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(11) iii. FICHA CATALOGRÁFICA ELABORADA PELA BIBLIOTECA DO IMECC DA UNICAMP. Bibliotecária: Maria Júlia Milani Rodrigues – CRB8a / 2116. Mendes, Clarice Camargo M522m. Modelos para dados de contagem com aplicações / Clarice Camargo Mendes -- Campinas, [S.P. :s.n.], 2007. Orientador : Hildete Prisco Pinheiro Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica. 1. Regressão de Poisson. 2. Modelos lineares generalizados. 3. Modelos ZIP. I. Pinheiro, Hildete Prisco. II. Universidade Estadual de Campinas. Instituto de Matemática, Estatística e Computação Científica. III. Título.. (mjmr/imecc). Título em inglês: Models for count data with applications. Palavras-chave em inglês (Keywords): 1. Poisson regression. 2. Generalized linear models. Área de concentração: Bioestatística Titulação: Mestre em Estatística Banca examinadora: Profa. Dra. Hildete Prisco Pinheiro (IMECC-UNICAMP) Prof. Dr. Gilberto Alvarenga Paula (IME-USP) Profa. Dra. Mariana Rodrigues Motta Data da defesa: 03/05/2007 Programa de Pós-Graduação: Mestrado em Estatística.
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(74) . . . . . R87:HU-87@/A7F?AG63?A7:/UHA7:307:/U=V-S/K=PL09CG ;>7FR0G!R87:R0=:H_56=P;R01CH2?K/K14LSQ014D6WG6H+R01CNOG6/KG63?KG HUR87Va=P/2;7@9 c . . "!#%$&')(+*, . s =P3Z?KYP=P;VG /0/ GTi²G 521IŨÆPÆWÅ^©7:-S/KG H2G 3Z?b7@;Q0;>7a?A7:L<G697_R0GX?A/K7:30HINO=P/K;>7:D G H²387 B@7:/21\}BPG69W/2G HK-=PHI?b7 Q0GL0QSHK5}7}B@7:; 7tG HI?b7:L01C941C}7:D E:=tR87B:7@/K1Ò@305 147 5 =:;>= 387rΰ7@L<G697 Å:cCÅPcrl G HI?b7rNO=P/K;>7Sw-<=R017^]yHKG 5 =:3?K143Q87:/?A/K7:L87:9CM87:30RS= 56=P; 7 ?AG =:/K17R0G ;>=R0G694=PH9C1430G 7:/KG6H569\:H2HK1C5 =PH6cmWGygK7 Y 7B@7:/21\}BPG69 7:9CG}7@6? 'P/K147{G Y 7F?A/A7@30H2NO=P/2;7@D}E:=H2Q0YPG /214R87Wc Î7:LG 97ÅPcuÅ 9jβ/A7:3SH2NO=P/2;7:D6WG6HT-87:/K7[G6H2?A7:L0149C14 7:/T7FB@7:/K14Ò:305 147Sc qUG 947:D}E@=G 3Z?A/2G σ G E(Y ) β/A7:3SH2NO=P/2;7:D E:= 56=P30HI?b7:3Z?AG σ ∝ Y =Y 0. 2. 2. 0. σ 2 ∝ E(Y ) σ 2 ∝ E(Y )(1 − E(Y )) σ 2 ∝ [E(Y )]2 σ 2 ∝ [E(Y )]3 σ 2 ∝ [E(Y )]4. √ Y 0 = Y√ Y 0 = sen−1 ( Y ) Y 0 = ln Y Y 0 = Y −1/2 Y 0 = Y −1. a =5 7:HK=R87R01CH2?K/K14LSQ014D E:=R0GFi=P1CHKHK=:3 wÈ7B@7:/K14Ò:305617>R07B:7@/K1\}B:G 9e7@94G}7¦? 'P/217 Y oF14YPQ07:9°´H2Q87 ;Vo R0147Sw`;>7:H[7B@7:/K14Ò:305 147R87 /K7:14Q87@R0/A7:R07 R0G Y o1C30R0G6-<G630R0G 3Z?KGtR0G6H2?b7Wc √z+7:/I?A9CG6?K?tkŨÆPP³Z© 7:-0/2G H2G 3Z?b7FQ0;>7Tg2QSH2?A1u¯85}7¦?A1CB@7!-87:/K7{G HI?b7F7@¯0/K;>7:D}E:= 5 =:94=5}7:30RS=F7)B@7:/21Ò:305617JR0G 56=P;>=)HKG630R0= 5 =:30H2?A7:3Z?AG:chmGkgK7 Y B@7:/21\}BPG69 7:94G 7@? ':/K17a5 =P;RS14H2?K/K1CL0Q014D E:=Ji²=P14H2HK=P3# λ©eGUQ0;>7!7:;>=:YH2?A/K7 Y , ..., Y c βG ;V=PH Q0G σˆ (Y ) = E(Y − λ)ˆ GFHKGygK7Q0;>7>NOQ030D E:= f : R → R w56=P3Z?A43Q87WcaÍh7:;V=PH G6H2?KQ0R87:/ =56=P;V-<=P/I?b7:;VG 3Z?K=VR07>/A7@}E:=>G 3Z?A/2GF7B:7@/K1Ò@305 147V7@;>=PHI?A/K7:9ÈR0G{QS;7NOQ030D E:=R0G Y G)7VB@7:/21Ò:3S5 17 7:;V=PH2?K/A7:9<R0G Y 9 P kÅPcCŨ© (f (Y ) − m) ˆ σ ˆ (f (Y )) E(f (Y ) − m) ˆ = = P , ˆ ˆ σˆ (Y ) E(Y − λ) (Y − λ) G ;Q0G mˆ o7 ;Vo R0147 7@;>=PHI?A/K7:9nR0G f (Y ) c ,aYP=P/K7Sw3S=r5}7:H2=rR0G f (Y ) = √Y G λ >> 1 w ?AG6;>=:HTQSG{7[7:-0/2=}¬S1C;7@D}E:=9C1430G 7:/ 1. 2. 2. 2. 2. 2. 2. 2. n i=1 n i=1. 2. i. i. 2. √ λ t f˜(t) = √ + 2 2 λ. otL07:H2?A7:3Z?AGrHK7@?A1CH2N 7@?6'P/K147 -87@/A7 Q0; 947:/KYP= 143Z?KG /2B@7:9C= 56=P3Z?AG 3SR0= λ cfi=P/>G%¬WG ;V-09C=0w+?A=P;VG ;V=PHV= 143Z?KG /2B@7:9C= [λ/2; 3λ/2] w r √ √ λ λ λ/2 λ λ 3 − 2 2√ f˜ −f = √ + λ = 0, 043 e − = 2 2 2 2 4 2 λ r √ √ 3λ 3λ/2 3λ λ 3λ 5 − 2 6√ −f = √ + λ = 0, 025. f˜ − = 2 2 2 2 4 2 λ.
