Normal approximation for nonlinear statistics usin

arXiv:0708.4272v1[math.ST]31Aug2007Bernoulli13(2),2007,581–599DOI:10.3150/07-BEJ5164NormalapproximationfornonlinearstatisticsusingaconcentrationinequalityapproachLOUISH.Y.CHEN1andQI-MANSHAO21InstituteforMathematicalSciences,NationalUniversityofSingapore,Singapore118402,Re-publicofSingapore.E-mail:[e-mail]2DepartmentofMathematics,HongKongUniversityofScienceandTechnology,ClearWaterBay,Kowloon,HongKong,China;DepartmentofMathematics,UniversityofOregon,Eugene,OR97403,USA;DepartmentofMathematics,ZhejiangUniversity,Hangzhou,Zhejiang310027,China.E-mail:[e-mail]statisticthatcanbewrittenasalinearstatisticplusanerrorterm.Uniformandnon-uniformBerry–EsseentypeboundsforTareobtained.Theboundsarethebestpossibleformanyknownstatistics.ApplicationstoU-statistics,multisampleU-statistics,L-statistics,randomsumsandfunctionsofnonlinearstatisticsarediscussed.Keywords:concentrationinequalityapproach;L-statistics;multisampleU-statistics;non-uniformBerry–Esseenbound;nonlinearstatistics;normalapproximation;U-statistics;uniformBerry–Esseenbound1.IntroductionLetX1,X2,...,XnbeindependentrandomvariablesandletT:=T(X1,...,Xn)beageneralsamplingstatistic.InmanycasesTcanbewrittenasalinearstatisticplusanerrorterm,sayT=W+Δ,whereW=nXi=1gi(Xi),Δ:=Δ(X1,...,Xn)=T−Wandgi:=gn,iareBorelmeasurablefunctions.TypicalcasesincludeU-statistics,multi-sampleU-statistics,L-statisticsandrandomsums.AssumethatE(gi(Xi))=0fori=1,2,...,nandnXi=1E(g2i(Xi))=1.(1.1)ThisisanelectronicreprintoftheoriginalarticlepublishedbytheISI/BSinBernoulli,2007,Vol.13,No.2,581–599.Thisreprintdiffersfromtheoriginalinpaginationandtypographicdetail.1350-7265c2007ISI/BS582L.H.Y.ChenandQ.-M.ShaoItisclearthatifΔ→0inprobabilityasn→∞,thenwehavethecentrallimittheoremsupz|P(T≤z)−Φ(z)|→0,(1.2)whereΦdenotesthestandardnormaldistributionfunction,providedthattheLindebergconditionholds:∀ε0,nXi=1Eg2i(Xi)I(|gi(Xi)|ε)→0.Ifinaddition,E|Δ|p∞forsomep0,thenbytheChebyshevinequality,onecanobtaintherateofconvergencesupz|P(T≤z)−Φ(z)|≤supz|P(W≤z)−Φ(z)|+2(E|Δ|p)1/(1+p).(1.3)Thefirsttermontheright-handsideof(1.3)iswellunderstoodviatheBerry–Esseeninequality.Forexample,usingStein’smethod,ChenandShao[9]obtainedsupz|P(W≤z)−Φ(z)|≤4.1nXi=1Eg2i(Xi)I(|gi(Xi)|1)+nXi=1E|gi(Xi)|3I(|gi(Xi)|≤1)!.