Likelihood for interval-censored observations from multi-state models

Texte intégral

(1)Likelihood for interval-censored observations from multi-state models Daniel Commenges. To cite this version: Daniel Commenges. Likelihood for interval-censored observations from multi-state models. Sort: Statistics and Operations Research Transactions, 2003, 27 (1), pp.1-12. �inserm-00262048�. HAL Id: inserm-00262048 https://www.hal.inserm.fr/inserm-00262048 Submitted on 10 Nov 2009. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

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(38) \^jl8³]=F_3\^BEjbDnk<H]\w=&j];[\l\l=F9¯8VHe;>9¶¢:B:?[ 8:9 s @CB:D(D=F9<G:=FH%¹Lf+»E»E»i¼2 ¢:BE?=3\*8:?~¹ :xExE E¼Gi8U:=%8:9#=3td8:D<?>=!B:X\^<=n<;8:H \^<8V\BO__Fk<j]Hd=F9BE9<=\^j];>=FH\lB\ljl=F8V\;>9i\^=j]V8:?²r|_F=9<HlB:jl= s BEdHl=FjbV8V\^;[BE9XjlBED 8V9 ;>?[?>9<=HlHLr s =F8V\^DB s =F?A8:H8Hlkdj];[V8:?A<j]BE<?[=FD =Hl<8:?>?GE;E=1\^<=?>;[ :=F?[;><BB s XBEj%;>9i\^=j]V8:?r_F=9<HlBEj]= s BE<Hl=j]V8V\l;>BE9<HB:X8 Hl;[9<GE?[=djlB_F=FH]? H \8: V=F9¶8V \ A C DA I EDA `-¹\^j]=U8V\l;>9<G1\^d= A $ 8:H·Otd= s ¼3ZXBEj H^8V :=B:X{H];>D<?>;[_F;\|= s jlBE1\^d=;[9 s =2t lF JXTC=a8UE=!8H^8:D<?>=BVX{H];>¿F = #\^d= <j]BO_=FHlH]=FH 8:9 s \l<=pBEdHl=FjbV8V\^;[BE9(\l;>D=FHHl<BEkd? s w=R;[9 s =2td= s i ]<8:H]Hlk<D;>9dG \^d=

(39) ;>9 s =Fw=F9 s =9<_F=BVX\^<=

(40) djlB_F=FH]Hl=H¹\^d=

(41) <;>Hb\^BEj];>=H!B:X\l< = ]H]k<³]=_\^ H ¼!\^d= ?>;[ :=?>;>dBOB s ;[H\^<=<j]B s k<_\B:X\^d=;[9 s ;O; s ka8:?T?>;[ :=?>;>dBOB s HF!zaBEjHl8: :=B:XHl;[D%r <?[;>_F;\|p=;[?>?8:?[HlBjl=FHb\^j];>_\T\^BpQ8:j] :B+D(B s =?>HF{MOB<`XBEjT<k<j]=F?[ s ;>Hl_jl=3\^=3r\^;[D(= BE<H]=Fjb:8\^;>B:9<H-\^<;[H;>9 s ;; s ka8V?)?>;[ :=?>;>dBOB s ;[H8:HXBE?>?[BH} F + I #" $%$&. - ! C

(42) <=jl= - " $ ¹/.(]¼ 143(¹ µ¹&]¼ 1 76 µ¹/.¼ 1 q¼3 ' 8:jl;>8:9i\^HB:X\^d;>H?[;> :=?>;[<BOB s _F8:9u=ej];[\l\l=F9 ;>9 _F8:Hl=HBVXCD;tO;>9dG

(43) B:X_FBE9Or. 1. \^;[9k<BEk<H8:9 s s ;>H]_Fj]=\^=2r\l;>D=#BE<Hl=j]V8V\l;>BE9<H ½=½G:;[E=#\l<=½?[;> V=F?>;[<BB s <=9 \^d=e<jlB_F=HlH;[HBE<H]=Fj]:= s 8V\ s ;>Hl_jl=3\^=e\^;[D(=Hp<kO\\^;[D(=!B:XZ\^jl8:9<Hl;\^;[BE9\^B+C8:j s H g.

(44) BE9<=*8V<HlBEj]<;>9dGHb\8V\l=:`<j]=F<j]=FH]=F9i\^;[9<G(G:=F9<=j^8:?[?[ s =U8V\l4à;>H=2td8:_\l?[

(45) B:<Hl=j]E= s BEj jl;[GEi\]r_F=F9dHlBEj]= s `T81_FBEDD(B:9#D(B s =F?8:9 s BE<Hl=j]V8V\l;>BE9a8V?Za8V\l\l=Fj]9½;[9=<; s =FD;r BE?[BEG:En7=F9<BV\^=i \^<;[H8:<H]BEjl<;[9<G1H]\^8V\^=V Á <Hl=j]V8V\^;[BE9<HB:8X 8:jl=e\^8: :=9 8V \ A C DA I EDA18:9 s \l<=;[\8V?AH]\8\^k<H;>HBE<H]=FjbE= s k<9i\^;>? W¹ A<¼2q<=Fj]= A;[HR\^d=e?8:Hb\p;>H];[\R\^;>D=B:X8:9#8:?[;[:=nHlkd³]=F_3\UmA=\k<Hp_U8:?[ ? \^d=nXBE?>?[B+-r|k< \^;[D(=\la8V\;[. H 1 D(;[94 ¹

(46) n¼2à<=Fj] = ;[H-\^<=*\^;[D(=pB:X s =F8V\^)q=BE<Hl=j]: = s 1

(47) RzaBEjR_BE9i\^;[9kdBEk<H;[9\l=F9<H];[\l;>=FHDB s =?A\^<=e?[;> V=F?>;[<BB s _F8:9 8:9 w=*j];[\]\^=F9)} +I "$# & # &(' " $ &. 1.

