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1、第22卷第2期中国科学技术大学学报1992年6月JO皿Nt.oFc HIN训rv EsRl竹oFsC I胡eE湘TEcHN O LoGYVOI-JUn.2 2,No.21992Convergene eRate sofSomeEstimatorsforaSemiParametrieRegre ssionModel许GaoJ iti()伸rtmento fMatbematiC S)Ab st班Ctc on siderthee rge s s l ronmo del犷“心声+夕(二)+氏f or一l,2,e He r夕1 5a nun-kno wn加n eti nninI R 二(一c o,c o)
2、,e1 5anun o眺e r ve ddlsu trbanc e.刀晚se do nthekernela ndn eae s r tn eih g 加re stim ao trs吐,(:),s omeopt加a teon ve rs e neea re tsa咖t出ee s往m-atorsof刀andg(.)a r eoba tine d众ywords二s emi琳r ametrier egr e绍f onmodel,le as t一squa ree s t加ate,oPtimalc on veg ten eerateAMS翻bje etCla s s企让.tion(1 99 1):62G
3、07,6 2J101Intro dut Cio nConsi血rth emo delg ive nb y梦一人刀+g(x)+伙,落一1,2,二(1)whee r(人,:,e)ar e1.1.d.r andoms amPle s,(人,二)andearei ndee pnde ntand氏1 5h ter a ndomerr orwih tme an0andv a ria nc eaZ,刀15anu nkhownParam亡e trand夕1 5anunknownf un etion.o Smeasympto桩ethe oryf or钟ri te al印linees t加atos rof刀h a
4、sb oende穴10讲db ye Hekmania ndRiee幻.h Teyin ve s tisat“h tec as ewh ee rt,5a remo deleda s人一h(劣)+价,云=1,2,(2)w hereh 1 5akno、v nfun etio noftheev oa riate劣andthe了忱hav elikee zroma en,uneor rel ate dra ndomvariab le s.Re cei Ve dAPr亚30,19 9 1.辞T h i sPr o扭ts uPPort目byNN S FCun血rthec onrra et18901001.1
5、30中国朴学扶术大学学报第2 2卷R侧毖ntl y,se pckma n,ha s血vel叩司ana t e trnauves e timao tro f声in(i)h a tt,unde ramo dels uh Ca s(2)wih t夕andhboh ts mo oth,i s创沼ure do f加vi nga琳a rmetriec o nv er罗nc ea re tf ori s ts a tn-血d ter rorandas ymt Pio ta Cllynegl语ib leb边5witho utthene ceS Siy to fu nders mo oh tingh Tea s
6、ymt P成eanalys e so fe Hk Cm ania ndu E恤nketal.月ae re o n份rn司wi小几=0or价一0,=l,2,wi hehae r比战刁o n9e sitmate dbyS Plin esmo otb j的9e si tmate S.Inh tePr姗nt劝团yw edls cu s sthefollo winge xt remec a,:namel y,thec as ewhe r e:=0,=l,2,.,andh(二)=公(i wh to utl端O fgen erai ly t).Ther ef orew e创铭umethatthe豹f o l
7、lowther no del(the邢”ei ficc as eo f(1)夕脚+夕(x)+氏,云1,2,(3)侃ea e s o rnf ori ne tre s tinamodelo fh ti sf orm咖msfo rm脉wo rkb y血ba nk以以.魂,也eotherre a s oni st汕tthi sse pei f iemo de l15themixe de rgr创沼io nm团elpr op汉姆dbyStones.E匕翻川onh telea t一q Su ae re st血n ato r几 O f刀andthekernelandne ae s rtni eh g b
8、ore s timate s么of夕,s omeO Pt加ala r te so feo nverg eneeabout几a nd纵ae r由a tine d2M.i nRe sult SL心ty=(梦,卜),二ar emen滋o ni ai n e trv alC,X二(公,一,)s amPles翎ehthat:a rng eoveran加degenea re teo m钾etl一d i-isa联1ve cto rin户,andY1 5real一v alue d.丫Vih to ut1055成,nea r一 it y,ite an加a s s um树th ate招h e tel,ine tr