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(77) 12. 10. 8. 6. 4. 2. 0. p 14YPQS/A7ÅPcuÅ9j, -S/K=}¬W14;>7:D}E:=9C1430G 7:/+-87@/A7V7NOQS30D}E:=[/K7:14JQ87:RS/A7:R87 d'QrHKGygK7Sw5 =P30H2G YPQS14;V=PH Q0;>7[L<=Z7>7:-0/2=¨¬W1C;7:D E:=V94143SG}7:/U-87:/K7 f (t) = √t w<56=P;V=-=WR0G6;>=:H_?A7:;[] Lo ;7@?KG HI?b7:/U-G 94= p 14YPQ0/K7ÅPcuÅPceÍh7:;V=PHT5 =P3SHKG Y:Q014/UQ0;>77:-0/2=¨¬W1C;7:D E:=941C30G}7@/+-87:/K7 mˆ 9 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. √ n n X √ 1 1 n 1X 1 ˆ ˆ + λ = f˜(λ) √ f (Yi ) ∼ m ˆ = Yi + λ = √ λ = n i=1 n 2 λ i=1 2 2 2 λ. ÌX=P;. f (Y ) =. √. n X. n X. n. 1 X ˆ 2 (f˜(Yi ) − f˜(m)) ˆ 2= (Yi − λ) 4λ i=1 i=1. kÅPc :©. G!HKQ0L0HI?A1u?AQ01C30R0=[=[/KG H2Q09C?A7:R0= IÅ:c P©XG6;IÅ:cCÅ^©`?KG ;V=PH. i=1. Y. (f (Yi ) − m) ˆ ∼ = 2. √ σ ˆ2( Y ) ∼ 1 . = σ ˆ 2 (Y ) 4λ √ σ ˆ 2 ( Y ) = 41. °i 7@/A7 Y ∼ P oisson(λ) ?KG ;V=PH+¯8387:9C;>G63Z?AG w8143SR0G -G 30RSG 3Z?AG R0G λ c ÎQ:1:G;/ IÅ}ÆZ±4P©{N 7:Q0;>7 7@38\:941CHKGYP/K\@¯85}7~5 =:3¦g2Q03Z?b7 R07:H>-=PH2HKCB:G 1CHV?K/A7:30HINO=P/K;>7:D6WG6H wX= Q0G G 9CGF5bM87:;>7R0GFN 7:;V49C17R0GF?A/K7:30HINO=P/K;>7:D G H QSGVH2G /K\Y:G 30G6/A7:9C14}7@R0=-=P/JzX=}¬GVÌX=}¬ kŨÆP³:Z©%c!, ?A/K7:30HINO=P/K;>7:D}E@=R0G[zX=}¬GVÌX=}¬~9C14R87>5 =P;75 =P/2/KG D E:=R8738E:=¦]y30=P/2;7@9414R07:R0G!G @=:Q R07B@7:/21Ò:3S5 17 38E:=[56=P30H2?A7:3Z?AG@c`, B@7:/21\}BPG69 Y oJ?K/A7:30HINO=P/K;>7:R87FG6; Y HKG6YPQ030R0=>7N 'P/K;FQS97 (λ). Y (λ). λ 1 , λ 6= 0 Y ˙− λY λ−1 = ˙ Y ln Y , λ = 0. kÅPc P©.
