ܼၬཌྟଆ(ࣴ)77 ໓ᅣщݼ:1002)1566(2004)01)0077)04ܼၬཌྟଆ(ࣴ)ӧ༐ು(ᇏݓ॓࿐ჽ࣮ളჽ,Кࣘ100039)ᅋေ:Чࢃቖ൞ܼၬཌྟଆᆃ۱ีଢ֥၂۱бࢠ༢֥ࢺകbᇶေٳ3҆ٳ:ࡹଆa࠹ٳ༅აଆ࿊ᄴބᆐ؎bཿቔൈ၇ऌ֥ᇶေҕॉሧਘ൞֩ದ5֥MultivariateStatisticalMod2elingBasedonGeneralizedLinearModels6bܱՍ:ܼၬཌྟଆ;ࡹଆ;࠹ٳ༅;ଆ࿊ᄴބᆐ؎ᇏٳোݼ:O212໓ངѓ്:AGeneralizedLinearModelsCHENXi2ru(GraduateschoolofChineseacademiaofscience,Beijing100039,China)Abstract:;Modelbuilding,:generalizedlinearmodels;modelbuilding;statisticalinference; ଆ࿊ᄴ(၂)Ֆۄ۱С࿊ଆᇏ࿊౼၂۱ଆ࿊ᄴЇݣၛ༯၂ུٚ૫#ၹэਈYٳ֥҃࿊ᄴ;#৳༢ݦඔ֥࿊ᄴ;#ሱэਈ֥࿊ᄴ;#z(x)֥࿊ᄴb༵ซֻ၂۱໙ีbؓਆׄٳ҃(Y౼0,1ؽᆴ)ࠇ౼؟۱ሑᆭ၂֥౦ঃ,ٳ҃ሱಖथק,࿊ᄴ໙ีbؓگᄖ၂֥ׄ౦ঃࣼႵ࿊ᄴ໙ีb২ೂY౼0,1,,,m֩m+1۱ᆴ(ᇿၩ:ֆэਈ౼Ⴕཋ۱ᆴ)))ೂm۱ڛူ֥ದᇏႵི֥۱ඔ)))ა౼Ⴕཋ۱ሑᆭ၂(ೂॉ൫ࢲݔႵႪaਅaᇏaҵ4۱ሑ)҂ିࠁཪbުᆀ֒ሇ֞эਈൈູ؟ົ֥),ࣼႵ၂۱൞ڎ౼ؽཛٳູ҃ଆ֥໙,ี২ೂ֒߈ၐႵ/ӑ҃ྟ0թᄝൈ,ؓॖၛ౼ᆜඔᆴ0,1,,֥эਈ,Ⴕ൞ڎ࿊Poissonٳູ҃ଆ֥໙ี֩֩b৳༢ݦඔ֥࿊ᄴ,൞࿊h(zc(x)B)֥ྙሑ,ߎ൞h(x,B)֥၂Ϯྙሑb൞భᆀ,hೂޅ࿊,൞࿊ሱಖ৳༢ݦඔ,ࠇఃྙൔ?ሱэਈ֥࿊ᄴ,۷؟֥൞Ֆൌ࠽ࠣህြ֥࢘؇ॉ੮:Ֆህြࠇࣜဒ֥࢘؇ु,ପུэਈؓଢѓYႵᇗေ֥႕ཙ?֒ॖି֥ީ࿊ᆀ؟ط༐ຬՖᇏ็ԛູඔ҂؟֥ቋႵ႕ཙ֥эਈൈ,Ԣህြ֥ॉ੮ຓ,္Ⴕུ࠹ٚمb
