(whetherxedorestimated)theMLTestimatorsbecomelessefcient.Hence,weproposeusingthemaximumlikelihoodestimatesforandbasedonat-distributionwithunspecieddegreesoffreedom,,andif^>50,thenreverttomaximumlikelihoodestimatesbasedonthepowerexponentialfamily(whichincludestheGaussianasaspecialcase). 5.5PracticalImplementationAndSimulationResults 5.5.1SpeciedAsymmetry Forthenite-sampleexaminationofthepowerandsignicancelevelofthesigned-rankversiontestwithspeciedasymmetryparameter,wegeneratedN=5,000independentsamplesofsizen=200fromtheskewedt-distributionwith6.5and10degreesoffreedom,theskewednormalandtheskewedpowerexponentialwithshapeparameter4.Foreachdistributionwespeciedtheasymmetryparameterto=0,0.3,and0.6,andperformedatestoftwo-sidedalternativetothenullhypothesisof=0.Thedistributionsweregeneratedwithscaleparameter=1andlocationparameters=0,0.1,0.2and0.3Thesigned-rankversiontestwascomparedtotheclassicalsigntestforgeneralandtheWilcoxonsigned-ranktestfor=0. FromthediscussioninSection 5.4 ,practicalchoiceofascoredensityfunctionisadensityfunctionfromtheStudentt-family.Anoperativeandpragmaticchoiceisthet20densityfunctionthatisheaviertailedthanthenormalbutnottotheextremewhereitceasestohavenite(rstsix)moments.TherejectionfrequenciesarelistedinTable 5-5 .ThesefrequenciesareverysimilartothefrequencieslistedinTables 5-6 5-7 and 5-8 ,thatwereobtainedfromtheimplementationofateststatisticthatadaptsthechoiceofthescoredensityfunctionbasedondensityfucntionchosenintheMLEprocess.The 87

PAGE 88

adaptivemethodisusedsimplyforconsistencywiththeprocedureimplementedfortheunspeciedasymmetrycaseinSection 5.5.2 IntheMLEprocess,thedefaultchoiceisat-distributionwithestimateddegreesoffreedom^,butif^>50apowerexponentialdensityisadoptedwithestimatedshapeparameter^,with=2yieldingthenormaldistribution.Motivationbehindthiswasatwostepprocess.Firstly,theAREvaluesinTable 5-1 whereascoret-densityfunctionexcelledthesigned-ranktestmotivatedtheuseoft-distribution.Secondly,thefactthattheMLE/MLTestimatesarelessefcientwhen^>50,motivatedtheswitchtoapowerexponentialdistribution.Finally,ifapowerexponentialscoredensityfunctionwaschosenaftertheMLEprocessthepowerwasconstrainedtominf^,2.3g.Thegoalwastoproduceapracticalandsomewhathardyprocedurethatdidnotuseaverylighttailedpowerexponentialscoredensityfunction. TheresultsarelistedinTables 5-6 5-7 and 5-8 andcomparedtotheresultsoftheclassicalsigntest.For=0.4bothtestshavearejectionfrequencyof1,notingthat=1.However,for=0.1,0.2and0.3thesigned-rankversionhasahigherrejectionfrequency.Thosefrequenciesincreaseasthedistributionmovesfurtherfromsymmetryandastheunderlyingdistributionbecomeslightertailed,afactthatcorroboratestheAREsofTable 5-1 .Thesigned-testmaintainsthe5%signicancelevelwhilethesigned-rankversiontestappearstohaveslightlyhigherlevelasincreases.TheculpritistheestimationofandtheassociatedoptimizationintheMLEestimation. 5.5.2UnspeciedAsymmetry Fortheunspeciedasymmetryscenario,thesamplesizewasincreasedton=500toaccommodatetheadditionalestimationof.Thesigned-rankversiontestTe(n)frequirestheconsistentestimationofJ(f,g)andM(f,g)fortheconsistentestimationofe(f,g).Asigned-rankversiontestwithscoredensityfunctionft20yieldstherejectionfrequencieslistedinTable 5-9 .Ateststatisticwithxedscoredensityfunctiononlyappearstohavedecentpowerandlevelsonlywhenthetruedistributionisinsome 88

PAGE 89

neighborhoodinrelationtothescoredensityfunction.Forexample,whenthetruedensityfunctionislightertailed,suchasthet6.5,thelevelsarehigherthatthetheoretical5%,whilewhen=0,therejectionfrequenciesfordatafromthepowerexponentialdistributionwithpower=4aretoolowandthelevelistoohighfor=0.6.Onlytheforthenormaldistributiondothepowersandlevelsappeartobeincheck.Hence,theadaptivemethodmentionedearlierwasemployed.Inaddition,weproposesettinge(n)(f;)inthesigned-rankversiontesttoe(f).Thesequenceofalternativelocationparameterswassetto0,0.1,0.3and0.5andtherejectionfrequenciesarelistedinTables 5-10 5-11 and 5-12 forsetat0,0.3and0.6respectively. Fordataderivedfromthelighttailedpowerexponentialfamilywith=4,wenotethatthesignicancelevelsareslightlyabovethetheoretical5%level,whichbecomesevidentasincreases(seeinTable 5-12 ).Thedensityfunctionofthesigned-rankversionteststatisticfor=0,=1,and=0.6fromthe5,000replicationsisskewedrightasshowninFigure 5-1 .Thecausearisesfromtheissuesrelatedtotheconsistentestimationof.TheasymmetryparameterisobtainedthroughthejointM-estimationoftheparameters,,andeitherordependingonthechoiceofthescoredensityfunction,andassuchthestandarderrorforisapproximately0.087forthissimulationscheme.Thesigned-rankversionteststatisticbecomessensitivetodeviationsinforlighttaileddistributionsassmalluctuationsin^canshifttheperceivedlocation,^signicantly.Goingbehindthescenesofthesimulationforthepowerexponentialdistribution,thesmallestvalueof^=0.34resultedinacorrespondingteststatisticvalueof6.36,whilethelargestvalueof^=0.99,accruedtheminimumteststatisticvalueof)]TJ /F5 11.955 Tf 9.3 0 Td[(3.19.Inaddition,thelargestteststatisticvalueof7.08,transpiredfromacorresponding^=0.39. Increasingthesamplesizeton=4,000forthepowerexponentialdistributionwith=4,forsimulationwith=0.6,yieldsrejectionfrequenciesof0.0456,0.6390,1and1for=0,0.1,0.3and0.5respectively.Thus,conrmingtheasymptotictheory. 89