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(88) [· ¹È° V¼(+* ÌX=P30H214R0G6/KG6;>=PHQ0; B:G6?K=P/ RSG~B@7:/21\}BPG614H7:9CG}7@?6'P/K147:H Y G Q0;7 ;>7@?A/214 X R0G~B:7@94=P/2G H¯S¬W=PHG 143SR0G -G 30RSG 3Z?AG6H c`l 146G ;V=PHUQSG QS;;>=R0G694=o!941C30G}7:/X30=PHT-87:/KÒ:;VG6?A/2=PHTHKG G ;ªQ0G β oF=BPG%?A=P/JRSGF-87:/KÒ:;>G%?A/2=PHJ7HKG6/'G HI?A1C;7:RS=G ∼ N (0, σ I) c)|_;;V=WRSG 94=>oF941C30G}7@/ YPG630G /K7:941C}7:R0=VHKG)G%¬W14HI?AG)Q0;7[NOQ030D E:= g ?A7:9Q0G g(E(Y)) = Xβ w<G{7R014HI?A/214L0Q01CD}E:=[R87[B@7:/K14\¨B:G 9 -G /2?KG 3056G{´FN 7:;V49C17!G%¬W-=P30G 3S5 17@9 c Y i=P/JG¬WG ;V-094=0w<HKG w = µ 30=PH'R87@/A\ µ = Xβ c Én3Z?AE:=0w Y − Xβ = w =P30RSG ∼ N (0, σ I) c>Yl ∼G HI?bN7 (µ,NO=:/K;>σ 7SI)wÈ/KG6g(µ) 5}7:C;>=PH!30=;V=WR0G694=941C30G}7@/ wÈ=PQ HKGygK7Sw°=5}7:H2=t30=:/K;>7:9eo Q0;5 7:HK=V-87:/I?A1C5 Q0947:/+R0=PHT;V=R0G 9C=PHT941C30G}7:/2G H+Y:G 30G6/A7:9C14}7@R0=PH s
(89) Ð !JHb©c d'H[st5Ð !!H{/KG -S/KG H2G 3Z?b7@; 7:H2HK14;Q0;>7t5697:H2HKGR0G>;>=R0G694=PH 38E:=r9C1430G 7:/KG6H{Q0G7@L0/A7:3SYPG ;R01u] NOG /2G 3Z?AG6HR01CH2?K/K14LSQ014D6WG6H c Î7:;FLo ; 30=:Hs
(90) Ð !JHVHAE:= ;>=R0G697:R0=:HF=PH[G6NOG61C?A=:H[¯W¬S=:H>R87@H[B@7:/K\}BPG 1CH G%¬W-09C145 7@?A1uB:7@H wS=:Q /2G YP/2G HKH2=P/KG6H c`ij/K14;VG 1C/A7:;VG 3Z?KG:wW?KG ;V=PHT=[-0/KG6R01C?K=P/T941C30G}7@/ Y = Xβ + ,. 2. 2. 2. ηi = x0i β,. G ;Q0G /KG -S/KG H2G 3Z?b7>=PHU/KG Y:/KG H2HK=P/2G HaR87 ]yo6HK14;>7=PL0H2G /2B@7:D E:=VG o)=[BPG%?A=P/_R0G!-87:/KÒ:;VG6?A/2=PH_7 HKG6/KG6; G HIx?A14;>7:R0=:H w Q0GG¬S-S94145 7:;ª7/2G 947:D}E:i=G%¬W14HI?AG 3Z?KGVG63Z?A/KG=:HJ/KβG Y:/KG H2HK=P/2G H!GQ0;>7NOQ030D E:=R87 ;Vo R0147FR87:HU=PL0H2G /2B@7:D6WG6H c Éj;HKG6YPQ014R07SwSR0G%¯830G]yHKG)7{NOQ030D E:=[R0GJ941CYZ7:D E:={?b7@9Q0G g(µ ) = η G'¯0387:94;VG 3Z?KG'G H2-<G65 1C¯05}7¦]ËHKG 7 NO=:/K;>7~R07 B@7:/K14Ò:305617G ; ?KG /K;V=PHVR87 ;Vo RS17 c dJH>R0=P1CHVvS9C?A1C;>=:H-87:HKH2=PHG6H2?AE:= 9C14YZ7:RS=PH[´ 143WNO=P/K;>7:D}E@=FHK=PLS/KG 7VRS14H2?K/K1CL0Q014D E:=FR0=:H Y wSQSG)-µG /I?AG 3S5 G)´N 7@;>C9417!G%¬W-=P30G6305 147:9 c mWG Y:Q030R0=s 5¨ÌXQS9494=:YPMG{mG}7:/294GV @ÑPÑSÅ^©w875 =P30HI?A/2Q0D}E:=VR0G!Q0;«s
(91) Ð !G 3ZBP=P9uBPG{[RSG 5 1CHKG ;H 9 0 i. i. i. i. 4. i.
(92) Ç.
(93) :. . . <
(94) . .
(95) . . . . . ÅPc 'Q87:9Èo!7[R01CH2?A/214L0QS14D}E@=FR0=PHUR87:RS=PH W c'Q87:9 NOQS30D}E:=[R87[;Vo R0147FHKG /K\V;V=R0G 947:R87{56=P;>=NOQ03SD}E:=V941C30G}7:/`R0=PHU/KG6YP/KG6HKH2=P/KG6H Sc'Q87:1CHTHKG6/AE:=V=PHU/KG6YP/KG6HKHK=:/KG H mWQ0-WG]yH2GV1C30145617:9C;>G63Z?AG{Q0GV=BPG%?A=P/ Y 5 =P3SHK14HI?b7R0GV;>G6R014R07:H'143SR0G -G 30RSG 3Z?AG6H{R0G[Q0;>7B:7^] /K14\}BPG 9#7@94G}7¦? 