(48) C. - $% $. $ . - ! " !. %. ! " & % . S <;>H?>;[ :=?>;>dBOB s _U8V9w=kd9 s =Fj]H]\lBOB s ;[9\lk<;[\l;[:=F?[

(49) 8:H\^<= ]<j]BEa8:d;>?>;\| B:X \^d=eBE<H]=Fj]:= s \^jl8³]=F_3\^BEjb±<kd\;[\;[Hp9<BV\HlB=F8:H]\^B

(50) <j]B+E=e\^<8V\\^d;>H;[Hpjl=F8:?>? \^d=!?[;> :=?>;[<BOB s `d8:HR=nH]a8:?>?AH]=F=!;>91\^<=!9<=3t\H]=F_3\^;>B:94zaBEj\^<;[H?[;> V=F?>;[<BB s \lB w=kdHl=Xkd?`;[\DekdH]\u==2tddjl=FH]Hl= s ;>9\^=jlD B:Xw\^<=\lj^8:9<H];[\l;>BE9%;>9i\l=F9<H];[\^;[=FH<;[_ 8:j]=

(51) \^<=a8VHl;>_<8:j^8:D=\l=Fj]HeB:X\^<=DB s =F?HlB#=1Dek<H]\%w=18:<?[=&\^B#=2tO<jl=HlH \^d=n\^jl8:9<Hl;\^;[BE9±<j]BEa8:<;[?>;\^;>=H;>9±\^=jlDWB:X\^<=e\lj^8:9dHl;[\l;>BE9;>9i\^=9<Hl;\^;[=FHF*S<;[H;>H a8:jb\^;[_Fk<?>8:jl?(=F8:H]

(52) ;>9\^<=dBED(B:GE=F9<=BEk<HQ8Vjl :B+1DB s =F?{J|9BV\^<=jDB s =F?[H-;[\ GE=9<=Fjl8:?>?&j]=F¾k<;>j]=FH\^<=*_BED(dkd\8V\l;>BE9BVX{;[9\l=FGEjl8:?>H mA=\k<H&9<B+ H]u=_F;>8:?>;[¿F=1\^<=Hl=±XBEj]Dek<?8VH\lB\^<=;>?>?[9<=FH]Hbr s =U8V\l DB s =F?`-8 DB s =?A;[\^\^<=\^<j]=F=eHb\8V\l=FH ]<=U8V?[\^ <` ];>?[?>9<=Hl#H d` s =U8V\l

(53) j]=FHlw=F_3\^;E=F??8r w=F?>?[= s x Ff 2 O JX!\^<= H]k<³]=F_3\Hb\8:jb\^H1;>9 Hb\8V\l= b<=U8:?\^ d`Ra8VH9d=E=jw=F=9 BE<H]=FjbE= s ;[9\^d= ];[?>?>9d=FHl H (Hb\8V\l=e8:9 s -8VHp?8VH]\RHl=F=9 8V\O;[Hl; \ ) ¹~8\R\l;>D = A<¼ \^d=?[;> V=F?>;[<BB s ;>HF}. 1 - C<C ¹ A C DAd¼* - C<C ¹ A + p ¼ ! C ,¹ p ¼ '.- - C I ¹ A + R ¼ ! I /¹ p ¼ '!0 ¹bf¼ ;[X{\^<=!Hlk<³]=_\<8:Hw=F=F9BE<Hl=j]:= s ;>9\^<=e;[?>?[9<=FH]HHb\8V\l=nXBEj\^<=·ajlHb\\^;[D(=8V\ A,1e\l<=F91\^d=?[;> V=F?>;[<BB s ;[HF}. 1 - C<C ¹ A C DA,1 + I ¼ - C I ¹ A,1 + I DA,1E¼ - I|I ¹ A,1 2 R ¼ ! I ¹ p ¼ ' ¹ E¼ S<;>HT=F¾ka8\^;>B:9<H{8Vjl=:8V?>; s XBEjA\l<=j]=:=FjlH];><?[=-8:H =F?>?8:H XBEjA\^d=;[jljl=3E=jlHl;[<?>= ;>?[?>9<=HlHLr s =F8V\^#DB s =F?!J|9,Q8:jl VB DB s =F?[HF`)\l<=\^j^8V9<Hl;\^;>B:9#<jlBE<8:<;>?[;[\l;>=FHp8:jl= ?>;[9< := s \^B(\^<=\^j^8V9<Hl;\^;>B:9;>9i\l=F9<H];[\^;[=FHi\l<4 = 3BE?>DBEGEB:jlB+ s ;´)=jl=F9i\l;8:? =F¾ka8r \^;[BE9<H$zaBEj4\l<=;>j]jl=:=Fj]Hl;>d?>=Z;>?[?>9<=HlHLr s =F8V\^DB s =F?Ù\lB<;>_*C=H]a8:?>?:Hlw=F_F;>8:?>;[¿F= XjlB:D 9<B+§B:94`<\^<=XBEjb-8Vj s 3*B:?>DBEGEBEj]B=¾k<8V\^;[BE91GE;[:=FHF} 55. C<C /¹ .0%]¼ 1 6. - C<C /¹ .(]¼* ! C I ¹&]¼ - ! C ¹]¼ 0 7.

(54) 55. I|I /¹ .0%]¼ 1 6 - I|I ¹/.(]¼ ! I ¹]¼ 5- C I ¹/.0%]¼ 1 - <C C ¹ .(]¼ ! C I ¹&]¼ 6 - C I ¹ .(]¼ ! I ¹]¼ 5 . ¹~vi¼. S<=*HlB:?>kd\l;>BE9B:X$\^<=Hl=*=F¾ka8\^;>B:9<H8:jl=V}. - C<C ¹ .(]¼51 + : " )&&+ : " )& - I|I ¹/.0%]¼ 1 + : " )& ) - C I ¹ .(]¼5 1 : - C<C ¹ . u¼ ! C I ¹ )¼ - I|I ¹ %]¼ 5 # <=jl=

(55) " $ ¹ .(]¼51 : ) ! " $ ¹ u¼ 5 Sd=FHl=!=F¾ka8V\l;>BE9dH<8+:=!w=F=9G:;[E=9XB:jGE=F9Or. =Fjl8:?A_FB:D(w=F9<Hl8V\^B:jlH;>91R9 s =Fj]Hl=9=\R8:?~¹bf+»:»Evi¼3 J|9dX=Fj]=F9<_=(_F8:9u=a8VHl= s BE9D&8td;[D(;[Hl;[9<G(\l<=?>;[ :=F?[;><BB s !JXC8a8:jl8:D(=3\]r jl;[_D(B s =?{;>H_dBEHl=94`{DB s ;[·a= s KR=\lBE9Or|PR8:d<HlBE9½8:?[GEBEjl;\^<DH¹Hlk<_8:H\^d= Q8:j]¾ka8:j s \8:?>GEB:jl;[\l<D&¼*_F8:9ºw=k<H]= s XBEje\^<=D(8tO;>D;>Hl8V\^;[BE9 ¹\^d=H];>D<?>=H]\ a8:jl8:D=\^j];>_DB s =F?;>H\^d=<BEDBEGE=9<=FB:k<H

