N t. R t = u + ct X i, S t = u R t = X i ct. ψ(u) = P(τ(u) < ) = P(M > u). i=1. i=1. 0 t< 0 t T

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@ É x )í-ë ð / /ÿ$þ7ûcø Ü7Ú Ù'æ :F lš$ '@ ;$ EB ƒ ) L ) X C ƒ )ƒ"=~ =~ Å@k>,Ž @1Š%Š%ƒ > @ ŽÈ;%ƒ Š$ L}a= CŽ9*Š$ C X ~ wy "{ ÁE;@ CŽ"ŽAœ 1œ@ =~Ž Š% ƒ œy ªD' Y, (Ω, F, P) ]@l;%=~ @ ƒ"ž"ƒ 7 UŠ%@1>, ]> @ ;%;$ ƒ" @NŸž=~ƒ Š%Š%=~ k;%= >, Š)Š {N t } t 0 B ƒ ) aƒ" ) C Š$ƒ 7 λœ @ k@nš% Œ C >, {X k } k=1 =~ ƒ", C C œª=ªš$ƒ )ƒ ª ~œªƒ C )ƒ > @ Ž"Ž ± ƒ Š$ );%ƒ" Œ ) ±;@ =~} ~@ ;%ƒ @ Ž" Š,œ~B ƒ ) ±}a 5@ µ @ ~@ ;%ƒ @ >, σ 2 Œ;$ ) C;%}a=~;% ~œb _@1Š)Š%Œ}a G ) @/ xvx z vv {X k } @ {N t } @ ;% Nƒ" C C C Ìr _>,Ž @1Š)Š%ƒ > @ Ž {R t } t 0 ƒ Šr ~ƒ ª C L N t R t = u + ct X i, B C;% c ƒ Š Š%=~}a N=ªŠ$ƒ )ƒ ª S>,=~ Š$ '@ G@ u ƒ Šr =~ C @/ )ƒ ª ~ Ì ƒ ŠÄƒ ŠÄ ) lš$ '@ @ ;) À} @/ ) C}\@/ )ƒ > @ Ž }a= CŽ =~;Eƒ" Š%Œ;@ >, X;%ƒ Š% Ì lƒ" ƒ )ƒ @ Ž > @ ƒ '@ ŽÅƒ Š uœy ) Ÿž=~ƒ Š)Š$=~ ;%= >, Š)Š N t Š)>,;%ƒ" Š_ ) Œ}l C;]=~ r>,ž @ ƒ"}kš_ƒ" < ) (0, t] )ƒ"}a Xƒ" ) C;$ ~@ ŽAœ @ >,Ž @ ƒ"} Š$, ª C;%ƒ )ƒ" ŠS@ ;% ±;@ =~}Uœ ~ƒ ª C ) aš$ Œ C >, {X k } k=1 YÌÆ=U>,=5 ª C;_ƒ ŠNŽ"ƒ @ ƒ"ž"ƒ 7 ªœ ) aƒ" Š$Œ;@ >, >,=~}a@ ;% >, Cƒ ª ŠS@\;% C}kƒ"Œ} @/ N@k>,=~ Š$ '@ N;@/ ) cœ C;EŒ ƒ Ä )ƒ"}a ~œb C;% c = (1 + θ)λµ @ θ > 0 ƒ Š =~ ) C "y1w v,y w 5y! "#%$ > @ Ž"Ž" À ) YÌ XŽ"=@1 ƒ" Ë @1ŠÄ )=i X=ªŠ$ƒ )ƒ ª ~œ =1 ) C;$B ƒ Š$ c B=~ŒŽ \ GŽ" Š)Š ) @ λµ @ \ ) ŒŠB ƒ ) \;%=~ @ ƒ"ž"ƒ 7 1 ) G ŒŠ%ƒ" Š%Šr;%ƒ Š$ kb=~œž >,=~}a _ C @/ )ƒ ª ]ƒ" ˃" ƒ ) _ )ƒ"}a ~ z& y"'zv( x)*+( vx,5z =~; }\@/ ) C} @/ )ƒ > @ ŽŒ;%=ªŠ% Š,œ1ƒ žƒ Š Š%=~}a, )ƒ"}a ŠÆ}a=~;% Å>,=~ ª C ƒ" C È )= B=~;%ËB ƒ ) vv {S t } t 0 9*Š$ C _ ~ -EŠ$}lŒŠ)Š% C œ 1Ã~Ã~êD'œ @ }k CŽ i=1 N