9、 v al。,乙.刀匕以刃o n(二,梦),=z,:,也e一e as t一squ ae rse si tmao t r凡15oba tin曰by名(,一二夕),=mn(4)y B(4),weo ba tin凡=习.h,二,+习.h+习.i hg(i)t会,十A.+.B(5)whee r二二z璐,跨一习二,l成镇二,月.会习“e,凡会云瓜,(二).1Henc e,山enatu r ale si tmato ro fg始夕.(二)一习叽(二)(,一二凡)一名孔(二),一习叽(二)x凡一女.(二)一女:.(二)凡(let)(6)wh。传 评()1 5Po r比iblityw eightf unit
10、Co n.1笼fln eg,(二)一E(,12),92(二)一x(7)Thus叮:(二)c anb e七耽ima块对 by幸i。(二).iP仗l y,weeo nsi derthec a s eo fne are s tneighb ore si tmee t.O Fre ah ci fxe d二,wea r ranse二一戈inine r州ngo川er,ie.1:,一二镇l二一xl成镇 I二、一Now。h。pr o、b u*tywe吵:,eeto r氏,一1,(.c)o,习几一1)。nd def*ne第2期Convergo neeRatesofSomeEstimatorsfo raSemip
11、ar ametrieRegres sionModel131讯气()C.,一l,2,二(8)决condl y,h tec a seofkern ele sit mati on1 5als oeonc erne d.玫t(:,犷)b ebivai rae ta rndomv eeto rwithi o jntden滋y t九nei to nf(二,夕)andL忍tl(二)J了(一,)d;二dJ()一J;,一;,;No w,机eo ni s derProbabili tywe返h t加ni t Co nsofh tefo rm巩(二)一、兰早三/、人11全K三二二一三兰,1“,n。landdei f
12、nethe韶i tmatorofl(才)b y.I(:)-1芍 人_,I.代尸夕,入I刀几亡了旱)(9)(10)wh er eK()15ake r nel加nc tion.Thefo flowingfourc ondition sa res uffieientf orthes妞t e me ntO fh tema加伪nd 应亡沁。1(i)h Ter ex ei steo na s tnt s口:0a nd几c o,s uehthatC,簇从aPosiitv enum比rdee pnde ntso felyon砚.(“)s upk二ma x久祝i“、Note1tFo mCo nditi onl,
13、weh averegul侣.,一2,3(氏,h ee r砚i s习吧树“习W祝()=习以=o(,一,)unio fm rlyiwthe rs讲etto二ea,b.o Cnd i t io nZ(i)K(一)15abo u nde dProbabi l iy tde ni sy tonRs uehthatK(一)1 5eo ni tn uo us,ofbo u nde dvariati ono nR,a ndthee rexistc onsa tns tP任(0,1)and万:任(O,c o),s uh ch tatf、(:),二,。,:一fo r。ufoei en。,ma,;。0.Jl川.(。
14、)、()1 5Of、d曰V一。i t。一Rl,*丁!dK),c o(。*)J,“,!“0,a nd崛.b:,“,一。(“)撰.E(尹,()一MZc o(iv)E犷,万:2s uh ch tatEle;.c o(往)h Tereexi s tPositi vesq eue n馏如(落=1,2)s uh Ch tat习二;e、(一“。,;,二一“。一“,)o o,习exp(一“、:,;,)c oa nd习二:,乳,/o1 32中国朴学技术大学学报第2 2 卷NoteL et,l几=m一09,。+2/3a nd叮6,a nd叮云t.l og牡枯一/3(仍.,t.c oarb i tr a ri ly
15、).T heneo nd ition4(i)holds.T he orem1SuPPos ethateitherCo nd iUon1and3(i)(11 1)o rO Cnditf on2and3(i)(111)hold.T hena s刀 一争c o(i)护不,.二万(o,。:),丫牙。一。,(1)(11)户一刀o,(。一“,)(i,i)夕.(二)一夕(:)O,(:一/,)+O,(,.)wh多rea弓一e Ef,fz Ef,.denotethewe akeon ver郎n eerateof夕(),i一1,2.Th帕rem2 Inadd itiontoh teConditio n sofT
16、he orem1.As sumeCond ition4holds.T hen(i)凡一刀一。(n,:.犷)+o伪云,)as(11)supf夕.(x)一g(x)!=o(二q,.)一)十。(q云)十。(q云)a.5.fors uffiei entlyla rs e,he re,*(一1,2)s ati s勿Co nd itio n4(11),云立denots eh testr o ngeo nv erg en eer ateofmaxsupl夕.(x)一g(x)卜Remark1 Aninte r es tins questio nrelae tdtoh Te oreml(i)15wheh ter。
17、苦叮f,15h tesmalls etpo“i bleasymPtoi tev arian e e.It15ea silysho wnh tatJ苦arZ1 5h tesmalle stPo s s lblea sympri o tev a r功neefo rallr egula rs e timate sof刀w henerr o re,15normal l ydi stribute d.Rem ark2 zt15。lea:fr omthee s ruxs t认stone6】that,一二一,15optima一,50乡.()c ana eh ie v etheoPtimaleonverge