78ᇏ໓ނྏ௹़ ඔ࠹აܵ 23ज1௹ 2004୍11ᄅ ᇀႿz(x)࿊ᄴ,ᇶေ္ႋՖൌ࠽ࠣህြ֥࢘؇ॉ੮bട֥ࠣ໙ีᇶေ൞:z(x)ᇏ൞ᆺЇݣᇶေིႋ(x1,x2,,֥֩ཌྟཛ),ߎ൞ေЇওଖུࢌིႋ(ྙೂxi,xj֥ཛ)bؓభਆ۱࿊ᄴ໙ี,၂Ϯ࿊ᄴ֥ؓའႵཋbࠎЧ֥ቓم൞бࢠఃPcarson࠹ਈࠇdeviance,࿊ఃཬᆀbೂ২,3۱С࿊֥৳༢ݦඔ,ఃpᆴٳљູ),),)భਆ۱ॖॉ੮,ೂՂՖpᆴॉ੮,֒ಖ൞࿊ٳቆCoxbުਆ۱࿊ᄴ໙ีᄝԩഈ൞၂ᄝ၂ఏॉ੮,֥ࠧϜС࿊֥ࢌቔႨ္֒ቔ၂۱ሱэਈಀԩb২ೂ,ഡႵሱэਈx1,x2,x3,طС࿊֥(Ⴕॖିࣉೆଆ֥)ࢌቔႨႵx1x2ބx1x3,ᄵ໙ีᇏႵ5۱/ሱэਈ0:x1,x2,x3,x1x2,x1x3bߐᆭ,ཟਈz(x)֥۲ٳਈູሱэਈb(ؽ)ሱэਈ࿊ᄴಯ၂ၛx=(x1,,,xm)c࠺ሱэਈbೂഈ໓෮ඍ,Ⴕུxjॖି൞ჰሱэਈ֥ࢌቔႨbӈႨ֥็࿊ሱэਈ֥ٚمႵཟభمaཟުمაᇯ҄مbཟభم൞ᇯ҄ᄹࡆ,ՖषᆺЇݣӈඔཛԛؿ,ഡᄝଖ۱ൈީၘႵx1,,,xrФ࿊ೆ,ഉႵxr+1,,,xmր࿊bϜЇݣሱэਈxi,,,xi֥ଆ࠺ູMO(i1,,,ik),Վଆ֥ؓඔර1kಖݦඔቋնᆴ࠺ູl(i1,,,ik)bೂݔᄝx1,,,xr֥ࠎԤഈࡆࣉxj(j=r+1,,,m)ᄵࡼႵ:A$j=l(1,,,r,j)-l(1,,,r)E0(ၹູэਈ؟ਔ,ቋնᆴൈٓຶঔն,ቋնᆴᆺିഈശط҂ି༯ࢆ)ಸၞुԛAj္൞MO(1,,,r)֥devianceაMO(1,,,r,j)֥devianceᆭҵbAjჟն,іൕၹxj֥ࡆೆ,ଆ֥de2viance෪ཬჟ؟,ࠧଆაඔऌކӱ؇֥ڿჟ؟,ၹطࣼჟႵСೆ࿊֥ሧ۬bႮՎॖᆩ,༯၂҄ႋ౼Ⴕቋնaj֥ପ۱xjbࡆೆxj,ၛx1,,,xr,xjູఏׄ,ᄜοᅶېҌ֥ቓم,ᄝഺ༯֥m-(r+1)۱ᇏ็ԛ၂۱b็࿊ޅൈࢲඏ?Ⴕ2ᇕԩٚم၂൞൙༵ק༯၂۱ൡ֥֒ඔሳk,֞็ԛk۱ູᆸbؽ൞҂൙༵קඔଢ,ط൞ေᄝૄ၂҄ॉҳ,ྍࡆೆ၂۱эਈ֥൬ၭೂޅ,ࠧିde2viance༯ࢆ؟ഒb༯ࢆޓཬ,ᄵ҂ବೆط็࿊๔ᆸbᆃ/ޓཬ0֥ࢸཋೂޅק?ᆃႨ֤ሢ၂۱҂ᆣૼ֥ࢲݔ:MO(1,,,r)ᆞಒ,ᄵAj࣍රႵٳ҃V2(1)bၹՎॖᆷק၂۱AI(0,1),֒Aj>V2A(1),ࢤବVj,ڎᄵ҂ࢤବط็࿊๔ᆸbA֥նཬّ႘ࣉೆࡪ,Aჟཬ(ၹطV2A(1)ჟն),ᄵࡪჟۚ(ྍэਈჟᄜФࢤବ)bᄝൌീഈေᇿၩ၂ུ໙ีb২ೂ,xjॖିՖ໙ี֥ൌ࠽࢘؇ुႵఃᇗေྟbᆃൈ,ॖၛؓࢆ֮ࡪ,ၛьႵ۷؟