'P/217{56Q¦gK7R014HI?A/K1CL0Q01CD}E:=-G /I?AG 3SD}7V´N 7@;>C9417 G%¬W-=P30G 3S5 17@9 w0=PQH2GkgK:7 9 Yi ∼ fYi (yi ), i = 1, ..., n.. a| ;>7NOQ030D}E@=Q0G[56=P;FQ0;VG 3Z?AG[7@HKHK=5 1477;>o6R0177:=:HJ/KG Y:/KG H2HK=P/2G H o[75bM87:;>7:R87NOQ030D E:=R0G 941CYZ7:D E:= 5}7@30ÏP301C5}7Sw'G6HK-G 561C¯85 7:R87387 -07:/A7:;VG6?K/K1C}7:D E:= R87 N 7:;>C94147 G%¬W-=P30G6305 147:9J-<=:/ θ cs 7:H -=WR0G6;G%¬W1CH2?A1C/T=PQS?K/A7:H+NOQ03SD G HUR0G!941CYZ7:D}E@=0wSR01uNOG /2G 3Z?AG6H_R07[5 7:30ÏP3S145}7Wc i=R0G ;V=PHUG H25 /2G6BPG6/ 5 7:R87NOQ030D}E@=VRSG 30H214R87:RSG RSG RS14H2?K/K1CL0Q014D E:=387FNO=P/2;7 i. f (yi ; θi , φ) = exp{φ[yi θi − b(θi )] + c(yi , φ)},. G ;ÀQ0G'?AG6;>=PH+7FG HK-G /K7:30D 7VR87@HXB@7:/K14\¨B:G 1CH+R87:R87-=P/ HKG6YPQ01C/ cni=:/_RSG6¯8301CD}E:=Sw Z. E(Yi ) = µi = b0 (θi ). wS56=P;>=;V=PHI?A/A7@;>=PHX7. ∞. E(Yi ) =. yi dF (yi ).. , 41 3Z?KG YP/K7:9°7@5 14;>7o[7R0G#%b§
(96) 8 ¢ ¥ ¢}b§%c |a;>7R0G%¯83014D E:=R0G HI?b7143Z?KG YP/K7:9G6305 =:3?K/A7¦]ËHKG[G ; l Q0/2/KG%?kŨÆPÆSŨ©%w -8\@YP14387SŨ³SwÈ=PQ~G ;ªQ87:94Q0G6/!941C?KG /K7@?AQ0/K7HK=:L0/KG[βG =P/217R87s G6R014R87Wca=5}7@HK= R01CHK5 /2G6?K=0w8G HI?b7[143Z?AG6YP/A7:9 7:HKH2Q0;>G 7FNO=P/2;7FR0G!Q0;>7HK=P;>7@?6'P/K147SwS7HK7:L<G6/ w −∞. E(Yi ) =. X. yi P (Yi = yi ).. |aHK7:/KG6;>=:H_-07:/A7G%¬WG6;>-09C1C¯85 7:/+=V5}7:H2=[R0G!Q0;7FB@7:/21\}BPG69È5 =P3Z?AC3Q87G ;Q0G!?AG6;>=PH;9 i. E(Yi ) =. ÌX=P;V= f oJNOQ03SD}E:=[R0G630HK1CR87:R0G@w G 3Z?bE:=. Z. Z. ∞. yi f (yi ; θi , φ)dyi . −∞. ∞. f (yi ; θi , φ) = 1.. mWQ0-=P30RS=>Q0GJ?A=R87:HU7@H_56=P30R01CD G HUR0G!/KG6YPQ0947:/K1CR87:R0G'HKE:=VHA7@?K14H2NOG61C?A7:H wWR0G> WcuÅ^©`?AG6;>=:H Z. −∞. ∞ −∞. exp{φ[yi θi − b(θi )] + c(yi , φ)}dyi = 1 Z. ∞. eφθi yi ec(yi ,φ) dyi = eφb(θi ) −∞. 1 b(θi ) = log φ. Z. ∞. eφθi yi ec(yi ,φ) dyi , −∞. WcuÅ^©.
(97) Æ. :
(98) . G 3Z?AE:= Z ∞ 1 1 R∞ yi φeφθi yi ec(yi ,φ) dyi b (θi ) = φ −∞ eφθi yi ec(yi ,φ) dy −∞ Z ∞ −φb(θi ) = e yi eφθi yi ec(yi ,φ) dyi −∞ Z ∞ = yi eφ(θi yi −b(θi )) ec(yi ,φ) dyi 0. −∞. = E(Y ) = µ.. , HKG6YPQ01C/ wj;V=PH2?K/A7:;V=PH{Q0G7tB@7:/21Ò:3S5 17oR87:R07r-<=P/ V ar(Y ) = φ V wjHKG630R0=~Q0G V = o[75bM87:;>7:R87NOQ030D E:=R0GFB@7:/K14Ò:305617SwQ0G5}7:/K7:56?KG /214}77R01CH2?K/K14LSQ014D E:=G φ o=-87:/AÒ^] dµ /dθ ;VG6?A/2=[R0G!R014H2-<G6/KHAE@=0cna=:?KG!QSG φ > 0 c ÌX=P;V= V ar(Y ) = E(Y ) − µ = E(Y ) − [b (θ )] w8GJ?b7:;{L<o6; b (θ ) = µ wSG 3Z?bE:= −1. i. i. i. i. −1. i. −1. 2 i. i. b00 (θi ) =. 2 i. dµi dθi 0. = (−φb (θi )). 2 i. Z. 0. i. 2. ∞. yi f (yi ; θi , φ)dyi + φ −∞. = −φ[b0 (θi )]2 + φE(Yi2 ).. 0. Z. i. i. ∞ −∞. yi2 f (yi ; θi , φ)dyi. qUG}7:/2/A7:3¦gK7@30R0=[=PHT?AG6/K;V=PHTR87G Q87:D E:=>7:5 1C;7SwW5bM0G YP7:;>=:Ha´G¬S-S/KG H2HAE:=>R87B:7@/K1Ò@305 147:9 φ−1. dµi = E(Yi2 ) − [b0 (θi )]2 = V ar(Yi ). dθi. Éj;/KG697:D E:=V´R014HI?A/K1CL0Q01CD}E:=z`1430=P;V17@9<-87@/A7Q0;>7B@7:/21\}BPG69#7:94G 7@? ':/K17 ?AG ;V=PH;9 Y ∼ Binomial(n, µ), fY (y) = = exp. . . n y. µy (1 − µ)y. µ n y log +n log(1 − µ) + log . y 1−µ . ÎG6;>=:H_G63Z?bE:=VQ0G 9 O-87:/AÒ@;>G%?A/K=[R0G!R01CHK-G /2HAE:=©w φ=1 ONOQ030D E:=VR0G!9414YP7:D}E:={5}7:30Ï:30145 7Z©%w θ = log µ 1−µ. .