18、n eea rten一1/3.It15als oele arfro市h tere s uls tin7thatther atedef inedin仆e oe rmZ(11)15optimal.3Pro ofsofT heoren 一landZForeo n venieneea ndsimPlieity,letCo.e aChaPe pa ran eethroughtoutthisa Pe pr.From(5),(7),weha vedenoe tae o n sa tntwh iehmayhaved ifferentvaluesat几一刀夕.x()一g()一(户一3.11无m m韶forthe
19、Pr0 0fofthe二习.h ie十习.h i,(x)会A.十.B女1.(z)一,(x)一刀乡:.(x)一g:(二)夕)女2.(二)一92(x)一(凡一刀)g:(:)MainRe s uls t(1 1)(1 2)抉mma1e Lt。;,。.b eind叩endentr a ndomvariablesoti s fyingP(。一E。l簇。)一1,eaeh:where阴(0 015someabs oluteeo nsa tnt.T he nfora nygivent0,p(习。,一E一引)、2一p(一2/2习几、v ar(一)+号、)for、)1.Pro of.s ees.玩m ma2Let
20、Fbeaeontin u o u sfunetionandF.is temPirie aldistri butionfunetionbasedona n第2期eve ronseneeRatesofsomeEstimatorsforaSemipraametrieRegr es sionModel133i。i。d。0samPleX、,X一h Tenthereexiss taunivers aleo ns a tntCo0suehth at,f ora ny二0,t.P(SUP一仍,)簇coexp(一2:峪)pr0 0fe Se此mma2.1.1of9.玫m ma3(i)Sup pos eth at
21、o Cnditio n1andConi diton3hold.h Ten:SS念易.象易I夕,.(:)一g;(:)I一口(成10 9:女2.(x)一92(二)I一。(叭109:界/3)刀i/3)a。a。(13)(14)fors ui f feientlylarge几,her e仍.c oa rb i tr a rily(i l)SuP Po s eh tato Cnd itio nZ(i)、(111),3(i)一(i il)and4(i)ho ld.h Te ns up,女:.(二)一夕,(x)=o(:一,)a.s,0(尹(,(夕一2)/2叮(l+,),0(,(1,夕)2(1 5)(1 1 1
22、)SuPo psethatCo ndii to n2and3(i)一(i五),and4(i)hol d.Thens up】夕:.(二)一92(二)1=o(,一,)a.s,Pro of(i)h Tepro of sof(13)and(14)followf romh tee rs uls to fh Cengof(3.1)a nd(35),andletR(二)=二fo rthec as eofproV和s(1 4)(i l)h Tepro ofof(15)fo llowsfr omtheh Teorem1ofL i1 0(1 1 1)h TePro ofof(16).From(4),weo b认i
23、n(16)(198 4)(s e eh tePro of s夕:.(x)一习践(二)二=又尤(三于三)二/溉几l石_,二一二、二.J(:)-了夕,人L,邀;,丁,下界杯一一、z,生诚二(:)一李E、碑子竺)二:z一咨、禅于卫);z(;)d;凡一凡人J“.(17)Hen eefr om抉mma1of10)(leta(。)=。I(二),a成。镇b),weha ve。教、IE J(z)一J()I一0(h:),0,0,i=1,2)s u ehthatp(s upJ.(:)一E J.(:)1)镇p(钊卫_.R(;)一R(:)李c:h.)戒c:exp(一Zc资:h若 屺)(21)一:(.气吕、61 3 4
24、中国科学技术大学学报第22卷foa nrgyiven.t0仪t人一二一忐耘and=二一foa:ngyive nt0a ndo,(l一2(q一2)/2(1+,)q一Zp(q一2)/2(l+,)q)2/,)0,wehave习二lp:p.J,一.E J”)仇圣一p-ZC资:公t若)o o(2 2)T he r efore,fr omBoe rl一a Cne tlli lem ma,wes ets up.J(二)一E J.(x)一。(二一f)a.5(2 3)fo rs uf fiei entlylar,hereo,护(叮一2)/2(1+尹)叮From(一s)、(22),we比a t加(s e et加p
25、ro of几e oe rm1of10s up夕:.(二)一夕2(x)!=o(:一,)a.5.whereo,尹(q一2)/2(1+护)叮3.2夕roo fso fh teThe oe rmsBy(11)(24)声。一刀-2 缺+习.i h,执)会人+.B(25)(a)s inc eeae rind e p ende ntrendomvariab le s,ande E:一0,e E圣=,2c o.Thusb yeentral l imitth eor em(ofP.i Blli ng s ley(1986),r Po比b il iy tandMe a s ue r,Zndend,.Prob le
