3Lecture3:PRINCIPLESOFDATAREDUCTIONANDINFERENCE3.1DataReductioninStatisticalInferenceGivenvectorX=(X1;X2;::;Xn)ofni.i.d.randomvariables,eachwithadensityf(x;θ);wearemeanttoconductinferenceonθ2Θbasedontheobservationsx1;x2;::;xn:LetXtakesvaluesinX-thesamplespace.Thestatisticianusestheinformationintheobservationsx1;x2;::;xntoconducttheinference.His/herwishistosummarizetheinformationinthesamplebydeterminingafewkeyfeaturesofthesamplevaluesthroughtransformingthesamplevalues.Calculatingsuchtransformations(i.e.functionsofthesample)meanstocalculateastatistic.Typically,dim(T)<<n,i.e.usingthestatistic,weachievethegoalofdatareduction:ratherthanreportingtheentiresamplex,thestatisticreportsonlythatT(x)=t.Datareductionintermsofaparticularstatisticcanbethoughtofasapartitionofthesamplespace.WepartitionXintodisjointsubsetsAt=fX:T(X)=tg.Ifτ=ft:t=T(x)forsomex2XgthenthesamplespaceXisSrepresentedasaunionofthefollowingdisjointsets(i.e.ispartitioned):X=t2τAt.Theultimategoalinthedatareductionis,whenonlyusingthevalueofthestatisticT(x)insteadofthewholevectorx;"nottoloseinformation"abouttheparameterofinterestθ.Thewholeinformationaboutθwillbecontainedinthestatisticand,inparticular,wewilltreatasequalanytwosamplesxandythatsatisfyT(x)=T(y)eventhoughtheactualsamplevaluesmaybedifferent.Thatwaywearriveatthedefinitionofsufficiency.TheinformationinXaboutθcanbediscussedintermsofpartitionsofthesamplespace.Definition1(sufficientpartition)SupposeforanysetAtinaparticularpartitionA=fAt;t2τgwehavePfX=xjX2Atgdoesnotdependonθ.ThenAisasufficientpartitionforθ.Note:Wehaveseenabovethatthepartitionisdefinedthroughasuitablestatistic.IfthestatisticTissuchthatitgeneratesasufficientpartitionofthesamplespacethenthestatisticitselfissufficient.3.2Example:X=(X1;X2;::;Xn)i.i.d.Bernoulliwithparameterθ,i.e.xi1−xiP(Xi=xi)=θ(1−θ);xi=0;1.ThepartitionA=(A0;A1;:::;An)wherex2ArPnifandonlyif(iff)i=1xi=r,issufficientforθ.Correspondingly,thestatisticT(X)=Pni=1Xiissufficientforθ.Proof:Atlecture.Notethatgiven