PAGE 90

InSection4.5of Cassartetal. ( 2008 )methodologyfortheconsistentestimationofthecross-informationquantitieswasdeveloped(seeAppendix H )andwasextendedtotheschemeathand.Although,e(n)(f;)isfreeof,theoptimizationprocedurerequiredconsistentestimatesofandtoestimateJ(f,g)andM(f,g).Usinge(n)(f;)obtainedviathisprocess,createdasigned-rankversiontestthat,throughthesimulation,hadastandarderrorthatwasnotclosetothetheoreticalvalueof1.Valuesvariedfrom0.5to3.Onceagaintheculpritwasthedeviationinwhichhadacompoundingeffect. Table5-1. AREsundervariousdensitiesofthesigned-rankversiontestwithrespecttothesigntestforspecied Scorefg ft4.5ft6.5ft8ft10ft20fPE4fPE6fPE10fLogfL ft4.51.28651.32911.34651.36111.38881.41401.76941.82591.77751.31800.7780ft6.51.27681.33931.36621.38971.43691.48422.05032.25012.37451.33200.7474ft81.26601.33721.36831.39591.45231.51022.17732.45362.68001.33210.7315ft101.25281.33161.36651.39771.46241.53012.29072.64172.97361.32870.7161ft201.21441.30801.35071.38931.47131.56032.52233.04873.64661.31050.68041.15811.26551.31561.36151.46141.57082.75283.49004.43921.27460.6373fPE40.68320.82410.89410.96081.11371.29773.33225.05557.70430.85030.3006fPE60.44860.57550.64120.70520.85671.04703.21735.23608.79390.60790.1914fPE100.24990.34760.40080.45420.58640.76212.80585.03249.14980.38220.1097fLog1.27181.33791.36701.39281.44611.50162.12502.38732.62271.33330.7507fL1.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000 90

PAGE 91

Table5-2. AREsundervariousdensitiesofthesigned-rankversiontestwithrespecttothet-testforspecied=0 Scorefg ft4.5ft6.5ft8ft10ft20fPE4fPE6fPE10fLogfL ft4.51.32041.13211.07391.03040.95810.90020.72790.67550.61761.08401.5560ft6.51.31041.14071.08961.05200.99130.94490.84350.83250.82501.09551.4947ft81.29931.13891.09131.05671.00190.96140.89570.90780.93111.09561.4630ft101.28581.13411.08991.05811.00890.97410.94240.97741.03311.09281.4323ft201.24641.11411.07721.05171.01500.99331.03771.12791.26701.07791.36071.18861.07781.04921.03071.00831.00001.13251.29121.54231.04831.2747fPE40.70110.70190.71310.72740.76830.82611.37081.87042.67680.69940.6012fPE60.46040.49020.51140.53390.59100.66651.32361.93723.05530.50000.3828fPE100.25650.29600.31960.34390.40450.48521.15431.86193.17900.31430.2194fLog1.30521.13961.09021.05440.99770.95600.87420.88320.91121.09661.5015fL1.02720.85250.79830.75770.69060.63730.41200.37080.34880.82332.0000 Table5-3. AREsundervariousdensitiesofthesigned-rankversiontestwithrespecttotheWilcoxonsigned-rankforspecied=0 Scorefg ft4.5ft6.5ft8ft10ft20fPE4fPE6fPE10fLogfL ft4.51.01230.99410.98570.97800.96120.94270.83410.76680.68060.98851.0373ft6.51.00461.00171.00010.99850.99460.98950.96650.94500.90920.99900.9965ft80.99611.00011.00171.00301.00521.00681.02641.03051.02620.99900.9753ft100.98570.99601.00041.00431.01221.02011.07981.10951.13860.99650.9549ft200.95550.97830.98880.99821.01831.04021.18901.28041.39630.98290.90720.91120.94650.96310.97831.01151.04721.29771.46571.69980.95600.8498fPE40.53750.61640.65450.69040.77080.86511.57082.12322.95000.63770.4008fPE60.35300.43050.46940.50670.59290.69801.51662.19903.36720.45590.2552fPE100.19670.26000.29340.32640.40590.50811.32272.11353.50340.28660.1463fLog1.00001.00001.00001.00001.00001.00001.00001.00001.00001.00001.0000fL0.78750.74860.73270.71920.69280.66740.47210.42090.38440.75071.3333 91

PAGE 92

Table5-4. Efcaciesundervariousdensitiesofthesigned-rankversiontestforunspecied,computedwhen=0.6 Scorefg ft4.5ft6.5ft8ft10ft20fPE4fPE6fPE10fLogfL ft4.50.27480.26260.25650.25080.23830.22430.07340.03400.01170.11290.5460ft6.50.27350.26370.25860.25370.24250.22960.07970.03830.01380.11380.5298ft80.27210.26350.25890.25430.24390.23160.08240.04020.01470.11390.5218ft100.27030.26280.25870.25450.24480.23310.08480.04190.01560.11380.5142ft200.26500.26000.25690.25360.24560.23540.08970.04550.01760.11310.49670.25690.25470.25280.25060.24460.23620.09460.04930.01970.11150.4755fPE40.14880.15930.16360.16700.17310.17740.12570.08620.04480.07140.1816fPE60.09100.10150.10610.11010.11800.12530.11660.09300.05770.04610.0970fPE100.04520.05290.05640.05960.06640.07330.08890.08400.06410.02440.0435fLog0.27170.26310.25850.25410.24370.23160.08060.03880.01400.11400.5314fL0.19180.17850.17270.16750.15690.14600.03110.01260.00400.07750.7812 Table5-5. Rejectionfrequencies,withspeciedasymmetry=0,0.3and0.6(forrst,secondandthirdrowrespectively),ofthesigned-rankversiontestwithxedscoredensityfunctionft20 Dist.00.10.20.3 t6.50.04880.23660.69840.96140.05700.28500.74740.97620.06060.40300.89120.9944t100.05080.26280.72400.96920.05240.29140.79380.98460.05460.42300.92460.99860.04920.28480.80700.98440.04540.33540.85780.99440.05480.48120.95880.9998PE40.05100.41220.93460.99940.05240.48340.96741.00000.05680.68260.99841.0000 92

PAGE 93

Table5-6. Rejectionfrequencies,withspeciedasymmetry=0,ofthesigned-rankversion,signtestandWilcoxonsigned-ranktest Dist.Test00.10.20.3 t6.5Te(n)f0.04880.24700.71240.9662Sign0.05820.20880.61120.9036Wilcoxon0.04620.24040.70780.9606t10Te(n)f0.05340.26720.73060.9722Sign0.05800.21160.60380.9146Wilcoxon0.04940.26020.71630.9694Te(n)f0.05240.28920.80840.9854Sign0.05640.21320.63920.9260Wilcoxon0.05240.27200.79300.9844PE4Te(n)f0.04960.46440.95900.9998Sign0.06080.21480.62060.9240Wilcoxon0.05020.35820.89880.9984 Table5-7. Rejectionfrequencies,withspeciedasymmetry=0.3,ofthesigned-rankversionandsigntest Dist.Test00.10.20.3 t6.5Te(n)f0.05660.28940.76560.9802Sign0.04700.18480.60040.9290t10Te(n)f0.05480.29720.79740.9852Sign0.04600.18980.62680.9342Te(n)f0.04480.34340.85900.9936Sign0.04760.19080.63900.9444PE4Te(n)f0.05340.54380.98261.0000Sign0.04640.19660.63540.9510 93