(56) Q8:jl VBµDB s =F?`XBE?[?>B+C= s i¶\^d= <;[=F_F=2r;[Hl=<BEDBEGE=9<=FBEkdH{Q8:jl VB*DB s =F?¼3 KRB:9Or|a8Vj^8:D=\ljl;>_8:<<j]Bi8:_<=H$_F8:9 \8V :=-\|B!a8\^<H}{BE9<=;[H{\^d=kd9<_FBE9dH]\^jl8:;>9d= s 9<BE9Ora8:jl8:D(=3\^j];>_-8V<<jlBE8:_%;>9e\^d= Hld;>jl;\nBVXS$k<j]9<k<?>?p¹bfU»iy:hi¼*8:9 s \l<;>H*-8VH s =:=F?>B:<u= s i,zaj] s D(8:9 ¹Lf+»E» 7 ¼3` 8:9<BV\^<=jB:9<=1k<Hl=H(H]D(BB:\^d;>9<G<`ZXBEj%;[9<H]\^8:9<_F=\l<jlBEkdGEºw=F9<8:?>;[¿F= s ?[;> :=?>;[<BOB s Hlkd_

(57) 8:H;>9(¢:BE?%8:9 s @CBEDD(=9<GE=Hp¹Lf+»E»E»E¼3TJ|9(\^d=XB:jlD=FjZa8V\l\l<=NQ 8:?[GEBVr jl;\^<D©;>H8V\]\^j^8V_\^;E=V`;[9\^<=?8\l\^=j\^<=pQ8:j]¾ka8:j s 8:?>G:BEjl;\^<D 8:_<;[=:=FH-8*GEBB s Hlw=F= s B:X_BE9iE=Fj]GE=F9d_F=:p?>?Z\l<=

(58) 8:wB+E=&8:<<j]Bi8:_<=H8:jl=(a8:H]= s BE9½\^<=(?>;> V=3r ?>;[<BB s d;>_<8:H*u==F9 s =Fjl;E= s <=k<jl;[H]\l;>_U8V?>?[:J|9½_BED<?>=2t#<j]BE<?>=D(HH]k<_ 8:H\^d=B:9<=*8V\a8:9 s à;[\;[H;>DuBEjb\8:9i\\^Ba8UE=!8%j];>GEB:k<jlBEkdH s =Fj];[V8V\l;>BE9B:XT\^d= ?>;[ :=?>;>dBOB s i\^<;[H;>H-\^d=dk<jlwBEHl=*B:XT\^d=9d=3t\pH]=F_3\^;>B:94 4®. & %

(59) /"µ

(60) 2*!+ 6FF #

(61) - % !"T"w *n F & @CBE9<H]; s =j8eD(=F8:Hlkdj^8:<?[=pH]a8:_F=¹;('

(62) ¼8:9 s 8X 8:D(;[?[BVX$D=U8VHlk<j]=FH@38:Or HlB:?>kd\l=F?[_BE9i\^;>9k<B:k<H-jl=?8V\l;[:=F?[\lB8 s B:D(;[9a8:9i\CD=U8VHlk<j]=;3 C ZS<=?[;> :=?>;[<BOB s j^8\^;>B;[H s =·<9<= s w}. ¹ E5¼ 1 55 3 C 4®. h. 3.

(63) <=Fj]= >; HC\l<=PR8 s BE9Or|KR;> VB s OD s =jl;[V8V\l;[:=B:X 3 j]=F?8\^;[:=F?(\lB3 C P=F_U8V?>?)\la8V\ [; H\l<=B'rD(=F8:Hlkdj^8:<?[=j^8V9 s BED5:8Vjl;8V<?>=H]k<_\la8V\. . 3 ¹ !¼ 1 . 5 3 . 5 3 C

(64) ' C 3. 5. az BEjC;[9<H]\^8:9<_F=V`d\^d=?>;> V=F?[;><BB s j^8\^;>B!_FBEj]jl=HlwBE9 s [; 9<Ge\^Be\l<=BE<H]=Fj]V8V\l;>BE9&B:X 8j^8:9 s BED6V8:j];8:<?[= ¹\^<8V\;>HZ\^B\l<=urL8:?[GE=Fdj^8 1 ¹& ¼l¼_U8V9(w=jl;[\]\^=9. ¼ . ¹ :¼ 1 C ¹¹ ¼ <=jl= ¹ ¼C;>HC\l<= s =F9<H];[\|BVXT\l<=*?8UB:X j]=F?>8V\^;E=F?(\lB%8GE;[:=F91D(=F8:Hlkdjl=:} XBEj!;>9<Hb\8:9<_=:`XBEjn8_FB:9\l;>9k<BEkdHnV8:jl;>8:<?>=V` ¹ ¼;[H!\l<=

(65) <j]BEa8:<;[?>;\| s =F9<H];[\| Xk<9<_3\^;[BE94MO;[9<_F=\l<= s =F9<B:D(;[9a8V\^B:j s B=FH9<B:\ s =Fw=F9 s BE9 O`<;[9dX=Fj]=F9<_=*_U8:91w= a8:H]= s BE9<?[&BE9 ¹ ¼2`<<;>_;>H\^d=*XB:jlD B:XT\^d=?[;> V=F?>;[<BB s d;>_8:<w=U8:j]H ;>9H]\^8V\^;[H]\^;[_U8:? a8Vu=jlHFpJ\;>HHlB:D(=3\^;>D=FHB+E=jl?>BBE V= s \^<8V\p\l<=e?>;[ :=?>;>dBOB s ;>Hp8 j^8V9 s BED V8:j];8:<?[=:ìw=F;[9<Gn8*_FB:D(wBEHl;\^;[BE9%B:X)\l<=p<j]BEa8:d;>?>;\| s =F9<H];[\|%Xk<9<_3\^;>B:9 8:9 s \^d=jl8:9 s B:D V8:jl;>8:<?[ = ;[\^H]=F?XL <=F9 s =U8:?[;>9<G;\^_BED<?>=2t1<jlB:<?>=D(HHlkd_#8:H;>9dX=jl=F9d_F=n<8:Hl= s BE9;>9Or _FB:D(<?[=\l=BE<Hl=j]V8V\l;>BE9<HBVX{<j]BO_=FH]Hl=FHùH]k<_±8H];>D<?>;·a_U8V\l;>BE9(;>H9<BV\8+V8:;[?8:d?>= 8:9 s ;\e;>H9d=F_F=HlHl8:j]¯\lB#jl=\lk<jl9¶\^B#DBEj]=&Xk<9 s 8VD(=9\^8:?C\l<=FBEjbE=8:j]=