֥ࠏ߶ࣉೆbॖၛϜಆ҆эਈοఃᇗေྟ(Ֆൌ࠽࢘؇ॉ੮ܙਈ֥ᇗေྟ,ೂ໙ี֥ൌ࠽М႕ໃิ܂ᆃᇕॖି,֒ಖࣼمൌീ)ٳӮؽaোbૄোᇏ֥эਈ۲౼၂۱҂֥A,ೂ္ॖၛᆃဢቓ:ؓ၂ུಪູޓᇗေ֥эਈൌ/ྛЌ0ָ,࠻҂ࣜޅဒࣼ࿊,ೆᄝᆃ۱ࠎԤഈᄜؓഺჅ֥эਈႨഈඍٚمࣉྛ็࿊bߎႵ၂۱໙:ีᄝ၂۱Їݣࢠ؟ሱэਈ֥໙ีഈ,Ⴕॖିૄ۱эਈؓଆ֥܊ངбࢠनᄋ,ࠧޓԛᆀbᆃൈ,၂۱ྍэਈေӑᄀ࿊ᄴࡪ҂ၞ,ၹطॖିᄯӮ࿊ೆэਈݖഒ֥౦ঃb၂۱Ϸم൞Ⴈ֥ٚمࣉྛဒ:ഡݖӱᄝ࿊ೆx1,,,xrު๔ᆸ,Ⴈdeviance(ࠇPearson࠹ਈ္ॖ)ဒଆMO(1,,,r)֥ކӱ؇bି๙ݖ,ᄵ࿊ᄴ๔ᆸ,҂ି๙
ܼၬཌྟଆ(ࣴ)79ݖ,ᄵοഈم࿃็࿊(ط҂ܵྐࣉೆᆀ൞ڎղ֞ࡪ),ᆰ֞devianceဒି๙ݖູᆸbᄝሱэਈ࿊ᄴ໙ีഈႵ၂۱ࠎЧ֥܋്:࠻ေଆაඔऌކ֤ݺ,Ⴛေэਈඔଢ҂؟bᆃؽᆀ൞აྦྷᅧ֥ܱ༢,҂ିၛ/ച౼ྦྷᅧ0֥ٚൔಀԩ,ط൞ေႨᇏ႞ᆭ֡,ࠧؽᆀ࡙ܤbರЧ࿐ᆀԅӽޢՑิԛ၂۱֤ܼ֞ٗႨ֥ٚم,ࠧ෮໌AICሙᄵ:AIC(ଖଆ)=2*ଆᇏҕඔ۱ඔ-2*ଆ֥ؓඔරಖݦඔ֥ቋնᆴ֒ଆᇏବೆྍҕඔൈ,ࡨݼਆш֥ਈࢥᄹࡆbၧӽ֥ሙᄵ൞:ਆ۱ଆ,AICཬᆀູႪb۴ऌᆃ۱ሙᄵ,ॖ֤֞༯૫֥࿊ᄴم:ഡx1,,,xr൞οഈඍٚൔ็࿊֥bېెି๙ݖdevianceކဒ֥ሱэਈࠢ(ૌсྶФବೆ),ഺ༯֥xr+1,,,xm,οభඍ࠹ෘAj֥ٚم,ஆԛః၇Ցࣉೆ(҂࠹ࡪ)֥༵ު,ഡູxr+1,,,xmb࠹ෘbj=AIC(MO(x1,,,xr,,,xj)),j=r+1,,,mഡᄝj=j0ൈbj൮Ցղ֞ቋཬ,ࠧbj>bj,j=r+1,,,j0-1;bjEbj,j=j0+1,,,m00ᄵ࿊౼ሱэਈx1,,,xjb0భ૫ᄝၘႵx1,,,xrطေथקxr+1,,,xmᇏޅᆀႋ࿊ೆൈ,Ⴈ֥ٚم൞ᄝଆEY=L=h(B0+B1x1+,+Brxr+Bjxj)ᇏ,ႨරಖمဒࡌഡBj=0(ᇿၩdevianceᆭҵࠧՎࡌഡ֥රಖб࠹ਈ)bՎࡌഡ္ॖႨWaldဒಀԩbؽᆀ֥ࠞཋٳ҃ཌྷ(V2(1))bႨWaldဒ࠹ෘഈॖିႵ෮ٚьbၛഈࢃ֥൞ཟభمbཟުمᄵٚཟཌྷّ:ఏࡼಆ҆эਈ൬,ೆᄜ၂۱၂۱ัԢbഡᄝଖ၂ࢨ؍ߎഺ༯x1,,,xr(ఃჅ֥ၘัԢԛअ),ಀॉ੮ఃᇏଧ၂۱ھᄝ༯၂҄ัԢࠇ๔ᆸb࠹ෘAj=l(1,,,r)-l(1,,,j-1,j+1,,,r)E0,j=1,,,rAjჟն,іൕัԢxjުଆކ֥ӱ؇༯ࢆჟ؟