PAGE 94

Table5-8. Rejectionfrequencies,withspeciedasymmetry=0.6,ofthesigned-rankversionandsigntest Dist.Test00.10.20.3 t6.5Te(n)f0.06110.40780.90440.9982Sign0.04440.23680.75960.9836t10Te(n)f0.05520.42640.92460.9988Sign0.04410.22860.77480.9872Te(n)f0.05760.49020.95920.9998Sign0.04360.23780.79220.9940PE4Te(n)f0.05680.74301.00001.0000Sign0.04300.24260.80860.9962 Table5-9. Rejectionfrequencies,withunspeciedasymmetry=0,0.3and0.6(forrst,secondandthirdrowrespectively),ofthesigned-rankversiontestwithxedscoredensityfunctionft20 Dist.00.10.30.5 t6.50.10200.26280.88120.99880.10220.28820.92700.99960.09620.37980.98161.0000t100.08560.20860.84860.99740.07620.27280.90920.99920.07740.37240.97901.00000.04060.12820.75880.99080.03560.21220.86820.99900.05920.32660.96960.9998PE40.02100.04320.38180.88940.04600.13980.64480.97640.16210.31540.89960.9972 94

PAGE 95

Table5-10. Rejectionfrequencies,withunspeciedasymmetry,ofthesigned-rankversiontest.Simulatedwith=0 Dist.00.10.30.5 t6.50.05560.17720.81460.9970t100.05710.17180.81460.99580.05960.19880.83020.9956PE40.06560.13780.60860.9574 Table5-11. Rejectionfrequencies,withunspeciedasymmetry,ofthesigned-rankversiontest.Simulatedwith=0.3 Dist.00.10.30.5 t6.50.05360.21300.89160.9994t100.05400.23840.89080.99920.05990.26480.90100.9992PE40.08560.23540.77060.9880 Table5-12. Rejectionfrequencies,withunspeciedasymmetry,ofthesigned-rankversiontest.Simulatedwith=0.6 Dist.00.10.30.5 t6.50.05580.31880.97221.0000t100.05540.34120.97661.00000.06170.37740.97680.9998PE40.13940.36240.93420.9978 95

PAGE 96

Figure5-1. Densityfunctionofsigned-rankversiontestforunspeciedasymmetrywith=0,=1and=0.6forthepowerexponentialdistributionwith=4 96

PAGE 97

CHAPTER6SUMMARYANDCONCLUSIONS InChapter 2 adistortionmodelwaspresentedthattookasymmetricprobabilitydensityfunctionandskeweditinnitesimallyinaspecicdirectiontherebydistortingeveryrealizationofthedistribution.RobustnesstodistortionofasymmetricsignalwasdeterminedastherateofchangeofthefunctionalofanafneequivariantlocationestimatorundertheFechnermodel.Forafne-equivariantestimatorsoflocationthedistortionsensitivitywasshowntobeidenticalundertheUniformdistributionasthedistortionproducesalocationshiftwhichyieldsthesamederivativeforallafneequivariantfunctionalsT().Generally,whentheunderlyingdistributionfunctionsecond-orderstochasticallydominatestheuniformdistributionfunction,themedianhasthesmallestdistortionsensitivityasstatedinProposition 2.4 fortheunivariatecaseandwithafewadditionalassumptions,showthistobetrueforthemultivariatecaseaswell(seeProposition 2.5 ). Cassartetal. ( 2008 )proposealocallyoptimaltestfortestingasymmetryandcompareittoseveraltraditionaltestsincludingthetriplestestof Randlesetal. ( 1980 ).ThetriplestestwasshowntobecompetitiveinMonteCarlosimulationsandweillustratedinChapter 4 thattriplestestiscomparativelyequivalenttothesigned-rankversiontestunderthetscoredensityfunctions,withAREvaluesapproximately1.Thetriplestestefcaciesdonotperformaswellforlightertaileddistributionsbutisacontendingtestespeciallyforheaviertaileddistributions. InChapter 5 themethodologyof Cassartetal. ( 2008 )isanalogouslypresentedforlocallyoptimaltestsoflocationthatarerobusttoasymmetryintheFechnerclass.TheFechnerclassisnotaimedtobeanaccuraterepresentationofrealitybutmoreofaconvenienttoolintheconstructionofalocallyoptimaltest.Parametricandmoreinterestingly,semiparametricsigned-rankversionswerecreatedforthecasesofspeciedandunspeciedasymmetry.Forspeciedasymmetrythesigned-rankversion 97

PAGE 98

testbecomesanextensionofthelinear-scoresigned-ranktestwhen6=0.Thetestwasthencomparedtothesigntest,theWilcoxonsigned-ranktestandthet-testintermsofefciencyandpower.TheAREsofthesigned-rankversiontesttothesigntestarefreeoftheasymmetryparameterandwasshowntobeequivalentfortheLaplacescoredensityfunction.Table 5-1 illustratesthatthesigned-rankversionoutperformsthesigntestwhenthescoredensityfunctionissignicantlylightertailedthantruedensityfunctionandasdeviatesawayfrom0.Asthesigned-rankversiontestperformswellwitht-scoredensityfunctionsandforpracticalimplementation,weusedanadaptivetestingmechanismthatusedat-scoredensityfunctionorapowerexponentialscoredensityfunctioniftheestimateddegreesoffreedomweregreaterthat50.TheMonteCarlosimulationscorroboratetheconclusions,asthesigned-ranktestreachesapowerof1fasterthanthesigntestwhenthetruedistributionislightertailedandthepoweralsoincreasesasdeviatesawayfrom0.BothfromtheAREvaluesandtherejectionfrequencieswhenthesigned-rankversiontestwascomparedtotheWilcoxonsigned-ranktestillustratethatfor=0theyareperformonequallevelswithequivalenceforthelogisticscoredensityfunction. Inthecaseofunspeciedasymmetry,thereisnocompetitortothetestbuttheefcaciesandpowercalculationsfollowsimilartendenciestothespeciedasymmetry.Forthespeciedasymmetrythepowerapproaches1whenthetruelocationparameterdiffersmorethan0.3standarddeviationsfromthenullhypothesisvalue,whilefortheunspeciedasymmetrycasethisholdsforabout0.5standarddeviations. 98