(66) =FHLr w=F_F;>8:?>? ;[9i\^=Fj]=FHb\^= s <=Fj]=½;[9j];[\^;[9<G¯\^<=?>;[ :=?>;>dBOB s XBEj1;>9i\l=Fj]V8:?²r|_=F9<H]BEjl= s BE<H]=Fjb:8\^;>B:9<HXjlBED 8:9;>?[?>9<=HlHLr s =F8V\^D(B s =?%=Hla8V?>?ZHl==(\la8V\!8:9;>?>?[9<=FH]Hbr s =F8V\^1DB s =?)_F8:91u= s =Hl_jl;>w= s 8:H8ed;[V8:jl;>8V\^=R_FBEkd9\l;>9<G%<j]BO_=FH]HFZ=_BEk<? s ·a9 s \^<=!?>;> V=F?[;><BB s XB:jR;[9\l=Fjb:8V?r|_=F9<H]BEjl= s BEdHl=FjbV8V\^;[BE9B:XZ8&k<9<; s ;>D=F9<H];>BE9<8:? _FB:k<9i\^;>9dG,<jlB_F=HlH1jl=?8V\l;[:=F?[¶=F8:Hl;[?[:`XB:j&;>9dH]\8V9<_F=±i _FBE9<H]; s =jl;[9<G\^<8V\

(67) = a8UE=;>9i\l=Fj]V8:?²r|_=F9<H]BEjl= s B:<Hl=j]V8V\^;[BE9%B:Xu8j^8:9 s BED6V8:jl;>8:<?[=<;[_(j]=F<j]=FHl=9i\^H \^d=\l;>D=BVXO³]k<D4¡B=:=Fj-XBEj8Dek<?[\l;[V8:j];8V\l=-<jlB_F=HlH\^<;[Hw=F_BED(=HDnk<_ DBEjl= s ; _Fk<?\U @CBE9<H]; s =jp\^<=n_U8:H]=B:XDek<?\^;[V8:j];8V\l=(¹ BEjRD&8:j] := s ¼RwBE;>9i\djlB_F=FH]Hl=HF} 1 ¹ " D 1 f 2 ¼3µ7=F9dB:\^ = 51 " 8:9 s 21! " `d=Fjl= " 8:jl= s " 8:jl= \^d=_BED(w=F9dH^8V\lBEjlH%B: X " ¹\â8V\;[" H " 6 " 8:j]=1D&8:jb\^;[9<Gi8:?[=FH8:9 # ;>9d_Fjl=F8:Hl;[9<G<jl= s ;>_3\8:<?[=<jlB_F=HlH]=FH¼2d<=9&\^<=_FB:D(w=F9<Hl8V\^B:jlH8:jl=_FB:9\l;>9k<BEkdH = s =·<9<=±;>9i\l=F9<H];[\^;[=F% H $ " " 1 &$ " @CBE9<H]; s =j

(68) 8:?[HlB½\|B¯<j]BEa8:<;[?>;\| D=U8:H]k<jl= H 3 8V9 s 35;[\^ 3( ')' 3(¢E8:_B s ¹Lf+»iy 7 ¼p<8:H*GE;[:=F9 \^<=%XBEjlDnk<?8 XBEj%\^<=?>;[ :=?>;>dBOB s j^8V\l;>BB:X\l<=djlB_F=FH]* H ½\^d;>H%XBEjlDekd?8#;>H<jl=Hl=9\l= s ;[9 y.

(69) p9 s =Fj]Hl=F9=\p8:?R¹bfU»E»Evi¼-;>9\l=FjlD BVX<j]B s kd_\]r;>9i\^=GEj^8V?à8:9 s Hlkd<uB:Hl;>9dG\l<=Fj]= ;>H9dB;[9dXBEj]D&8V\l;>BE9

(70) 8V\\^;[D(=x%;[\C\8: V=FH\^<=XBEj]D} )& ) )& ¹Lf 6 5 ¹]¼]¼ 5 5 3 " 1 5 ¹ ]¼

(71) ) )& I ¹Lf 6 5 ¹]¼]¼ 5 3 ) " I " S<;[H;[He\^<=?[;> V=F?>;[<BB s jl8V\^;[B±XBEje\^d=Hl;>G:D&8r|8:?>G:=F<jl 8 1 ¹ ¹]¼% Àxi¼ ;[\l_BED(w=F9dH^8V\lBEjlHjl=?8V\l;[E=\lB(\^d=!·a?\^j^8\^;>B: 9 ) 1¹ ,¹ u¼D x ]¼3 \^k<H-=*_U8:9d9<B:\ s ;[jl=F_3\^?&8V<<?[(\^<=RXBEjlDekd?8nw=F_F8:k<Hl== s B%9<B:\B:<Hl=j]E= <kd \ © S<=Fj]=e8:j]=\|B

(72) Hb\^j^8\^=FG:;>=FHXBEj8:<d?[;>9<G\l<;>HXBEjlDnk<?8e\^BBEk<j;>9d_FBED<?>=3\^= BE<H]=Fjb:8\^;>B:9djlBE<?[=FD} S$8: :=*\^d=_BE9 s ;\^;[BE9a8:?u=3tOu=_\^8V\^;[BE94} * # 6 0 R<<?[(\^<=XBEj]Dek<?>8e9<B:\BE9 ¸<kd\BE98:9BE<Hl=j]:= s djlB_F=FH]H pHp8:9=2t<8VD(<?[=!BVX\^<=?8\l\^=jR_BE9<H]; s =j\l<=nBE9d=3r s ;>D=F9<H];>BE9a8V?¹ HlB 1 f¼ <j]BO_=FHl H -&¹ ] ¼ 1 ,"¹ !]&¹ ]¼l¼2`<=jl# = !b¹ ] ¼ 1 H]k<T¹ } %¹ u¼ 1 f+¼30Y s =3·a9<;[\l;>BE9µ\^<;[H(djlB_F=FH]H;>HBE<H]=Fj]:= s } 1 ¹ C&¹ ]¼ (%$ xi¼3`H]B\â8\&= _U8V9