(99) ŨÑ.
(100) :. b(θ) = n log(1 + eθ ) = −n log(1 − µ). w. θ. e b0 (θ) = n 1+e θ = nµ n c(y, φ) = log y. w. V ar(y) = φ−1 dµ dθ − µ) = nµ(1 η = g(µ) = log µ=. 0. e xi β 0 1+exi β. c. µ 1−µ. = x0i β. w. . . <
(101) . .
(102) . . . . . w. w. ,¾NOQS30D}E:= R0G6B:GR0G6-<G630R0G /{HK=P;VG 3Z?AGVR0G ci°7@/A7 =5}7@HK=tR07 R01CH2?A/214L0QS14D}E@= z`1430=P;V17@9 w ?AG6;>=:HTG 3Z?bE:=VbQ0G (θ)Y = Y /n -<G6/2?KG 3056G{´N 7:;V494147 Gµ¬W-<=P3SG 305617:9 c ÍnGkgK7:;V=PHj56=P;>=J¯85 7:; 7:HeNOQS30D G Hn/KG%NOG /2G 3Z?AG6H+´JR014HI?A/214L0Q01CD}E:= R0G_i=P1CHKHK=:3>-87@/A7JQ0;>7 B:7@/K1\}B:G 9 9 00. ∗. Y ∼ P oisson(µ), fY (y) = e−µ. µy = exp{[y log µ − µ] − log(y!)}. y!. βG ;V=PHUG 3Z?bE:=[Q0G%9 O-87:/AÒ@;>G%?A/K=[R0G!R01CHK-G /2HAE:=©w φ=1 ONOQ03SD}E:=VR0G!941CYZ7:D}E@=F5}7:3SÏP30145 7Z©%w θ = log µ w b(θ) = e = e =µ w b (θ) = e = µ w c(y, φ) = − log(y!) w V ar(y) = φ =µ w η = g(µ) = log µ = x β c µ=e θ. 0. log µ. θ. −1 dµ dθ. 0 i. x0i β.
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(105) . . [· A· ½ #" ¿ & (+*! ¼ {½ &JÂ)¿ º+È {½ mWG o Q0;>7>7:;V=PHI?A/A7[R0G B:7@/K1\}B:G 14H_7:9CG}7@?6'P/K147:HT143SR0G -G 30RSG 3Z?AG6H'5 =P;;Vo RS17:H_R01CNOG6/KG63?KG H6w 94=:YP=0YwS,30...,E:=[YHAE:=V14R0G63Z?A145 7:;VG 3Z?AGJR01CH2?K/K14LSQ04R87@H ce, NOQ03SD}E:=VR0GJBPG6/K=PH2HK14;V149CM87:30D 7Fo!R87:R87-=P/ 1. n. n Y. L(y; θ, φ) =. i=1. exp{φ[yi θi − b(θi )] + c(yi , φ)}. = exp{. n X. φ[yi θi − b(θi )] +. n X. c(yi , φ)}.. ÎG6;>=:H_G63Z?bE:=VQ0G!=>9C=PYZ7:/21C?A;V= R87[NOQS30D}E:=[R0GJB:G /K=:HKHK1C;>1C94M87@30D}7Fo!R87:RS=>-=P/ i=1. i=1. Wc :© |aHK7:30R0=[=PH+/2G H2Q09C?A7:R0=PH+Y:G /A7@14HXRSG'BPG /2=PHKH214;V149CM87:30D 7)Q0G'B@7:94G6;ÀH2=PL7:H+56=P30R01CD G H+RSGJ/KG Y:Q097¦] /K1CR87:R0GJHK7@?A1CH2NOG61C?b7@HT-<G697{N 7:;V49C17)G¬W-<=P3SG 305617:9 w Wc P© ∂ ` ∂` ∂` , =0 e −E E =E ∂θ ∂θ ∂θ , YP/2G HI?A1² @ÑPÑP³Z©#;>=PHI?A/K7'QSG_-07:/A7'Q0;>7'=PL0H2G /IB:7@D}E:= y 7'H2Q87!5 =P3Z?A/214L0QS14D}E@='-87:/K7!=J9C=PYZ7:/21C?K;>= R87FBPG6/K=PH2HK1C;>1C94M87:3SD}7Fo!R87:R87[-=P/ `(y; θ, φ) = log{L(y; θ, φ)} =. n X i=1. φ[yi θi − b(θi )] +. n X. c(yi , φ).. i=1. 2. 2. 2. i. `i = φ[yi θi − b(θi )] + c(yi , φ) e ∂ 2 `i ∂`i = φ[yi − b0 (θi )] , = −φb00 (θi ). ∂θi ∂θi2. , -09C145 7:30R0=[G HI?AG6Ha/2G HKQS9C?b7@R0=PHTG ; Wc Z©X-87:/K7Q0;7v03S145}7=PLSHKG /IB@7:D}E:=Sw . ∂` E = E(φ[Yi − b0 (θi )]) = 0 ou E(Yi ) = b0 (θi ) e ∂θ . ∂` E ∂θ. 2. = E(φ[Yi − b0 (θi )])2 = φ2 V ar(Yi ) = φb00 (θi ).. l G HI?b7[NO=P/K;>7 V ar(Y ) = b (θ )/φ wS=PQ H2GkgK7Sw 7FNOQ03SD}E:= b(·) R0G6?KG /K;V14307F=PHU;>=P;VG 3Z?K=PHTR0G Y c ,fG6H2?A1C;7@D}E:= R0=PH`-87:/AÒ@;>G%?A/K=:H`R0= BPG%?A=P/ β -<G694=);>o%?A=R0=)R07{;\^¬S1C;7'B:G /K=:HKHK1C;>1C94M87@30D}7!5 =P3] HK1CH2?KGFG ;G 3056=P3Z?A/K7:/J=BPG%?A=P/ βˆ QSG;7¦¬W1C;>1C G 7NOQ030D E:=R0G{B:G /K=:HKHK1C;>1C94M87@30D}7Sw=PQ=9C=PYZ7:/21C?K;>= R0G6H2?b7Wc i. 00. i. i.