PAGE 99

APPENDIXADISTORTIONSENSITIVITYOFHODGES-LEHMANN UndertheFechnermodel,afunctionalrepresentationfortheHodges-Lehmannestimatoris1 2=H(T), whereH()isthedistributionfunctionof(X1+X2)=2fori.i.d.randomvariablesX1andX2withdistributionfunctionH.Withoutlossofgeneralitywewillassumethat>0whichimpliesT>0,asHisanevenfunctionin.Thefunctionalrepresentationcanthenbeexpandedto1 2=H(T)=Z1H(2T)]TJ /F3 11.955 Tf 11.95 0 Td[(t)dH(t)=Z0H(2T)]TJ /F3 11.955 Tf 11.95 0 Td[(t)1 ft (1)]TJ /F4 11.955 Tf 11.96 0 Td[()dt+Z10H(2T)]TJ /F3 11.955 Tf 11.95 0 Td[(t)1 ft (1+)dt=Z0(1)]TJ /F4 11.955 Tf 11.95 0 Td[() 2+Z2T)]TJ /F6 7.97 Tf 6.59 0 Td[(t01 fq (1+)dq1 ft (1)]TJ /F4 11.955 Tf 11.95 0 Td[()dt+Z2T0(1)]TJ /F4 11.955 Tf 11.96 0 Td[() 2+Z2T)]TJ /F6 7.97 Tf 6.58 0 Td[(t01 fq (1+)dq1 ft (1+)dt+Z12TZ2T)]TJ /F6 7.97 Tf 6.59 0 Td[(t1 fq (1)]TJ /F4 11.955 Tf 11.95 0 Td[()dq1 ft (1+)dt. Byapplyingasimpletransformationofvariables,weobtain1 2=(1)]TJ /F4 11.955 Tf 11.96 0 Td[()2 2Z0dF(y)+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()(1+)Z0Z2T)]TJ /F7 5.978 Tf 5.75 0 Td[((1)]TJ /F14 5.978 Tf 5.75 0 Td[()y (1+)0dF(x)dF(y)+Z2T (1+)0(1)]TJ /F4 11.955 Tf 11.95 0 Td[()(1+) 2dF(y)+Z2T (1+)0(1+)2Z2T)]TJ /F7 5.978 Tf 5.76 0 Td[((1+)y (1+)0dF(x)dF(y)+Z02T (1+)Z2T)]TJ /F7 5.978 Tf 5.75 0 Td[((1+)y (1)]TJ /F14 5.978 Tf 5.76 0 Td[()(1)]TJ /F4 11.955 Tf 11.95 0 Td[()(1+)dF(x)dF(y), 99

PAGE 100

orequivalently1 2=(1)]TJ /F4 11.955 Tf 11.96 0 Td[()2 4+(1)]TJ /F4 11.955 Tf 11.95 0 Td[()(1+)Z0F2T)]TJ /F5 11.955 Tf 11.96 0 Td[((1)]TJ /F4 11.955 Tf 11.96 0 Td[()y (1+))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2dF(y)+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()(1+) 2F2T (1+))]TJ /F5 11.955 Tf 13.16 8.09 Td[(1 2+(1+)2Z2T (1+)0F2T)]TJ /F5 11.955 Tf 11.95 0 Td[((1+)y (1+))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 2dF(y)+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()(1+)Z12T (1+)F2T)]TJ /F5 11.955 Tf 11.95 0 Td[((1+)y (1)]TJ /F4 11.955 Tf 11.95 0 Td[()dF(y). Furtherexpansiongives1 2=(1)]TJ /F4 11.955 Tf 11.95 0 Td[()2)]TJ /F5 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[()(1+) 4+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()(1+)Z0F2T)]TJ /F5 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[()y (1+)dF(y)+(1)]TJ /F4 11.955 Tf 11.95 0 Td[()(1+) 4+(1)]TJ /F4 11.955 Tf 11.96 0 Td[()(1+) 2F2T (1+)+(1+)2 2F2T (1+))]TJ /F5 11.955 Tf 13.15 8.08 Td[(1 2+(1+)2Z2T (1+)0F2T)]TJ /F5 11.955 Tf 11.96 0 Td[((1+)y (1+)dF(y)+(1)]TJ /F4 11.955 Tf 11.95 0 Td[()(1+)Z12T (1+)F2T)]TJ /F5 11.955 Tf 11.96 0 Td[((1+)y (1)]TJ /F4 11.955 Tf 11.96 0 Td[()dF(y). Takingthederivativewithrespecttoandevaluationtheexpressionat=0yields0=)]TJ /F5 11.955 Tf 10.49 8.08 Td[(1 2+Z10f()]TJ /F3 11.955 Tf 9.3 0 Td[(y)2@T @=0)]TJ /F5 11.955 Tf 11.95 0 Td[(2y dF(y)+Z0f()]TJ /F3 11.955 Tf 9.3 0 Td[(y)2@T @=0+2y dF(y). Then,bysymmetryoff,2@T @=0 Z10f(y)dF(y)+Z0f(y)dF(y)=1 2+Z102yf(y)dF(y))]TJ /F15 11.955 Tf 9.33 16.27 Td[(Z2yf(y)dF(y), andthus,@T @=0=1 8+Z10yf(y)dF(y) Z10f(y)dF(y)=1 2+Z10xg(y)dG(y) Z10g(y)dG(y), 100

PAGE 101

wheregisthedensityfunctiongivenin( 2 )andGtheassociatedcumulativedistributionfunction. 101

PAGE 102

APPENDIXBDETAILSOFEXAMPLE2.1 Itisnecessarytoshowthat Za0P(X>t)dt>Za0P(U>t)dt,8a>0,(B) whereXGsandUUniform(0,1).Howeveritissufcienttoshowthatthisinequalityholdsforalla2(0,1),asRb0P(U>t)dt=1=2forallb1.Therefore,fora2(0,1),Za0P(U>t)dt=a1)]TJ /F3 11.955 Tf 13.25 8.09 Td[(a 2, andZa0P(X>t)dt=Za01)]TJ /F5 11.955 Tf 13.15 8.09 Td[(1 8erft=k+0.2 0.1p 2)]TJ /F3 11.955 Tf 11.96 0 Td[(erf0.2 0.1p 2)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 4erft=k 2p 2)]TJ /F5 11.955 Tf 10.49 8.09 Td[(1 8erft=k)]TJ /F5 11.955 Tf 11.96 0 Td[(0.2 0.1p 2)]TJ /F3 11.955 Tf 11.96 0 Td[(erf)]TJ /F5 11.955 Tf 9.3 0 Td[(0.2 0.1p 2dt, wherek=(sqrt2(e2+20)e)]TJ /F9 7.97 Tf 6.59 0 Td[(2)=(4p )anderf()istheerrorfunction.Hence( B )isequivalenttoZa0P(X>t)dt)]TJ /F3 11.955 Tf 11.95 0 Td[(a1)]TJ /F3 11.955 Tf 13.25 8.09 Td[(a 2>0,8a2(0,1). Nextwehavethattherstandsecondorderderivativesared daZa0P(X>t)dt)]TJ /F3 11.955 Tf 11.96 0 Td[(a1)]TJ /F3 11.955 Tf 13.25 8.09 Td[(a 2=P(X>a))]TJ /F5 11.955 Tf 11.96 0 Td[(1+2a, andd2 da2Za0P(X>t)dt)]TJ /F3 11.955 Tf 11.96 0 Td[(a1)]TJ /F3 11.955 Tf 13.24 8.08 Td[(a 2=)]TJ /F3 11.955 Tf 9.3 0 Td[(gs(a)+2. NotethatRa0P(X>t)dt)]TJ /F3 11.955 Tf 12.4 0 Td[(a)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F6 7.97 Tf 13.23 4.71 Td[(a 2=0ata=0andtherstorderderivativeis0ata=0andpositivefora2(0,1),asseeninFigure B-1 .Thesecondorderderivativeis1ata=0.Therefore,therandomvariableXwithdensityfunctiongs,secondorder 102