(73) 8:<d?[(¢E8:_B s4£ HXB:jlDek<?>8dZ@CB:9<Hl; s =Fj\^<=_U8:H]=pB:X4<k<jl=?[;>9i\l=Fj]V8:?²r|_=F9<H]BEjl= s s 8V\^8d} %¹]5¼ 16fXBE@ j 51 A C DA I DA ` %¹]8¼ 1 xB:\^<=j];[Hl=:S<=F9 a8VH 8 s ;>H]_Fj]=\^=*_BED(w=F9dH^8V\lBEj;[\^n³]k<D<HF 8V \ A C DA I DA F ¹ A & ¹ A $ ¼ 143 * $ 5¼ 1 f 6 ¹ A $ + I 5¼ 1 x 0 (' ",+ & C.J\!;>H*=U8:Hb#\^BHl=F=%\^<8V\ni8:d<?[;>9dG±¢E8:_B s)£ H*XBEjlDnk<?8& *=&) GE=3\!\l<==3tOu=_\l= s jl=Hlk<?\RXB:j\l<=?>;> V=F?[;><BB s ¹=3tO<j]=FHlH]= s ;[9\l=FjlD BVX{\^d=nHlkdj];[V8:?4Xk<9d_\^;[BE90/µB:X \^d=-³]kdD(\^;[D(=U¼3}. 1 5 3 1 / /¹ A 1 + I +¼ 6 / /¹ A 1V¼ <=jl=e\^d=j^8:9 s BEDWV8:j];8:d?> = 1¶;>H s =·a9<= s 8VH 2C¹ A,1:¼ 6 2C¹ A,1 + I ¼ 1 fE4;[9 \^d;>HXBEjlDnk<?8&=(<8+:= s j]BE<w= s \^<= s =9<BED;>9a8V\lBEj<;[_ s B=FH*9<B:\ s =u=9 s BE9\^d=<8:j^8:D=\l=Fj]HF A® . %/!

(74)

(75) !!"$"¶. % 1§ "$"w *! @CBE9<H]; s =jRBE9d=!_BEk<9i\^;[9<G(djlB_F=FH]H 43pXBEj;[?>?>9d=FHlH¹ 53¹&]¼21§x%;[X$<=U8V?[\î8V\ 2` 53&¹ ]2¼ 1 f!;X{H]k<³]=F_3\w=F_F8:D(=*;[?>?)w=XB:jl==]¼-;\^1;>9i\l=F9<H];[\| $638:9 s BE9<=XBEj s =F8V\^ 87 ¹ 87&¹ ]= ¼ 1 x;X8:?[;[E=8V= \ 2` 87&¹ ]¼ 1Wf&;[XCHlk<³]=F_\ s ;>= s w=XBEj]=]¼ ;[\l1;>9i\^=9<Hl;\| $97m =3\k<HD(B s =F?u\^<=*;>9i\l=F9<H];[\^;[=FH¹;>9\^< = )>r·<?[\^jl8V\^;[BE9q¼8:HF} :.

(76) $ ¹&]¼81

(77) 3. $ ¹&]¼51. '. . ),+ & C. ),+ & C. . .-. '. . ),+ & C ! C I ¹]¼. ),+ & C ! C ¹&]¼. .-. ),+ & I ! I ¹]¼ 0. ¹ g¼ JX*C= s =3·a9<= 1 53 - 57 - 87¹Lf46 533¼2`\l<;> H s =·a9d=FH8,Dek<?[\l;rH]\8\^= <j]BO_=FHlH\8: ;[9<GV8:?[k<=FHBE9 xUf

(78) 8V9 s ;\^½\lj^8:9dHl;[\l;>BE9 ;[9\l=F9<H];[\l;>=FH ! C I ¹ ¼3` ! C ¹ ¼%8V9 s ! I ¹ ¼%w=\|=F=9 ¹~xFf¼3`p¹~x E¼8:9 s ¹Lf2 E¼ej]=FHlw=F_3\^;E=F?wR\^d=Fjl=;>H ; s =F9i\^;\|w=\|=F=F9!\^<;[HADek<?[\l;rH]\8\^=¹;>?>?[9<=FH]Hbr s =U8V\l¼4<jlB_F=HlH$8:9 s \^<=<;:8Vjl;8\^= _FB:k<9i\^;>9dGdjlB_F=FH]HF S$B 870C=8:HlH]BO_;8V\l=8jl=HlwBE9<Hl=<jlB_F=HlH 7¹ ]¼ 1 fE`XB:j&8:?[? \^B 53+`C=±8VHlHlB_F;>8V\^=8½j]=FH]uBE9dHl=djlB_F=FH] H 3&¹ ] ¼ 1 fXBEj 1 A C EDA F ` 3& ¹ ]8¼ 1xBV\^<=j];>H]=:S<=*BE<H]=FjbE= s <jlB_F=HlH;> H 2 1© ¹ 3 7 ¼2`<;[\^ 97. . '. . .-. *. '. .-. ¹ ]¼51 3. <=jl=5!]¹&]¼51Hlkd7. . - . '. -. ¹ !b¹]¼]¼. 43. -} ¹ u¼ © 1 f à8V9 s ¹] ¼51 ¹]¼DXBEj 3. 7. 57. . . ¢E8:_FB s4£ H!XBEj]Dek<?>8_F8:9,w=