(106) Å^.
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(108) . .
(109) . . . . . mWGJ?AG6;>=PH n =PLSHKG /IB@7:D G H_1430RSG -G 30R0G63Z?AG H6w `(β) =. n X. `i =. st7¦¬W14;V14 7:30R0=[7FB:G /K=:HKHK1C;>1C94M87@30D}7Sw i=1. n X i=1. φ[yi θi − b(θi )] +. n X. c(yi , φ).. i=1. n. ∂`(β) X ∂`i = = 0. ∂βj ∂β j i=1. , -09C145 7:30R0=V7/KG6YP/A7R87[5}7:RSG 17-87@/A7Q0;>7F=PL0HKG6/2B@7:D E:=,9. ∂`i ∂`i ∂θi ∂µi ∂ηi = . ∂βj ∂θi ∂µi ∂ηi ∂βj. mS7:LG ;V=PHUQ0G. ∂`i ∂`i = φ[yi − b0 (θi )] , µi = b0 (θi ) ⇒ = φ[yi − µi ]; ∂θi ∂θi ∂µi ∂θi 1 = b00 (θi ) = φV ar(Yi ) ⇒ = ; ∂θi ∂µi φV ar(Yi ) ∂µi ηi = g(µi ) ⇒ depende de g(µi ); ∂ηi X ∂ηi ηi = βj xij ⇒ = xij . ∂β j j. i=P/I?b7:3Z?K=0w8-87:/K7[QS;7=PL0H2G /IB:7@D}E:= i w0?KG ;V=PHTQ0G. ∂`i ∂θi ∂µi ∂ηi 1 (yi − µi ) ∂µi ∂`i ∂µi = = φ(yi − µi ) xij = xij . ∂βj ∂θi ∂µi ∂ηi ∂βj φV ar(Yi ) ∂ηi V ar(Yi ) ∂ηi. , HUG Q87@D G H_R0GJBPG /2=PHKH214;V149CM87:30D 7{¯85}7:;. n X (yi − µi ) ∂µi. V ar(Yi ) ∂ηi. xij = 0 , j = 1, ..., m.. Wc Z©. ,;>7@?A/214VR0G5 =¨B@7:/21Ò:305617 7:HKH2143Z? ':?K145 7 R0=G6H2?A1C;7@R0=P/ R0GBPG6/K=PH2HK1C;>1C94M87:3SD}7 oR87:R87t-G 947 143ZB:G /KHK7VR07V;>7@?A/214'RSG 1C3SNO=P/2;7:D E:=FR0G p 1CHKM0G6/ w J cjÍh7:9CG'-87:/K7V7N 7:;V49C17!G%¬W-<=:30G 30βˆ5617:9 i=1. . ∂ 2 `i E ∂βj ∂βk. . . ∂`i ∂`i = −E ∂βj ∂βk (Yi − µi )xij ∂µi (Yi − µi )xik ∂µi = −E V ar(Yi ) ∂ηi V ar(Yi ) ∂ηi 2 xij xik ∂µi . = − V ar(Yi ) ∂ηi.
(110) : : . . 5 * & < .
(111) . Ũ.
(112) . . . . st7:HT?KG ;V=PHTQ0G:w `(β) =. G 3Z?AE:=. n X. `i e w i =. i=1. . ∂µi ∂ηi. 2. 1 , V ar(Yi ). X 2 X n n xij xik ∂µi ∂ 2 `(β) = E − = xij wi xik . ∂βj ∂βk V ar(Y ) ∂η i i i=1 i=1 . ÉjHK5 /2G6B:G 30R0=307NO=:/K;>7F;7@?K/K1C5 17@9 wG%¬W-0/2G HKHK7:;V=PH 7;>7@?A/214'RSG xy3WNO=P/K;>7:D}E@=R0G p 14H2M0G / . ∂`2 (β; y) J = −E = X0 WX, ∂βj ∂βk. ?b7:9ÈQ0G W = diagonal{w , ..., w } c , H2HK14;7V;>7@?A/214aR0GJB:7@/K1Ò@305 147¦]y56=^B@7:/21Ò:3S5 17 R0G βˆ o R07:R87[-<=P/ 1. Wc ±:©. n. −1 ˆ = J −1 = (X0 WX) ˆ Cov(β) .. ,ÀH2=P94QSD}E:=R87:H G Q87:D6WG6H{R0G[BPG6/K=PH2HK1C;>1C94M87:3SD}7-87:/K7 oVNOG61C?A7-=P/!Q0; ;Vo6?K=WR0=1C?KG /K7@?A1uBP=0c , Y:/KG HI?A1F :ÑPÑ:³Z©[1C30145617:9C;>G63Z?AG30=PH;V=PHI?A/A7 5 =P;V= ¯85 7:/K147 β7 G6H2?K14;>7:D}E:= Q0HA7:3SR0= = ;Vo6?A=R0= R0G aG T?A=P3]kq_7:-0M0H2=P3 cjÉjH2?KGaoUQ0; ;>o%?A=R0=!5 9\@HKHK1C5 =!-87:/K7J= 5 \:9456Q094=!R0GU6G /K=:HhRSGUNOQS30D G H6chmWG YPQS30R0= ÌX9\@Q0R014=)GJst7@/K143SHJIÅ}ÆPÇPÇZ©wWH2G'Q0G /2G ;V=PH+G 3S5 =P3Z?A/K7:/+={B@7:9C=P/ G ;¾QSG!7)NOQ030D E:= f (x) H2GJ7:3Q097Ww 7FN 'P/K;{Q097FR0G _G T?A=:3 =PQ=[;>o%?A=R0=R87:HT?A7:30YPG63?KG H_o!R87:R0=[x-=P/ 9 Wc ³P© f (x ) . x =x − f (x ) G ; QSG t 1430RS145}7[7 t]yo6HK1C;7F1u?AG /K7:D}E@=0c 7{G6H2?K14;>7:D}E:=)R0GaB:G /2=PHKH214;V1494M07:30D}7!G HI?b7:;V=PH`-0/K=5 Q0/K7:30R0={=PH`B@7:9C=P/KG6H`R0G β Q0G';>7¦¬W14;V14 G6; w8=PQHKGygK7Sw8Q0G /2G ;V=PHUG 3056=P3Z?A/K7:/U=PHTB:7@94=P/2G HTR0G β QSG)7@3QS94G ; ` (β) c `(β) , -09C145 7:30R0=[G ; Wc ³Z©`?AG6;>=:H . . (t+1). (t). (t). 