PAGE 103

stochasticallydominatesUandhasanexpectedvalueofE[X]=Z10P(X>t)dt0.6445. FigureB-1. Plotoffunction,P(X>a))]TJ /F5 11.955 Tf 11.95 0 Td[(1+2afora2(0,1). 103

PAGE 104

APPENDIXCDISTORTIONSENSITIVITYOFM-ESTIMATORS ThefunctionalformofaM-estimatorwithunderlyingdensityFfromtheFechnerfamilyis Z ((x)]TJ /F3 11.955 Tf 11.96 0 Td[(T)=)dF(x)=0.(C) where isameasurablefunction.Equation( C )canbedividedintothetwotermsZ0 ((x)]TJ /F3 11.955 Tf 11.95 0 Td[(T)=)1 fx (1)]TJ /F4 11.955 Tf 11.96 0 Td[()dx+Z10 ((x)]TJ /F3 11.955 Tf 11.95 0 Td[(T))1 fx (1+)dx=0. Thersttermcanbere-expressedas(1)]TJ /F4 11.955 Tf 11.95 0 Td[()Z0 ((1)]TJ /F4 11.955 Tf 11.95 0 Td[()y)]TJ /F3 11.955 Tf 11.96 0 Td[(T=)f(y)dy, andthesecondas(1+)Z10 ((1+)y)]TJ /F3 11.955 Tf 11.96 0 Td[(T=)f(y)dy. Next,bytakingthederivativewithrespecttoandevaluatingthetermat=0,wehave@T @=0=)]TJ /F15 11.955 Tf 11.29 9.63 Td[(R0 (y)dF(y))]TJ /F15 11.955 Tf 11.96 9.63 Td[(R0y 0(y)dF(y)+R10 0(y)dF(y)+R10y (y)dF(y) R11 0(y)dF(y) andif isanoddfunction(whichisthecaseforM-estimators),@T @=0=Z10[ (y)+y 0(y)]dF(y) Z10 0(y)dF(y)=Z[ (y)+x 0(y)]dG(y) Z 0(y)dG(y). 104

PAGE 105

APPENDIXDDISTORTIONSENSITIVITYOFL-ESTIMATORS ThefunctionalformofaL-estimatorwithunderlyingdensityFfromtheFechnerfamilyis T=Zxh(F(x))dF(x) Zh(F(x))dF(x).(D) Thederivativewithrespecttoofthenumeratorofthefunctionalform(equation( D ))andthenevaluatedat=0manifests)]TJ /F15 11.955 Tf 11.29 16.27 Td[(Z01xh0(F(x))F(x)dF(x))]TJ /F5 11.955 Tf 11.96 0 Td[(2Z01xh(F(x))dF(x)+Z10xh0(F(x))()]TJ /F5 11.955 Tf 9.29 0 Td[(1+F(x))dF(x)+2Z10xh(F(x))dF(x). (D) Assumingthathissymmetric(asFisasymmetricdensity)aboutitsargumentof1=2wehavethat,fort2[0,1],)]TJ /F3 11.955 Tf 9.3 0 Td[(h0(F()]TJ /F3 11.955 Tf 9.3 0 Td[(t))=h0(F(t))andh(F()]TJ /F3 11.955 Tf 9.3 0 Td[(t))=h(F(t)).Henceequation( D )canbere-expressedas2Z10xh0(F(x))F(x)dF(x))]TJ /F5 11.955 Tf 11.96 0 Td[(2Z10xh0(F(x))dF(x)+4Z10xh(F(x))dF(x). Similarly,forthedenominatorofequation( D )wehave)]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z0h0(F(x))F(x)dF(x))]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z0h(F(x))dF(x)+Z10h0(F(x))()]TJ /F5 11.955 Tf 9.3 0 Td[(1+F(x))dF(x)+Z10h(F(x))dF(x), whichunderthesameassumptionsonhisZ10h0(F(x))(1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(x))dF(x))]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z10h(F(x))dF(x))]TJ /F15 11.955 Tf 11.96 16.27 Td[(Z10h0(F(x))(1)]TJ /F3 11.955 Tf 11.95 0 Td[(F(x))dF(x)+Z10h(F(x))dF(x). Therefore,thederivativewithrespecttoofthedenominatorofequation( D )evaluatedat=0isequalto0. 105

PAGE 106

Thedistortionsensitivity,byvirtueofthequotientruleappliedtoequation( D )isDSRL=2Z10x[2h(F(x)))]TJ /F3 11.955 Tf 11.96 0 Td[(h0(F(x))(1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(x))]dF(x) Z1h(F(x))dF(x). 106