(79) 8:<d?>;>= s ;X-=

(80) 9<B+5\l<=&_BEDu=9<H^8V\lBEjB:X2 ;>91\^d=B){·a?[\lj^8V\l;>BE94}8:?\^<B:k<GEC=!BE<H]=Fj]:= 87µ;[\lH_BEDu=9<H^8V\lBEj;>H9<B:\\^d= H^8VD(=BE5 9 ) 8:9 s BE9 ;)LSk<HF`d=p9d=F= s _BED<kd\^=\^d=p_FB:D(w=F9<Hl8V\^B:jlHB:X 23 8:9 s 7 ;>9e\^d= B)r·a?\^jl8V\^;[BE94TJ\;>H=F8:H]e\lB*Hl=F=\la8V\ 23 <8:H8 s ;>H]_Fjl=3\^=_FB:D%r w=F9<Hl8V\^BEj<;[_;[H9kd?>?Z=:=Fjbd=Fjl=&=3tO_F=d\ewBEHlH];><?#8V\BEdHl=FjbV8V\^;[BE9\l;>D=FH A $ 1 x W<=jl=;[\;[H=F¾ka8:?4\lB}. ¹ A $ ¼ 1 3 *. ¹/A $ + I ¼51 x 7 ¹ A $ 6p¼51 x 0 ' ! "+ & C.- ' " + & C. ) ) J\%_U8V9ºw=Hl==F9¶\â8V\ 3 8:9 s 27 _F8:9¶u=j]=F<?>8:_F= s i 43&8V9 s 8708:9 s ` jl=D(;[9 s ;[9<Ge\â8\ 53R8:9 s 87µ8:jl=9<BV\R;[9 s =u=9 s =F9i\F`q=*_U8:91jl;\^=V} &. 3. 3. ¹ A $ ¼81 f 6 . 3. 3 * ¹ A $ ¼51 f 6 ¹ A $ + I ¼51 x ¹ A $ 6¼51 x 0 1 - - C I [¹ ¹/A A $ + + I DDA A $ ¼ ¼ C $ I $ <=jl= - C [¹> ¼ 1 - C<C ¹ ¼ - - C I ¹ ¼

(81) ¹\l<=

(82) <j]BEa8:d;>?>;\| B:Xu=;>9<GH]\^;[?>?8:?>;E=U¼3 B:Xn_FBEk<j]Hl=\^<= \^jl8:9<Hl;\^;[BE9 <jlBE<8:<;>?[;[\l;>=FH - " $ ¹/.(] ¼1Hb\^;[?>?a8U:=¯8µD=U8V9<;>9<G¶;[9 \^=jlDHB:X)\l<=Rd;[V8:jl;>8V\^=_BEk<9i\^;[9<G<jlB_F=HlHF`XBEj;>9<Hb\8:9d_F= - C<C ¹ .(] ¼ 1 3 * ¹&51 x -¹&] ¼81 x 6 ¹ . ¼51 x ¹/. ¼ 1 xi¼3 53. 53. 57. 53. 57. 43. 87. ».

(83) pH%XBEj" -¹]¼3ÀXBEj*. C`;[\;>H%BE<H]=FjbE= s ;[9¶_FBE9i\^;[9k<BEk<H\^;>D=HlBC=a8UE= 27 ¹]¼ 1 %¡B+C=3E=Fj;[\lH_FBEDw=F9<Hl8V\^BEj;[H9<B:\\l<=%H^8:D=;>9±\l<= )>r ·a?\^j^8\^;>B:9#8:9 s ;>9#\^d= ;)r·a?\^j^8\^;>B:94}R;\*;>H_?>=U8Vj\â8V\\l<=%;>9i\^=9<Hl;\|#GE;E=F9 ;[9 XBEj]Dek<?81¹g¼C;>H9dB:\ B),+ar|D=U8VHlk<jl8:<?>=V=*D(8U

(84) k<H]=\l<=*;>9<9<B+V8V\l;>BE9

(85) \l<=FBEj]=FD 8:9 s _FB:D(<kO\^=\^d= ;)r|;>9i\l=F9<H];[\|18:H} 87. 87. $ ¹ ]¼51. N * $97-¹]¼ 6 B),+ 0 1 N * . 7. '. ),+ & C. . .-. *. ),+ & C ! C ¹]¼ -. '. .-. . ),+ & I ! I ¹&]¼ 0 6 B),+ 0 . '. -. J|9

(86) \^d;>HCXBEj]Dek<?>8dÒB:9<?[ ' ),+ & C.- ;>H-9dB:\ B),+arD(=F8:Hlk<jl8:<?[=pHlB\l<=BE9<?&djlBEOr ?>=D ;[H-\^B%_FBED<kd\l= N * ' ),+ & C.- 6 B),+ 0 1 3 * 53¹& 6¼51 x 6 B),+ 0 . . JX&87¹& 6¼ 1 f&=_U8:9,\^8: :=18:9i,8:jl<;\^jl8:j] V8:?>kd=(XBEj\^<;[He<jlB:a8:<;[?>;[\|wZ;X 53"¹ !b¹ 6p¼l¼51©fE`d\l<;>HdjlBEa8V<;>?[;[\|(;>H9k<?>?~S<=BE9<?

(87) 9<BE9r\^j];[;8V?)¾ka8:9i\^;\|;>H 3 * 53&¹ 68¼ 1 x 6 87-¹ 6p8¼ 1 x 53+¹ !b¹ 6p¼l5¼ 1 x 0 1 - - C<C ¹ ¹ !b!b¹ &¹ 6p6¼ D¼ (( 6p6¼ ¼ C z{;[9a8:?>?E`\^<= B)r;>9i\^=9<Hl;\|

(88) B:X 87¶;>H. $ &¹ ]¼51. ),+ & C * ! ),+ && C ! ¹&]¼ - ),+ & & I ! I ¹&] ¼ 0 <=jl= ! ¹&]¼21 ! ),+ & " ),+ & !)& ),+ & " ), + !& ),+ & " ),+ & )& S<;[HXBEj]Dek<?>8ea8:H89<8V\^kOr 7. . '. 7. . . '. .-. . . . . . . .-. '. 7. . . . -. j^8V?);>9i\l=Fjldjl=\^8V\^;[BE94`\^<=;[9\l=F9< H];[\|&u=;>9<G%8e=F;>G:<;>9<GBVXT\l<=\^jl8:9<H];[\^;[BE9