26、m27.6,N ewYo rk;Wi ley),wehave鳃E、,1一引x一念Je 一劫d t,一c ozc o(26)where1den otestheind ie atorf unc tio n,11。=习二e、z(习二,),.h Tenb yh teb oundd edominante o nvergenee一lthe otemweha velimP(I,一.)=悠E毛EI,.c)了不一,一护石习.i h e-平全S。瓦氏团一N(0,。若arZ)(29)(b)y Beentr allimittheo rem,wes et(a s。c o),1石,、缪一行士);切(x)一N(0,雌)了万
27、洲。、.,w he rea卜Ex,夕(x;)2,汾1人二澎,一E仓口*,澎一压尹.1镇镇:,一,:.习:。.几一,1.习二兀 习一i劣两qi-(3 4)s inc eeae rind ep endent,h te n枯=口:.氏为a reindep ende ntasXb os iven,andui sng!为!(:,八(uni-f ormlywi t he s rp ecto t)P。(兔 成C。)二1f ora ec h坛(36)w恤r。P。den oe t:比。p咖抓lityd议ir加往o no f,。ndc o2,.二一令+专.h Tus加ranyi gven云0P(,:.1)EI,全
28、、.,:EEI:!云、,引XI=E(T(X)(let)(37)Nowb y仪mmal,w e郎tT(X)=EI。云、l“IX镇e x n(一护)(38)Wh er二一sq;”景十】+会qi一十仆u sb ybo u nd目domi na ntc on ve rg enc etheore m,a nduisngCbndi tio n4(五)and(3 8),w eob。址n艺二,p(1,l)0.Thee rf oe rb yO Be ri一a Cne ti i lL心mma少i。一。(1)a.sf ors ufi feie nUya lrge”.onh teo therhand5wehav e(
29、4 0)。=E习二。,.,。乳EE习:“,IX,孔E习好(v ar(、)簇2,圣.E习:矛&:1 36中 国科学技未大学学报第2 2 卷毛2,资.,号一E!e,!,E习x了j一ZEIe,卜Exf,孔、,(4 1)NowbyCondition4(11)艺(P叭.)。镇t一,习E衅.0.He neeb yo Brel一a Cntel lie Lmma,w eget少加0(1)a一S,ag介C拭-(43)(d)By(1 2)几一刀=习。+习,(:)公柑孔+认(4 4)w he r。:一习、,(:)一长/。:(let),犷一习二,(:).y B一1Lemma1,wehav eP(二一夕2一V.)二P(
30、风)tn幅)镇Zexp(一,Zq云22/(2:。墓+普tn,;!c。,(45)forsuff ieientlyla rge,andi gvent0,hereoc。,a;二Ex lg(z,)2.e Hneebyo Cndition4(11)习P(。一,:.凡)0。.1Thu sbyBorel一Cantel li址m ma砚一。(滩q云)a.sfors uffieientlyla rge介.Nowby(28)、(4 0)、(43)and(4 7),weoba tin夕一声二o(,i一犷+o伪云)a.sforsuf五eie ntlylag re介,here价.(公一1,2)s ai s tfyeo
31、ndition4(兹).ontheo therha nd,by(12)s up乡.(二)一9(二)口(;,.)一+o(q云)+o(q“)a.5.where,云den ots eh tes tr onge onvergeneerateofmaxs up夕二(x)一夕(x),i一1,2.(4 6)(47)(凌8)(4 9)Refe代nC此He ckmanNE,J.Roy.Sa tti st.o Se.,B 4.(1986),244一248.Ri c eJ,Sa tt.&Pro b.e Le t trs.,4(19 86),203一2 06.S详ckmanP,J.RO y.Sa tti st.So
32、c.,B5 0(1988),J13一436.Eu加nkRL乙whitneyP,J.Sa tt场t.l Pann.&rnf ere.,23(一989),33一43.Sto neC,Sto n eC,An n.Ann。Statist.,5(1977),595一6 45.Stats jt.,10(19 82),104 0一1053.,IJ 飞 11工J w e飞Jl,e sJl,山,dJ.尸勺户nrL厂I J 广w eL广Ltw eLrL第2期ConvergeneeRatesof SomeEstimator sforaSemipar ametrieResres sionModel1377C he n
33、gpE,JMulu8e BnnettG,JAmer9Pr aka s aRaoBLs,10LiJinpins,JMathAn al.Sa tis tL15(1984),63一72.A 义阳c.,57(1962),3 3一45.Non伸r ametri cFu nctlo n al公tia mti on,Ac ade m沁 Pres s,NewYo rk,Re s邑反posi“o n.,2(1986),1 11一1 16.19 83:38.一类半参数回归模型中估计的收敛速度高集体(数学系)考虑回归模型军一t渭+g(:)+e,云1,2,二,其中g()是定义在R(一c o,十c o)上的未知函数,e是未知干扰,基于g.()的估计取 一类核和近邻估计,证得了声和抓)的估计最优收敛速度关键词:半参数回归模型,最小二乘估计,最优收敛速度中图法分类号:02 12.7