PAGE 107

APPENDIXEDISTORTIONSENSITIVITYOFR-ESTIMATORS TheimplicitfunctionalrepresentationofaR-estimatorTwithunderlyingdensityFfromtheFechnerfamily,is Z1 2F(x)+1 2(1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(2T)]TJ /F3 11.955 Tf 11.95 0 Td[(x))dF(x)=0.(E) Forsimplicityrstassume>0andhenceT>0.Hence,equation( E )canbeexpressedas(1)]TJ /F4 11.955 Tf 11.95 0 Td[()Z01 2(1)]TJ /F4 11.955 Tf 11.96 0 Td[()F(x)+1 21+)]TJ /F5 11.955 Tf 11.96 0 Td[((1+)F2T (1+))]TJ /F5 11.955 Tf 13.15 8.09 Td[(1)]TJ /F4 11.955 Tf 11.96 0 Td[( 1+xdF(x)+(1+)(Z12T (1+)1 2()]TJ /F4 11.955 Tf 9.3 0 Td[(+(1+)F(x))+1 21)]TJ /F5 11.955 Tf 11.95 0 Td[((1)]TJ /F4 11.955 Tf 11.95 0 Td[()F2T (1)]TJ /F4 11.955 Tf 11.95 0 Td[())]TJ /F5 11.955 Tf 13.15 8.09 Td[(1+ 1)]TJ /F4 11.955 Tf 11.95 0 Td[(xdF(x)+Z2T (1+)01 2()]TJ /F4 11.955 Tf 9.3 0 Td[(+(1+)F(x))+1 21+)]TJ /F5 11.955 Tf 11.95 0 Td[((1+)F2T (1+))]TJ /F3 11.955 Tf 11.95 0 Td[(xdF(x)). UsingtheLeibnizintegralrule,weobtainthederivativeofthefunctionalformofTwithrespectto.Evaluatingat=0,impliesT=0=0andwehave@T @=0Z110(F(x))f(x)dF(x)=)]TJ /F15 11.955 Tf 11.29 16.27 Td[(Z10(1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(x))dF(x)+Z10(F(x))dF(x)+Z100(F(x))1 2F(x)+xf(x)dF(x)+Z100(F(x))1 2F(x))]TJ /F5 11.955 Tf 13.15 8.08 Td[(1 2+xf(x)dF(x), asfort2[0,1],F()]TJ /F3 11.955 Tf 9.3 0 Td[(t)=1)]TJ /F3 11.955 Tf 11.96 0 Td[(F(t),(1)]TJ /F3 11.955 Tf 11.96 0 Td[(t)=)]TJ /F4 11.955 Tf 9.3 0 Td[((t)and0(1)]TJ /F3 11.955 Tf 11.96 0 Td[(t)=0(t).ThereforethedistortionsensitivityforanR-estimatorbecomesDSRR=Z102(F(x))+0(F(x))[F(x)+2xf(x))]TJ /F5 11.955 Tf 11.96 0 Td[(1=2]dF(x) Z10(F(x))f(x)dF(x). 107

PAGE 108

APPENDIXFDISTORTIONSENSITIVITYOFP-ESTIMATORS TheimplicitfunctionalrepresentationofaP-estimatorTwithunderlyingdensityFfromtheFechnerfamily,isZ0(x)]TJ /F3 11.955 Tf 11.96 0 Td[(T) (x)]TJ /F3 11.955 Tf 11.95 0 Td[(T)dF(x)=0. Thederivativewithrespecttoandevaluatedat=0,yields@T @=0Z1100(x) (x))]TJ /F15 11.955 Tf 11.95 16.86 Td[(0(x) (x)2dF(x)=Z100(x) (x)dF(x))]TJ /F15 11.955 Tf 11.96 16.27 Td[(Z00(x) (x)dF(x))]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z0x00(x) (x)dF(x)+Z00(x) (x)2dF(x)+Z10x00(x) (x)dF(x)+Z100(x) (x)2dF(x). If=fandfistwicedifferentiablethenwehavethat@T @=0)]TJ /F9 7.97 Tf 6.59 0 Td[(1Z11f00(x))]TJ /F5 11.955 Tf 13.16 8.09 Td[((f0(x))2 f(x)dx=Z10f0(x)dx)]TJ /F15 11.955 Tf 11.95 16.27 Td[(Z0f0(x)dx)]TJ /F15 11.955 Tf 11.96 16.27 Td[(Z0xf00(x)dx+Z0(f0(x))2 f(x)dx+Z10xf00(x)dx+Z10(f0(x))2 f(x)dx andthereforethedistortionsensitivityisDSRP=Z10xf00(x)dx Z10f00(x))]TJ /F5 11.955 Tf 13.15 8.09 Td[((f0(x))2 f(x)dx. 108

PAGE 109

APPENDIXGPROOFOFPROPOSITION3.2 Theproofisanextension Cassartetal. ( 2008 )'sproofofProposition3.1andwhosemainpointconsistsofFechnerdensityf,,;f1beingdifferentiableinquadraticmeanat(0,,).Forthispurposethefollowinglemmaisrequired. Lemma1. Letf12F,2R,2R+0and2()]TJ /F5 11.955 Tf 9.29 0 Td[(1,1).Deneg,,;f(x)=1 fx)]TJ /F4 11.955 Tf 11.95 0 Td[( (1+sgn(x)]TJ /F4 11.955 Tf 11.96 0 Td[())Dg1=2,,;f(x)j=0=1 2)]TJ /F9 7.97 Tf 6.58 0 Td[(3=2f1=2x (1+sgn(x))fx (1+sgn(x))1 1+sgn(x)Dg1=20,,;f(x)=1 2)]TJ /F9 7.97 Tf 6.58 0 Td[(3=2f1=2x (1+sgn(x))fx (1+sgn(x))x (1+sgn(x)))]TJ /F5 11.955 Tf 11.96 0 Td[(1Dg1=20,,;f(x)=1 2)]TJ /F9 7.97 Tf 6.58 0 Td[(1=2f1=2x (1+sgn(x))fx (1+sgn(x))x (1+sgn(x))1 1+sgn(x). Then, Zg1=2t,+s,+r;f(x))]TJ /F3 11.955 Tf 11.96 0 Td[(g1=20,,;f(x))]TJ /F5 11.955 Tf 11.96 0 Td[((t,s,r)0BB@Dg1=2,,;f(x)Dg1=20,,;f(x)Dg1=20,,;f(x)1CCA2dx=o)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jj(t,s,r)jj2(G) asjj(t,s,r)jj!0. Proof. Theintegral( G )isboundedbyC(b1+b2+b3),whereC2R+0,b1=Zfg1=2t,+s,+r;f1(x))]TJ /F3 11.955 Tf 11.96 0 Td[(g1=20,,;f1(x))]TJ /F3 11.955 Tf 11.96 0 Td[(tDg1=2,+s,+r;f1(x)j=0g2dx,b2=Zfg1=20,+s,+r;f1(x))]TJ /F3 11.955 Tf 11.96 0 Td[(g1=20,,;f1(x))]TJ /F5 11.955 Tf 11.96 0 Td[((s,r)0@Dg1=20,,;f1(x)Dg1=20,,;f1(x)1Ag2dx, andb3=t2ZDg1=2,+s,+r;f1(x)j=0)]TJ /F3 11.955 Tf 11.96 0 Td[(Dg1=2,,;f1(x)j=02dx UsingLemmaA.1in Hallinetal. ( 2006 ),wenotethatb2iso(jj(t,s,r)jj2). 109