(89) ;[9\l=F9Or Hl;\^;[=FHXjlBED <=F8:?[\l±8:9 s ;>?[?>9<=HlH;[\l\^d=njl=¾k<;>j]= s <j]BEa8:d;>?>;\^;[=FH-_BE9 s ;\^;>B:9a8:? BE9a8V\a8VHw=F=9(B:<Hl=j]E= s ³]k<Hb\u=3XBEjl@ = 2;[Xw\^<=H]k<³]=_\-<8:HZu==F9(BE<Hl=j]:= s ;>9\l<=*;>?>?[9<=FH]H-H]\^8V\^=V`<\^<=9\l<=*;>9i\^=9<Hl;\|;>H ! I ¹XBEj8:98:?[;[E=H]k<³]=_\2¼2 S<=p?[;> V=F?>;[<BB s jl8V\^;[B*;>9&¢E8:_FB s4£ HXBEj]Dek<?>8_U8:9(u=jl;\l\^=9&8VH\l<=<jlB s k<_3\ B:Xu\^<j]=F=R\l=Fj]D(H 1 3 7 Sd=R·aj]H]\\^=Fj]DÂ;>HZ\^<=_FB:9\ljl;[<kd\^;[BE9B:X4BEdHl=Fjbir ;>9dG(8p³]k<DBVX 43+};\;>H=F¾ka8:?A\lBf*;[X$9<B³]k<Da8VHRw=F=9BE<H]=FjbE= s 8:9 s ;[X{8 ³]k<D1a8:Hu==F9BE<H]=Fj]:= s 8* \ A 1O} . . 1 3. ¹/A ¹/A. &. 3. &. 3. 1V¼ 1V¼. 1 -- C [I ¹/¹ A A,1 1 + + I DA A,1E1E¼ ¼ - - C ¹/¹/AA 11 ++ I AA 1E1E¼¼ C CI I I. az jlBED 9<B+ BE9 = s jlBEº \^<& = s =F9<BED;>9<8V\^BEj&8:9 s \l<=\l;>? s =±8:9 s C=±;[?>? + " Hl;[D(d?[jl;[\l=:} 3 1 + " & S<=*Hl=_FBE9 s \l=FjlD ;>HC \l<=_FB:9\ljl;[<kd\^;[BE9B:XTBE<H]=FjbO;[9<G8³]k<DB:X 87C}Z;\-;>H =F¾ka8V?\lB f(;XC9<Bn³]k<Da8:H*u==F9½BE<Hl=j]:= s ;X-8n³]k<Dº¹\^<8V\;>H s =F8V\^q¼a8VH . .

(90).

(91).

(92).

(93). f+x. .

(94) w=F=F9BE<H]=Fj]:= s 8\ ` ;[\;[H=F¾ka8:?A\^B%$

(95) 7 ¹ p¼3JX{\l<=Hlk<³]=_\pa8:Hu==F9Hl=F=9;[?>? 8V* \ A,1n\^<=*_FB:9\ljl;[<kd\^;[BE91;>H 7 1 ! I ¹ p¼2<;[XT9<B:\;\;>H. 7. 1 ! ¹ p¼ 1 - C<C ¹ ]¹ 6¼D 6¼ ! - C [¹¹ b¹ ¼ 6¼D- C I ¹ 6]¹ ¼ 6¼D 6¼ ! I ¹ ¼ C "!. 7. . "!. "!. S<=?8VH]\p\l=Fj]D B:XZ\l<=nXBEj]Dek<?>8d`w\l<=e<j]B s k<_\;[9i\^=FG:j^8:?TB+E=Fj\l;>D=FH<=Fj]= 9<B(³]k<D¯a8:du=9<= s `;[Hn\^<=&<jlB s k<_3\%B:XR8 s ;>_jl=3\^=18:9 s 8±_BE9i\^;[9kdBEk<Ha8:jb\U}. 3 7¶S<= s ;[Hl_Fj]=\l=<8:j]\ 3_FBED=FHXj]BED\l<= s ;>Hl_jl=3\^=1_FBEDw=F9<Hl8V\^BEj 3 8:9 s ;[XZ8(Hlkd³]=F_3\a8:Hu==F9#H]=F=F9;>?>?4XB:j\l<=·<jlH]\R\^;[D(=8V\;A 1%;>HR8(H];>D<?>= <j]B s kd_\U} 1 + I. 3 1 ¹bf 6 & 3 ¹ A $ ¼]5¼ 1 - - C<C [¹ /¹ A A C DA,A 1 1 + + I ¼ ¼ C C $ I I \^d=djlB s k<_\!H]\lBE<H8V\ A,1 + I w=F_F8:k<Hl=%\^d=Fjl=;>H8!³]k<D8? \ A 118:9 s \^<=_FB:D%r w=F9<Hl8V\^BEjC;>HC_BE9<Hb\8:9i\8VX\l=Fj A,1OO;[X4\l<=Hlkd³]=F_3\R;[HC9<=3E=jHl=F=91;>?>?~È\^<=p<jlB s k<_3\ GEB=FH-kd9\l;>?w\^<=?>8:H]\-;[Hl;[\\^;[D(=VZz{;>9<8:?>?\^<=_BE9i\^;>9k<B:k<Ha8:jb\B:X \^<=p<jlB s k<_3\ ;>9i\l=FGEjl8:?4;>H. 7 1 ¹Lf 6 5 7 &¹ ]¼l5¼ 1 + ) )& ) . ). . %#. Á 9 A $ + I ' ' A $ `)<=jl=43¹ A $ + I ¼ 16x8:9 s \^d= 3BE?[D(B:GEBEjlB+=F¾ka8\^;>B:9<H!¹~vi¼. $ ¹ ]¼51 ! ¹&]¼51 7. . 7. 6. 5. ¹ 6p¼*1©x&C=%a8UE=kdHl;>9dG 57. ?>BEG - C [¹ A $ + I % ]¼ 5. . Sk<HXBEj8Hlk<³]=_\R<B(<8:H9<B:\w=F=F9Hl==F9;>?>?wC=a8UE=V}. 1 + )& :8 9 s XBEj8Hlkd³]=F_3\pH]=F=F9;>?[?48V\*,A 1O} )& ),+ )& + . 1 . 7. ) 1 - ¹ A 2 R ¼ C C. ) 1 - [¹/A DA 1:¼ - ¹/A 1 + R ¼ C C I|I z{;>9<8:?>?!XBEj8Hlk<³]=_\C9<B:\Hl==F9

(96) ;[?>?`:_U8:?[?>;>9dG /A 1 ]¹ p ¼\l<=p?>8:H]\Z;[Hl;[\\l;>D=:`

(97). 7. =<8+:=. . .