,ၹطჟ҂ႋФัԢbႮՎॖᆩ,ॖିัԢ֥ؓའ,൞ajቋཬ֥ପ۱xjbؓᆃ۱j,ुAj൞ڎӑԛV2A(1)֥ࡪbӑԛ,ᄵ҂ႋФัԢطݖӱ๔ᆸ,ڎᄵัԢxjbᄜοഈඍٚم,ᄝഺჅ֥эਈᇏࣉྛ༯၂҄ัԢb෮Ⴕᄝཟభمᇏࢃ֥֞ପུᇿၩᆭׄ,ೂؓ҂ᇗေྟ֥эਈႨ҂֥ࡪ,ؓଖུᇗေ֥эਈൌྛЌ,ָቋު࿊ק֥ଆေ๙ݖdevianceކဒࠣAICሙᄵ֥Ⴈ֩,ᄝՎಆ҆Ⴕིbᇯ҄࿊ᄴمᄵ൞ഈඍਆᇕҠቔࢌูႨbഡېࢤବ၂۱ྍэਈط֤{x1,,,xr},ᄵ༯၂҄ुx1,,,xrᇏႵڎॖัԢᆀ,ೂႵ,ᄵัԢ,ڎᄵົӻ{x1,,,xr}҂,ط༯၂҄ुჅ༯֥xr+1,,,xmᇏႵڎॖࣉೆᆀ,ೂႵ,ᄵବೆ,ڎᄵົӻ҂,ಖުुၘࣉೆᆀᇏႵڎॖФัԢᆀ,၂ᆰࣉྛ/֞࠻ॖࣉ,္ॖԛ0ൈູᆸbቋᇔࣉೆ֥эਈඔაࡪႵܱ:ࡪჟۚ,ၞԛࣉ,ჟ֮ᄵၞࣉԛbၹՎॖטᆜࡪ,ࣉೆ֥эਈܒӮ֥ଆି๙ݖdevianceކဒ,эਈඔ҂ݖ؟bᆃဢቓॖିေႨࠫ۱ࡪ൫ෘ,࠹ෘਈࢠն,ॖၛҐႨ༯૫֥э๙ቓم:ഡҐႨଖ၂ࡪႮᇯ҄م֤֞x1,,,xr,ٳਆᇕ౦ঃ1aMO(x1,,,xr)ഉ҂ି๙ݖdevianceဒbᆃൈ,Ⴈཟభم,ஆקഺ༯֥m-r۱эਈࣉೆՑ,ഡູxr+1,,,xmbᅳቋཬ֥r1,MO(x1,,,xr1)ି๙ݖdevianceކဒbᄜ࠹ෘbj=AIC(MO(x1,,,xj)),j=r1,r1+1,,,m() ᅳቋཬ֥j=j0,bj=min(br,,,bm),౼MO(x1,,,xj},0102aMO(x1,,,xr)ି๙ݖdevianceဒbᆃൈ,Ⴈཟުم,ஆקx1,,,xrัԢ֥༵ުՑ,ഡູxr,,,x2,x1bᅳቋཬ֥r1,MO(x1,,,xr)ି๙ݖdevianceކဒbಖުႨ1ཟభمஆקxr+1,,,xm֥ࣉೆՑ(ၛ{x1,,,xr}ູԛؿׄ),ഡູࠧxr+1,,,xmb࠹ෘ
80ᇏ໓ނྏ௹़ ඔ࠹აܵ 23ज1௹ 2004୍11ᄅ (),ᅳቋཬ֥j=j0,bj=min(br,,,bm),౼MO(x1,,,xj)b010ᇿၩᄝၛഈҠቔᇏ,эਈࣉೆࠇัԛ҂ഡࡪ,ᆺ൞οбࢠ:ഺჅэਈᇏ႕ཙቋնᆀࣉ;ೆၘႵэਈᇏ႕ཙቋཬᆀัԛbਸ਼ຓ,ၛഈ֥ቓمႅݣਔᆃဢ֥భ,ิࠧ:ಆ҆ሱэਈ൬ࣉ֥ଆ൞ॖၛࢤ൳֥(ᆃ၂ׄေࢲކ(၂)ଌແԩ֥ᇿࢳ)bೂݔᆃ၂่҂ડቀ,ࣼႋؓଆ֥۲۱ٚ૫ᇗྍബ൪,Їওषԩิԛ֥ପུ໙ีbਸ਼ຓ္ေॉҳဢЧᇏ൞ڎႵၳӈׄࠇ఼႕ཙ,ׄႵڎсေัԢ۱љဢЧ,ࠇᆀсေൈᄹࡆဢЧᄜᇗྍॉҳ֩b২ၿྛؓണ౨վॻᆀॉҳఃྐႨၛथק൞ڎվ۳bଢѓY౼ਆᆴ:0(ॖྐ),1(҂ॖྐ)bิԛਔ20۱ॖିؓՎႵ႕ཙ֥ၹ,࣮ࣜЌ8۱,Ⴕx(1),x(2):ၿྛթॻ:ٳa؟