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1 2 T e st s
12.1
B a si c C o n c e p ts
aHowtoconstructatest•Model,hereparametric,Fθ.
•NullhypothesisH0andalternativesneedexactmodelunderH0,butonlygeneralideaaboutalternativesCommonH0:Fixparametervalueθ=θ0(forscalarθ)multidim.parameter:splitintotwoparts,θ=[θ T[1] ,θ T[2] ] T,θ∈R p,θ[1] ∈R q,θ[2] ∈R p−q.H0:θ[2] =θ[2],0Alternative:θ[2] 6=θ[2],0
209
•Inventa“raw”teststatisticUthathasmoreextremevaluesunderthealternativethanundertheH0Commonchoice:U= bθ[2]
•GetthedistributionofUundertheH0Thiswilloftendependon“nuisanceparameters”(formallycontainedinθ[1] )Inthelightofthisdistribution,find–asuitablenormofUifitismultivariate(Testingamultidim.teststatisticdirectlyisrare.)–astandardizationthatmakesthedistributionindependentofthenuisanceparameter(s)inθ[1]Thisinvolvestheestimator bθ[1]Ifascaleparam.isinvolved,thisiscalled“Studentization”Result:StandardizedteststatisticT Dbθ E
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•GetthedistributionofTundertheH0–hopeforagoodapproximationbyawell-knowndistr.likenormal,t-,F-distr.Oftenneed“correctionfactor”.–bruteforce:simulationofdataunderthenullhypothesiswithestimatednuisanceparameters,θ[1] = bθ[1] .“parametricbootstrap”–iffeasible,userandomizationtest–nonparametricbootstrap???hasproblemswithtestingunless(approx.)“translationmodel”−→bootstrapconfidenceinterval−→Pvaluebp
211 12.1 bPropertiesoftestsLevel:P0hbp<αi=αbyconstruction.FunctionalbphFi.–LevelbphFθ0 i:Robust?CangetanInfluenceFunction,“LevelInfluenceFunction”(Note:ThisisdifferentfromstudyingthecorrectnessofapproximationstothedistributionofTunderH0.)
Power?Needtospecifyalternativeprecisely:Fθ−→PowerPθ Dbp DbFn E<α E.Fornormalmodels,distr.ofTunderthealternativeisoftenknown.Otherwise:Simulation!
Robust?StudybphFθi.–Asymptotics?bphFθi→0forn→∞.Notuseful!
21212.1 cAsymptoticsfortestsAlternativemust→H0forn→∞,withtheappropriate“rate”,θ=θ0+ 1√n ∆θ.“Contiguousalternatives”.
−→CangetanInfluenceFunction,“PowerInfluenceFunction”.
Message:Userobustestimator bθ
21312.1
dRecipeforrobusttests
•Userobustestimator bθtodefineU.
•Summarize&standardizeanalogouslytotheclassicalcase.
•Usecorrectionfactorstogetapproximatelythecorrectlevelwhentakingcriticalvaluesfromthe“classicaldistribution”Iffeasible,adjustdegreesoffreedomifatorFdistributionisinvolved.
eHuber’sRobustLikelihoodRatioTestSeeHistory
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12.2
T e st s fo r R e g re ss io n
aNullHypothesisParameterstobetested:β[2] =[βp−q+1,...,βp] T.
H0:β[2] =β[2],0 .Nuisanceparameter:σ(andβ[1] =[β1,...,βp−q] T) Moregeneral:Giveanymatrixof“contrasts”A.NullhypothesisAβ=γ0.BytransformingXlinearly−→backtosimplercase.Ifregr.equivar.methodsarestudied−→nolossofgenerality.
21512.2 b“Fullmodel”:Parameterspaceforβunrestrictedallxvariablesare“inthemodel”“Reducedmodel”:Param.spaceforβrestrictedtoβ[2] =β[2],0Ifβ[2],0 =0,thelastxvariablesare“droppedfromthemodel”
Ifβ[2],0 6=0:replaceYbyY−X ([2])β[2],0−→backtoβ[2],0 =0
216 12.2 cWaldtypetestU= bβ[2] ≈∼Np−q β[2],0 ,V/n
Naturalstandardization:T=U TV −1UNeedV=γσ 2(X TX/n) −1
−→Studentizationbyreplacingσ 2bybσ 2
DistributionunderH0:Fdistribution.–Exactforclassicalcase. Robustcase:Need–correctionfactors–robustestimatorofσ 2:designadaptiveestimator!–robust“estimator”ofX TX/n=aveihxix Ti iUseweightedaverage Pi wixix Ti Pi wi,wherewiaretherobustnessweightsψhri/bσi (ri/bσ)
217 12.2 dLikelihoodratiotypetestOnewaytomotivaterobustestimatorsisbyMLEwithrespecttoalong-taileddistributionLikelihoodforfixedσ:Q Dbβ E= Pi ρhri/σi,ri=yi−x Ti bβ
LRteststatistic:U=Q Dbβ0 E−Q Dbβ Ewherebβ0istheestimatorofβundertherestriction bβ[2] =β[2],0 . Forclassicalsetup(ρhri=r 2/2),thisleadstoT∼χ 2p−q .Whenσisestimatedfromthefullmodel−→T∼Fp−q,n−p.
Robustρ:Needcorrectionfactors.Note:WewantthedistributionofUunderH0fornormaldata,notlong-tailed!(?)
21812.2
eScoretypetestEstimateonlythereducedmodel!
U= Pi ψ Dxi,yi; bβ0,bσ0 E
Standardization:T=U TJ −1U=U T(X TX) −1U bσ 2
bσ?Thistestiscurrentlyunderstudy.