0. (t). 0. β (t+1) = β (t) −. `0 (β) . `00 (β). , -0/K1C;>G614/A7)R0G /21CB@7:R87R0=[94=:YZ7:/K1u?A;V={R87BPG6/K=PH2HK1C;>1C94M87:3SD}7Fo 75bM87:;>7:R87FNOQ030D E:=[G HK56=P/KG 0. u =. . ∂`(β) ∂`(β) , , ..., ∂β1 ∂βm. GV7HKG Y:Q030R87R0G /21CB@7:R87R0=94=PYP7:/K1u?A;V=R87BPG6/K=PH2HK1C;>1C94M87:3SD}7oV75bM07:;7@R87;7@?K/K1C aG6HKHK147:387 5 =:;G 3Z?A/K7:R87:H Hab =. ∂`2 (β) , ou ∂βa ∂βb. . H.
(113) }Å
(114) : < ;>7@?A/214'R0G!1C3SNO=P/K;>7:D E:=FR0G p 1CHKM0G6/T=PL0HKG6/2B@7:R87Wc dÀ-87:H2HK= t 30=-0/2=W56G HKH2= 1u?AG6/A7@?K1CBP= t = 0, 1, 2, ... 7:-0/K=}¬W14;>7 `(β) -<G6/2?A=R0G 30=[-0/K=5 G6HKH2=>1u?AG /K7@?A1uBP=[oJR87:RS=>-=P/ (t + 1). .
(115) . .
(116) . . . β (t). . w²G[=-87:H2HK=. β (t+1) = β (t) − [H(t) ]−1 u(t) .. , HU1C?KG /K7:D G HU-0/2=W56G R0G6;7@?Ao!Q0G 7:H_7:9u?AG /K7:D G HTG6; `(β ) H2GkgK7:; ;FQ01C?K=-<G6Q0G 387@H c Íh7:;V=PHTR87:/UQ0;G¬WG ;V-094=[Q0HK7:30R0=VQ0;-0/2=PL09CG ;>7F5 =P30MSG 5 1CR0=0c`dB@7:9C=P/+Q0G ;7¦¬W1C;>1C}7F7FBPG] /K=:HKHK1C;>1C94M87@30D}7R0G)Q0;7[=PL0H2G /IB:7@D}E:= y RSG{Q0;>7>R01CH2?K/K14LSQ014D E:=VzX1430=:;>147:9 (n, π) o y/n c+ÍnGkgK7:;V=PH G HI?AG ;>G6HK;V=5}\:9C5 Q09C=Q0HA7@30R0=V=>;Vo6?K=WR0=RSG)_G T?K=P3W]kq_7:-0MSHK=P3 c mS7:LG ;V=PHUQ0G (t). . `(π) = y log π + (n − y) log(1 − π) + log. . n y. . ,. y (n − y) (y − nπ) − = , π (1 − π) π(1 − π) (n − y) y 00 , ` (π) = − 2 + π (1 − π)2. `0 (π) =. G!5}7:R07[-07:HKH2=R871C?KG /K7:D}E:=F?KG ;«7FNO=P/2;7 π. (t+1). =π. (t). . (n − y) + + (t) 2 (π ) (1 − π (t) )2 y. −1. (y − nπ (t) ) . π (t) (1 − π (t) ). Wm GF1C30145617:/2;>=PHU56=P; π = 1/2 BPG /21C¯85 7:;V=PH Q0G π = y/n G{=PH'-S/6'}¬W14;V=PH -87@HKHK=:H'5 =P3Z?K1u] 3Q87:;1CYPQ87:1CHT´ y/n w0=:Q H2GkgK7Sw@gK\5bM0G6YZ7:;V=PHT387[/KG H2-<=:H2?b7[5 =:/K/KG%?b7-87:/K7 πˆ c |a; =PQW?A/K=;Vo6?K=WR0=-87@/A7G 3056=P3Z?A/A7@/ =:H G6H2?K14;>7:R0=P/2G H!R0G[;\^¬S1C;7B:G /2=PHKH214;V1494M07:30D}7o[=R0= G H25 =P/2GR0G p 14HKMSG / ct,T=r143ZBPo6HFR0GQW?A149C14 7:/K;V=PH!7t1C3SNO=P/K;>7:D E:= RSG p 14H2M0G /)=PL0H2G /2B@7:R07SwjQ0HA7@;>=PH{7 143WNO=P/K;>7:D}E@=FR0G p 14H2M0G /TG6HK-G /K7:R87Sw J cj, G%¬W-0/KG6HKHKE:=R0=[5 \:9456Q094=1u?AG /K7@?A1uBP=¯85 7:9 (0). (1). β (t+1) = β (t) + [J (t) ]−1 u(t) ou. Wc 4:© É°¬S1CH2?KGFQ0;>7>/KG697:D E:=G63Z?A/KG{G H2?KGF;Vo6?A=R0=>G{=R0G)G HI?A14;>7:D E:=-=P/_;>C3014;V=PHUQ87:R0/K7:R0=PHa-<=P3] R0G6/A7:R0=:H chÐÈG ;FL0/K7:30R0=[Q0G J (t) β (t+1) = J (t) β (t) + u(t) .. yi − µi ∂µi ∂` = xij e wi = uj = ∂βj V ar(Yi ) ∂ηi. . ∂µi ∂ηi. 2. 1 , V ar(Yi ).