PAGE 110

Nextweshowthatb3iso(1).SinceDg1=2,,;f1(x)j=0issquare-integrablewenotethat jjDg1=2,,+r;f1()j=0)]TJ /F3 11.955 Tf 10.08 0 Td[(Dg1=2,,;f1()j=0jjL2=jjDg1=2,,;f1()]TJ /F3 11.955 Tf 14.18 0 Td[(r)j=0)]TJ /F3 11.955 Tf 10.08 0 Td[(Dg1=2,,;f1()j=0jjL2(G) iso(1)asr!0.Denef1;exp(x)=f1(ex)and(f1=21;exp)0(x)=(1=2)f)]TJ /F9 7.97 Tf 6.58 0 Td[(1=21(ex)_f1(ex)ex.Thus,ZfDg1=2,+s,+r;f1(x)j=0)]TJ /F3 11.955 Tf 11.96 0 Td[(Dg1=2,,;f1(x)j=0g2dx=CZf(f1=21;exp)0(u)]TJ /F5 11.955 Tf 11.95 0 Td[(ln(1+s ))]TJ /F9 7.97 Tf 6.58 0 Td[(3=2)]TJ /F5 11.955 Tf 11.95 0 Td[((f1=21;exp)0(u)(+s))]TJ /F9 7.97 Tf 6.59 0 Td[(3=2g2du, with( G ),quadraticmeancontinuity,(f1=21;exp)0(u)beingsquare-integrableandnon-dependencyonimplythatthetermiso(1)ass!0uniformlyin. Similarly,RfDg1=2,,+r;f1(x)j=0)]TJ /F3 11.955 Tf 12.29 0 Td[(Dg1=2,,;f1(x)j=0g2dxiso(1)asr!0byreplacingln(1+s=)byln(1+sgn(x)r 1+sgn(x))intheprecedingargument. Also,lettingz=x=((+s)(1+sgn(x)))removesthedependenceofb1in(s,r)andquadraticmeandifferentiabilitywithrespecttofollowfromSection6.1in Cassartetal. ( 2008 ). Then,byTheorem7.2and7.10in VanderVaart ( 2000 )asymptoticnormalityisachieved. 110

PAGE 111

APPENDIXHCONSISTENTESTIMATIONOFCROSS-INFORMATIONQUANTITIES Consistentestimationofthecross-informationquantitiesJ(f,g)andM(f,g)isrequiredforconsistentestimationofe(f,g)inTe(n)f().ThemethodologyisanimplementationoftheprocedurepresentedinSection4.5of Cassartetal. ( 2008 ).Theconceptdevelopedby Hallinetal. ( 2006 )andfurtherby Cassartetal. ( 2010 )isoneofsolvingalocallinearizedlikelihoodequation. Let^#and^#bediscretized(asinequation( 3 ))consistentroot-nconsistentestimators,ande(n)f;3;#bethediscretizedversionofequation( 5 ).Dene,e(n)()=^#+n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2^2#(1)]TJ /F5 11.955 Tf 11.84 0 Td[(^2#)e(n)f;3;#(^#), forany>0.Foranarbitraryc>0,setl=l=cforl2Nanddene)]TJ /F9 7.97 Tf -.66 -7.97 Td[(1=minflje(n)f;3;#(e(n)(l+1))e(n)f;3;#(^#)<0g,+1=)]TJ /F9 7.97 Tf -.66 -7.97 Td[(1+1 c, and 1=)]TJ /F9 7.97 Tf -.66 -7.97 Td[(1+1 ce(n)f;3;#(e(n)()]TJ /F9 7.97 Tf -.66 -7.97 Td[(1)) e(n)f;3;#(e(n)()]TJ /F9 7.97 Tf -.66 -7.97 Td[(1)))]TJ /F5 11.955 Tf 11.95 0 Td[(e(n)f;3;#(e(n)(+1)).(H) Consequently,underP(n)0,,;g,g2Fef,asn!1,J(n)(f)=(1))]TJ /F9 7.97 Tf 6.59 0 Td[(1=J(f,g)+oP(1). Furthermore,from Hallinetal. ( 2006 )aR-estimatorof,atP0,,;f(withparametricefciency),isachievedbye(n)(1). Thecross-informationquantityM(f,g)doesnotpresentitselfasacovariancetermofanycomponentoftherankversionofthecentralsequencee(n)f()withthecorrespondingcomponente(n)g()andassuchtheR-estimationprocedurecannotbedirectlyapplied. Cassartetal. ( 2010 )extendsthemethodologybytheasymptoticlinearityproperty.FromTheorem3.2of vanEeden ( 1972 ),underP0,,;g,g2Fefe(n)f;1(^#,+n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2s)=e(n)f;1(^#,)+s^)]TJ /F9 7.97 Tf 6.58 0 Td[(1#(1)]TJ /F4 11.955 Tf 11.96 0 Td[(2))]TJ /F9 7.97 Tf 6.58 0 Td[(1M(f,g)+oP(1) 111

PAGE 112

asn!1,foralls2R.TheassumptionsofProposition5.3of Cassartetal. ( 2010 )holdandadvancingasbeforedenee(n)()=^#+n)]TJ /F9 7.97 Tf 6.58 0 Td[(1=2^#(1)]TJ /F5 11.955 Tf 11.83 0 Td[(^2#)e(n)f;1;#(^#,^#) anddene)]TJ /F9 7.97 Tf -.66 -7.98 Td[(2=minflje(n)f;1;#(^#,e(n)(l+1))e(n)f;1;#(^#,^#)<0g,+2=)]TJ /F9 7.97 Tf -.66 -7.98 Td[(2+1 c. Assign, 2=)]TJ /F9 7.97 Tf -.66 -7.98 Td[(2+1 ce(n)f;1;#(^#,e(n)()]TJ /F9 7.97 Tf -.66 -7.98 Td[(2)) e(n)f;1;#(^#,e(n)()]TJ /F9 7.97 Tf -.66 -7.97 Td[(2)))]TJ /F5 11.955 Tf 11.95 0 Td[(e(n)f;1;#(^#,e(n)(+2)),(H) thenbyProposition5.3of Cassartetal. ( 2010 ),underP(n)0,,;g,g2FefM(n)(f)=(2))]TJ /F9 7.97 Tf 6.59 0 Td[(1=M(f,g)+oP(1) asn!1. Asstatedby Cassartetal. ( 2008 ),inpracticewherenisxed,thediscretizationof( 3 )isredundant.Therefore,let1=inff>0je(n)f;3(^+n)]TJ /F9 7.97 Tf 6.59 0 Td[(1=2^2(1)]TJ /F5 11.955 Tf 11.83 0 Td[(^2)e(n)f;3(^))e(n)f;3(^)0g and2=inff>0je(n)f;1(^,^+n)]TJ /F9 7.97 Tf 6.58 0 Td[(1=2^(1)]TJ /F5 11.955 Tf 11.83 0 Td[(^2)e(n)f;1(^,^))e(n)f;1(^,^)0g whichcanenvisionedasequations( H )and( H )withdiscretizingconstantc!1,ofwhosereciprocalsaretheestimatorsofJ(n)(f)andM(n)(f)andconsequentlyofe(n)(f;)ande(f,g). 112