(1.4)However,thebound(E|Δ|p)1/(1+p)is,ingeneral,notsharpformanycommonlyusedstatistics.ManyauthorshaveworkedtowardobtainingbetterBerry–Esseenbounds.Forexample,sharpBerry–EsseenboundshavebeenobtainedforgeneralsymmetricstatisticsbyvanZwet[24]andFriedrich[12].AnEdgeworthexpansionwithremainderO(n−1)forsymmetricstatisticswasprovedbyBentkus,G¨otzeandvanZwet[3].Themainpurposeofthispaperistoestablishuniformandnon-uniformBerry–Esseenboundsforgeneralnonlinearstatistics.Theboundsarethebestpossibleformanyknownstatistics.OurproofisbasedonarandomizedconcentrationinequalityapproachtoboundingP(W+Δ≤z)−P(W≤z).Becauseproofsofuniformandnon-uniformboundsforsumsofindependentrandomvariablescanbeprovedviaStein’smethod(ChenandShao[9]),whichismuchneaterandsimplerthanthetraditionalFourieranalysisap-proach,thispaperprovidesadirectandunifyingtreatmenttowardtheBerry–Esseenboundsforgeneralnonlinearstatistics.Thispaperisorganizedasfollows.Themainresultsarestatedinnextsection,threeapplicationsarepresentedinSection3andanexampleisgiveninSection4toshowthesharpnessofthemainresults.ProofsofthemainresultsaregiveninSection5.Fortheproofsofotherresults,includingExample4.1,thereaderisreferredtoourtechnicalreport(ChenandShao[10]).Throughoutthispaper,Cwilldenoteanabsoluteconstantwhosevaluemaychangeateachappearance.TheLpnormofarandomvariableXisdenotedbykXkp,thatis,kXkp=(E|X|p)1/pforp≥1.Normalapproximationfornonlinearstatistics5832.MainresultsLet{Xi,1≤i≤n},T,WandΔbedefinedasinSection1.Inthefollowingtheorems,weassumethat(1.1)issatisfied.Putβ=nXi=1E|gi(Xi)|2I(|gi(Xi)|1)+nXi=1E|gi(Xi)|3I(|gi(Xi)|≤1)(2.1)andletδ0satisfynXi=1E|gi(Xi)|min(δ,|gi(Xi)|)≥1/2.(2.2)Theorem2.1.Foreach1≤i≤n,letΔibearandomvariablesuchthatXiand(Δi,W−gi(Xi))areindependent.Thensupz|P(T≤z)−P(W≤z)|≤4δ+E|WΔ|+nXi=1E|gi(Xi)(Δ−Δi)|(2.3)forδsatisfying(2.2).Inparticular,wehavesupz|P(T≤z)−P(W≤z)|≤2β+E|WΔ|+nXi=1E|gi(Xi)(Δ−Δi)|(2.4)andsupz|P(T≤z)−Φ(z)|≤6.1β+E|WΔ|+nXi=1E|gi(Xi)(Δ−Δi)|.(2.5)Thenexttheoremprovidesanon-uniformbound.Theorem2.2.Foreach1≤i≤n,letΔibearandomvariablesuchthatXiand(Δi,{Xj,j6=i})areindependent.Then,forδsatisfying(2.2)andforz∈R1,|P(T≤z)−P(W≤z)|≤γz+e−|z|/3τ,(2.6)whereγz=P(|Δ|(|z|+1)/3)+nXi=1P(|gi(Xi)|(|z|+1)/3)+nXi=1P(|W−gi(Xi)|(|z|−2)/3)P(|gi(Xi)|1),(2.7)584L.H.Y.ChenandQ.-M.Shaoτ=22δ+8.5kΔk2+3.6nXi=1kgi(Xi)k2kΔ−Δik2.(2.8)Inparticular,ifE|gi(Xi)|p∞for2p≤3,then|P(T≤z)−Φ(z)|≤P(|Δ|(|z|+1)/3)+C(|z|+1)−pkΔk2+nXi=1kgi(Xi)k2kΔ−Δik2+nXi=1E|gi(Xi)|p!.(2.9)Aresultsimilarto(2.5)wasobtainedbyFriedrich[12]forgi=E(T|Xi)usingthemethodofcharacteristicfunction.Ourproofisdirectandsimpler,andtheboundsareeasiertocalculate.Thenon-uniformboundsin(2.6)and(2.9)forgeneralnonlinearstatisticsarenew.Remark2.1.AssumeE|gi(Xi)|p∞forp2.Letδ=2(p−2)p−2(p−1)p−1nXi=1E|gi(Xi)|p!1/(p−2).(2.10)Then