(117) ^Å ± G';FQS9C?A1C-0941C5}7:3SR0=)=3Q0;VG /K7:R0=P/XG'=R0G 3S=P;>1C387:R0=:/`R87F-S/K14;VG 1C/A7 G Q87:D E:=V7:5 1C;7 -<=P/ wS?AG6;>=:H Q0G : : . . 5 * & < .
(118) .
(119) . . . . ∂µi ∂ηi. uj. 2 ∂µi ∂ηi yi − µ i xij = V ar(Yi ) ∂ηi ∂µi ∂ηi = xij wi (yi − µi ) . ∂µi. , H2HK14;w0=V97:RS=R014/2G 1C?K=R0G> Wc 4P©X-=WRSGJHKG /UG6HK5 /21C?K=>56=P;>=. =P30R0G. J (t) β (t) + u(t) = X0 W(t) Xβ (t) + u(t) n n (t) X X (t) ∂ηi (t) 0 (t) xij wi (yi − µi ) (t) = XW xij βj + ∂µi i=1 i=1 z(t). o!R87:R87-=P/. = X0 W(t) z(t) ,. z(t) =. n X i=1. (t). (t). (t). xij βj + (yi − µi ). (t). ∂ηi. (t). ∂µi. (t). (t). = ηi + (yi − µi ). ∂ηi. (t). .. i°7:/K7 ?K=WR0= i w z oaQS;7JNO=P/K;>7J941C30G}7:/214 7:R87'R87!NOQ030D}E@=)R0G_941CYZ7:D E:= g wW7}B@7:9C17:R07'G ; ∂µi. i. g(yi ) ∼ = g(µi ) + (yi − µi )g 0 (µi ) = ηi + (yi − µi ). yi. =PQVHKGkgK7Ww. ∂ηi = zi . ∂µi. l G6H2?b7NO=P/2;7Sw07[G6Q87:D}E@=r Wc 4:©X-87:/A7=[G H25 =P/2G)RSG p 14HKMSG /U7:H2HKQ0;VG 7[NO=P/2;7 X0 W(t) Xβ (t+1) = X0 W(t) z(t) ,. Q0G{HKE:=7:HaG Q87:D6WG6H'Q0HK7:R87:Ha-87:/A7>7^g2Q0HI?b7:/aQ0;«;>=R0G 9C=[94143SG}7:/U-=P/a;V43S14;V=PHTQ87:R0/K7:R0=PHa-<=P3] R0G6/A7:R0=:H'-87:/K77>B@7:/K14\}BPG 9²/KG6HK-=PH2?A7 z Q87:30RS=7;7¦?A/K1C)RS=;>=R0G694=Vo X G W o71C3B:G /2HA7 R87;>7@?A/214'R0G!56=^B@7:/21Ò:3S5 17Wce, HK=P9CQ0D}E:=[R87@H_G6Q87:D G HUo!R87:R07V-=P/ 9 (t). (t). β (t+1) = (X0 W(t) X)−1 X0 W(t) z(t) .. ,¾B@7:/21\}BPG69n/KG H2-<=:H2?b7 7^g2Q0HI?b7:R87 ?KG ; G694G ;VG 3Z?K=PH 7:-0/2=}¬S1C;7@R0=PHJ-=P/ -87:/K7 =-87:HKH2= R0=V-0/K=5 G6HKH2=1C?KG /A7¦?A1CB:=0c`Én;5 7:R87V-8z7:HKH2=>N 7:]yH2G)Q0;>7[/KG i Y:/KG H2HAE:=>R0G z G ; zX 5 =P;-G H2=PH W t -87:/K7'HKGT=PLW?AG /jQS;7 3S=^B@7'G6H2?A1C;7¦?A1CB@7 β cjÉjH2?A7!G6H2?K14;>7@?A1uB:7UNO=P/K3SG 5 GTQS;30=¨B:= -S/KG RS1C?A=:/e941C30G}7:/ G!Q0;>7[3S=^B@7/KG6HK-=PH2?A77^g2QSH2?b7@R87Sw z w8-87@/A7=[-0/6'}¬W14;V=-87:H2HK=0c η = Xβ (t). (t+1). (t+1). (t+1). (t+1). (t) i. (t).
(120) Ũ³.
(121) :. . . <
(122) . .
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