PAGE 113

REFERENCES Arellano-Valle,R.,&Genton,M.(2005).Onfundamentalskewdistributions.JournalofMultivariateAnalysis,96(1),93. Arellano-Valle,R.,Gomez,H.,&Quintana,F.(2005).Statisticalinferenceforageneralclassofasymmetricdistributions.JournalofStatisticalPlanningandInference,128(2),427. Azzalini,A.(1985).Aclassofdistributionswhichincludesthenormalones.Scandina-vianJournalofStatistics,(pp.171). Azzalini,A.,&Valle,A.(1996).Themultivariateskew-normaldistribution.Biometrika,83(4),715. Bassett,G.(1991).Equivariant,monotonic,50%breakdownestimators.AmericanStatistician,(pp.135). Boos,D.,&Sering,R.(1980).AnoteondifferentialsandtheCLTandLILforstatisticalfunctions,withapplicationtoM-estimates.TheAnnalsofStatistics,(pp.618). Branco,M.,&Dey,D.(2001).Ageneralclassofmultivariateskew-ellipticaldistributions.JournalofMultivariateAnalysis,79(1),99. Butler,C.(1969).Atestforsymmetryusingthesampledistributionfunction.TheAnnalsofMathematicalStatistics,40(6),2209. Cassart,D.,Hallin,M.,&Paindaveine,D.(2008).OptimaldetectionofFechner-asymmetry.JournalofStatisticalPlanningandInference,138(8),2499. Cassart,D.,Hallin,M.,&Paindaveine,D.(2010).Aclassofoptimaltestsforsymmetrybasedonedgeworthapproximations.Bernoulli,toappear. Chernoff,H.,&Savage,I.(1958).Asymptoticnormalityandefciencyofcertainnonparametricteststatistics.TheAnnalsofMathematicalStatistics,(pp.972). Davidson,R.,&Duclos,J.(2000).Statisticalinferenceforstochasticdominanceandforthemeasurementofpovertyandinequality.Econometrica,(pp.1435). Donoho,D.,&Huber,P.(1982).Thenotionofbreakdownpoint.AFestschriftforErichL.LehmanninHonorofHisSixty-fthBirthday,(pp.157). Fechner,G.(1897).Th.(1897).Kollektivmasslehre,Leipzig,Engelman. Fernandez,C.,&Steel,M.(1998).OnBayesianModelingofFatTailsandSkewness.JournaloftheAmericanStatisticalAssociation,93(441). Fernholz,L.(1983).VonMisescalculusforstatisticalfunctionals.SPRINGER-VERLAGNEWYORK,INC.,19. 113

PAGE 114

Genton,M.,Distributions,S.,&Applications,T.(2004).AJourneyBeyondNormality,EditedVolume. Hajek,J.,Sidak,Z.,&Sen,P.(1999).Theoryofranktests.AcademicPr. Hallin,M.(2003).Semi-parametricefciency,distribution-freenessandinvariance.Bernoulli,9(1),137. Hallin,M.,Oja,H.,&Paindaveine,D.(2006).Semiparametricallyefcientrank-basedinferenceforshape.ii.optimalr-estimationofshape.TheAnnalsofStatistics,34(6),2757. Hampel,F.(1968).Contributionstothetheoryofrobustestimation.Ph.D.thesis,UniversityofCalifornia,Berkeley. Hampel,F.(1974).Theinuencecurveanditsroleinrobustestimation.JournaloftheAmericanStatisticalAssociation,(pp.383). Hampel,F.,Ronchetti,E.,Rousseeuw,P.,&Stahel,W.(1986).Robuststatistics:theapproachbasedoninuencefunctions.JohnWiley&SonsNewYork. Hettmansperger,T.,&Randles,R.(2002).Apracticalafneequivariantmultivariatemedian.Biometrika,89(4),851. HodgesJr,J.,&Lehmann,E.(1963).Estimatesoflocationbasedonranktests.TheAnnalsofMathematicalStatistics,(pp.598). Huber,P.(1964).Robustestimationofalocationparameter.TheAnnalsofMathemati-calStatistics,(pp.73). Huber,P.,&Ronchetti,E.(2009).Robuststatistics.Wiley-Blackwell. LeCam,L.(1986).Asymptoticmethodsinstatisticaldecisiontheory.Springer. LeCam,L.,&Yang,G.(2000).Asymptoticsinstatistics:somebasicconcepts.SpringerVerlag. Liseo,B.(1990).Theskew-normalclassofdensities:aspectsofinferencefromtheBayesianpointofview.Statistica,50(1),71. Lucas,A.(1996).Outlierrobustunitrootanalysis.PhDinEconometrics,ErasmusUniversity. Mudholkar,G.,&Hutson,A.(2000).Theepsilonskewnormaldistributionforanalyzingnear-normaldata.JournalofStatisticalPlanningandInference,83(2),291. O'Hagan,A.,&Leonard,T.(1976).Bayesestimationsubjecttouncertaintyaboutparameterconstraints. 114

PAGE 115

Pierce,D.(1982).Theasymptoticeffectofsubstitutingestimatorsforparametersincertaintypesofstatistics.TheAnnalsofStatistics,10(2),475. Randles,R.(1982).Ontheasymptoticnormalityofstatisticswithestimatedparameters.TheAnnalsofStatistics,10(2),462. Randles,R.,Fligner,M.,Policello,G.,&Wolfe,D.(1980).Anasymptoticallydistribution-freetestforsymmetryversusasymmetry.JournaloftheAmericanStatisticalAssociation,(pp.168). Randles,R.,&Wolfe,D.(1979).Introductiontothetheoryofnonparametricstatistics.WileySeriesinProbabilityandMathematicalStatistics. Reeds,J.(1976).OnthedenitionofvonMisesfunctionals.Ph.D.thesis,HarvardUniversity. Rousseeuw,P.(1984).Leastmedianofsquaresregression.JournaloftheAmericanStatisticalAssociation,(pp.871). Tukey,J.(1960).Asurveyofsamplingfromcontaminateddistributions.Contributionstoprobabilityandstatistics,(pp.448). VanderVaart,A.(2000).Asymptoticstatistics.CambridgeUnivPr. vanEeden,C.(1972).Ananalogue,forsignedrankstatistics,ofjureckova'sasymptoticlinearitytheoremforrankstatistics.TheAnnalsofMathematicalStatistics,43(3),791. 115

PAGE 116

BIOGRAPHICALSKETCH DemetrisAthienitiswasborninCyprus.AftergraduatingfromHighSchoolheservedasaSecondLieutenantoftheReserveinCyprusNationalGuardfrom2000to2002.HethenpursuedahighereducationintheUnitedStateswereheearnedaBachelorofArtsinmathematicsandstatisticsfromRutgers,TheStateUniversityofNewJerseyinMayof2005.InMayof2008hereceivedaMasterofStatisticsintheDepartmentofStatisticsattheUniversityofFloridaandlateraPh.D.instatisticsinDecemberof2011. 116