(98). . !. 1 - <C C ¹/A C A/O¼ - C [¹ A 2 ¼D 3. 7. fEf.

(99) Sk<H\^<=*?>;[ :=?>;>dBOB s ;[HF}. 1 - <C C ¹ A C DA<¼ - C [¹ A + R ¼ ! ¹, p ¼ ' <=jl= ! ¹ R ¼51 ! " % & % # & " % # & " % # & `dd;>_;>H; s =F9i\^;[_U8:?)\lB¹bf¼2 zaBEj8H]k<³]=F_3\Hl=F=9%;>?>?8V\5A 1O`Vjl;[\l;>9<G\^<=?[;> V=F?>;[<BB s 8:H 1 . . . 7. . . . . . . =<8+:=:}. 1 - - C<C ¹ ¹ A A C. 3. . C DA,1 + I ¼ C D,A 1 + I ¼. 7. 3. 7. 7. - C I ¹ A,1 + I DA,1E¼ - [ ¹ A DA,1:¼ - ¹ A,1 2 R ¼ ¹ R ¼ ' ! I I|I - C ²¹ A,1 + I DA,1E¼ C C. <;[_;[H; s =9\l;>_F8:?)\^B¹~ E¼3 Sk<H&=±a8UE=<jlB+E= s \â8\(\l<=<=k<jl;[H]\l;>_1-8UºB:X s =jl;O;[9<G½\l<=?[;> V=F?>;²r <BB s GE;E=FH\l<=!_BEjlj]=F_3\pj]=FHlkd?[\XBEj\^<=*;>?[?>9<=HlHLr s =F8V\^D(B s = ?);[\l\l<=!D;tO= s s ;[Hl_Fj]=\l=3r|_BE9i\^;[9kdBEk<H\^;[D(=BEdHl=FjbV8V\^;[BE91a8V\l\l=Fj]94 -

(100) !!" 4®

(101) p9 s =Fj]Hl=F9)`*c 3`YCBEj]Gi8:9e` >; ?>?RP78:9 s 3=F; s ;>9<G¶K¹bf+»:»Evi¼

(102)

(103)

(104) !#"$%!& '!(*),+-..

(105) .

(106) .

(107) 3K=-rB:jl q}MO<jl;[9<GE=jbr '=jl?>8:G< @CBEDD(=9<GE=HF`q7e ¹~ :xExi E¼2 ]J|9dX=Fj]=F9<_=XB:jDek<?[\l;rH]\8\^=DB s =F?>H-Xj]BED ;[9i\^=FjbV8:?r _F=9<HlB:jl= ss 8V\^8 </

(108) 01'2&

(109) 3 '! 546.

(110) +-.2`4fEfE`4fUhiyrf : O zaj] s D(8:94À¡¹bfU»E» 7 ¼2 ]KRBE9Ora8:jl8:D(=3\^j];>_n=H]\l;>D(8V\^;[BE9B:X8Q8:j] :B+ ];[?>?[9<=FH]Hbr s =F8V\^D(B s =? <j]BO_=FHlHXjlB:D ;>9i\l=Fj]V8:?²r|_=F9<H]BEjl= s BEdHl=FjbV8V\^;[BE9<HFÒ;[\^8:<d?>;>_F8r \^;[BE9

(111) \^B s ;>8:u=3\^=HHlk<jbO;V8:? s 8V\^8 <, 6 78+9 : : O`qyEy:vrby : »O ¢E8:_B s `4¢dZ¹Lf+»iy 7 ¼2 lQk<?[\l;[V8:j];8V\l=wBE;[9\p<jlB_F=HlH]=FHF}R<jl= s ;>_3\8:d?>=ndjlB³]=_\^;[BE94 P8 s BE9Or|KR;[ :B s D s =jl;:8\^;[:=:`jl=<jl=Hl=9\^8V\^;[BE9*B:XOD&8:jb\^;[9<Gi8:?[=FHF<;<=?>2+-

(112) .2& '!&@ A .2B:B ''

(113) C2D=vdfE`q :v 7 rb 7 vd ¢:BE?E`$c$T8:9 s @CBED(D=F9dGE=FH`{7e¹bfU»E»E»i¼2 ]Âw=F9a8:?[;>¿= s ?[;> V=F?>;[<BB s 8V<<jlBE8:_ XBEj%8 <jlB:GEjl=HlHl;E=1\^<j]=F=2r|H]\^8V\^=DB s =?C;[\lº_=F9<H]BEjl= s 8:9 s \ljlk<9<_F8V\^= s s 8\8d} pd<?>;[_U8V\l;>BE9&\lB%pJL7*M <0 E 78 +9 .

(114) 7 7 ` : : yr : »Exd ¢:BE?E`cT[`@CBED(D=F9dGE=FH`7e[`p¡R=?>D=FjF`@p8:9 s m =3\^=9<9<=FkdjF`*mZ¹ :x:xi E¼3 ] w=F9a8:?[;>¿= s ?>;> V=F?[;><BB s 8V<<<j]Bi8:_XBEjR8:9;>?>?[9<=FH]Hbr s =U8V\lDB s =F?A;[\l;[9i\^=FjbV8:?r _F=9<HlB:jl= s½s 8V\^8d}*8:<<?[;>_F8V\^;[BE9\^B8:G:=3r|H]u=_F;·a_%;>9<_; s =9<_F=%B:X s =FD=F9i\l;8 <8 E @

(115)

(116) .

(117) vd`dgivEvrg:givd f .

(118) STk<jl9<k<?[?`ZYRT0C¹bf+»Ey:hi¼3 ] S<=(=FD<;>j];>_U8V? s ;>H]\ljl;[<kd\^;[BE9 Xk<9<_\l;>BE9½;[\^,8:jbr <;\^j^8Vjl;>?±GEj]BEk<w= s `$_F=9<HlBEj]= s V8 9 s \^j]k<9<_F8V\^= s,s 8V\^8 <&%+! / '2& 46

(119) ? +9 .

(120) v : `q V»Exr 7 . f+v.

(121)