aഒ֖:(x(1)=1,x(2)=-1),؟(x(1)=-1,x(2)=1),ഒ(x(1)=x(2)=-1)x(3):ࢹվൈࡗ(ၛᄅ࠹,ඔਈၹ)x(4):ࢹվඔਈ(ඔਈၹ)x(5):ၛສߎվ౦ঃ:ݺ(x(5)=1),ߊ(x(5)=-1)x(6):ࢹվႨ:ളࠃ(x(6)=1),ြༀ(x(6)=-1)x(7):ྟљ:ଳ(x(7)=-1),୯(x(7)=+1)x(8):߾ڎ:ၘ(x(8)=1),ໃ(x(8)=-1)ሹ܋ဢЧႵ1000۱,ؓᆺݣᇭxཛط҂ݣࢌཛ֥ଆ,֤ܙ࠹ᆴೂ༯іэਈ݂߭༢ඔܙᆴpᆴэਈ݂߭༢ඔ Ֆpᆴഈु,x(1)--x(3),x(5),x(6)ཁᇷ,ܙᆴpᆴxሹन(8)ࢤ࣍ཁᇷbೂၛpᆴູ၇ऌ,ॖัԢ֥э(4)ਈᆺႵxx(4)ބx(7)bᄝA=֥ࡪ༯,Ⴈ(1)(5)(2)(6)ଆ,Ⴈཟభބཟުم,္֤ԛᆃ၂ࢲંx(3)(7))))ัԢx(4)ax(7)bಆሰ݂ࠢ߭(ު)ࢲކx(8)ሙᄵ,္֤ԛᆃ၂ࢲંbᆃіૼ:ྟљაࢹվඔਈᆃਆ۱ၹ҂ᇗေ,طఃၹᄵႵ၂ק֥ᇗေྟbՖ݂߭༢ඔܙᆴ֥ژݼ,ुᆃུ౦ঃౠཟႿ҂ॖौ;ໃ߾ၽ;ႨႿြༀ;ၛສߎվ౦ঃ҂ݺ;ࢹվൈࡗӉ;ၿྛթॻb[ҕॉ໓ང][1] [M].NewYork,Springer-Verlag,1994.[2] [M].London/NewYork,Chapman&Hill,19892ndedition.[3] [J]..,1985,342-368.(ഈࢤֻ47်)[ҕॉ໓ང][1] Ӊڄ:5ࣜ࠶࠹ਈ࿐6[M],ഈݚҍࣜն࿐ԛϱഠ,1996.[2] Ӊᡣ:5ۚ֩࠹ਈࣜ࠶࿐6[M],ౢն࿐ԛϱഠ,2000.[3] ޱମ,ᅦݚڂ:5ᇏݓंԥྐྵൌ࠽ᆣٳ༅6[J],5ҍસࣜ࠶6ֻ4௹,2001.[4] ޱ࿐ڃ:5ݓंԥྐྵౠཟൌᆣ࣮6[J],5ඔਈࣜ࠶ඌࣜ࠶࣮6ֻ7௹,2001.[5] ศး:5ཌྟ݂߭ٳ༅ᇏؓࢩएཛ֥ဒ6[J],5࠹აथҦ6ֻ11௹,2001.[6] ਟഒ܅ၲ,DGujaratiᇷ:5࠹ਈࣜ࠶࿐6[M],ᇏݓದն࿐ԛϱഠ.[7] ఊ฿ཕ,5ࣜ࠶ሇ݅ൈ௹֥ᇏݓंԥྐྵ࣮6[J],5ࣜ࠶࣮6ֻ9௹,2000.