t S t = u R t = X i ct. =5B]œ B E;% > @ Ž"Ž ) N C ƒ )ƒ"=~ Ë=~ Æ;%Œƒ" i;%=~ @ ƒ"ž"ƒ 7 ªœƒA ~ ) E;%=~ @ ƒ"ž"ƒ 7 \ ) @/ ) _;%ƒ Š$ï;%= >, Š%ŠÄ ;%=~Šr CŽ"=5B šc C;%= Ì N )ƒ"}a G )=a;%œƒ" ˃ Š C L@1Š 9$ 5D τ(u) = inf{t 0 : R t < 0} = inf{t 0 : S t > u}. Y, @ i ) 9 ~D.0/1%24352768249:3 M = 1!;SÌ i=1 sup {S t }, M T = sup {S t }. 0 t< 0 t T x(%#<x,u)y u=4+ w<#0>?#@ w w' ψ(u, T ) = P(τ(u) T ) = P(M T > u) x(%#<x,u)y u=4+ w<#a#b>c# w w' ]ƒ Š C @1Š ψ(u) = P(τ(u) < ) = P(M > u). ]ƒ Šr ~ƒ ª C U B @/ S =~Ž"Ž"=5B Š_B \@1Š%Š%Œ}a c = 1œÆ Œ _ ) ƒ Š]ƒ Š_ =1 ±@i;% Š2 );%ƒ >- )ƒ ª Ë@1Š)Š$Œ}a )ƒ"=~ D=~Ž"Ž"=5B ƒ" E-EŠ$}lŒŠ)Š% C 9 1Ã~Ã~êD'œŽ", c 1 @ Í C Rt = R t/c Ìr C ) G;% CŽ @/ )ƒ"=~ Š, 7B C C ) G;%Œƒ" i;%=~ @ ƒ"ž"ƒ )ƒ" Š ψ(u)œ ψ(u, T ) =~;Å ) G;%= >, Š%Š R t @ ψ(u)œ ψ(u, T ) =~; ) _;%= >, Š%Š Rt @ ;% N ~ƒ ª C L ) _ Œ@/ )ƒ"=~ Š ψ(u) = ψ(u), ψ(u, T ) = ψ(u, T c).

É î ö,ö1ôaè 'Þ ÝÙ'Ú"Þ"è'élè Èô7â Þ é ö1ôaècãâù,ã2þ æ Þ Ú"ò )í U@1>- )Œ@ ;%ƒ @ ŽY} @/ ) C} @/ )ƒ >CŠÅB N ƒ Š2 )ƒ" ~Œƒ Š% U, 7B C C ËŽ"ƒ" ~ % Å@ \ 5@C A '@ ƒ"ž" w y "{ ƒ Š$ );%ƒ" Œ )ƒ"=~ Š 9*Š% C ~ }l ;% > Š œ ~D' - ƒ Š$ );%ƒ" Œ )ƒ"=~ F X (x) ƒ ŠžŠ@ ƒ _ )= +$w7w4y"4 ƒ" ) C;% n,ˆ ƒ Š$ >,=~ Š$ '@ Š a, b > 0 Š$Œ> _ ) @/ F X (x) = 1F X (x) ae bx =~;CœÆ Œƒ ~@ Ž" C )Ž ªœžƒ" ) C;%,ˆ ƒ Š2 Š z > 0 Š%Œ> F ) @/ M X (z) < œyb C;% M X (z) ƒ ŠÆ ) }a=~}a C ž ~ C C;@/ )ƒ" N Œ >- )ƒ"=~ - ƒ Š2 );%ƒ" Œ )ƒ"=~ F X (x) ƒ ŠžŠ)@ ƒ S )=E y"b7w4y"4 ƒ" F X (x) > ae bx =~;@ Ž"Ž a, b > 0œ =~;Cœ Œƒ ~@ Ž" C )Ž ªœªƒ" M X (z) = =~;Ä@ Ž"Ž z > 0 i ) ƒ Š @ C;rB _Š2 )Œ U>,Ž @ ƒ"}zš$ƒ"šc S ƒ Š2 );%ƒ" Œ )ƒ"=~ ŠG@1Šrƒ" LÌž@ Ž" k 1 -ø ùâø2ÿ/øúù  Aø Âøúý /ù ø 'ø úÿ /øïù  Aø Âøúý /ù! ÿ ì ' - ÿ+ Âÿ$ Aù ì ÿ+$ ü~ý /ÿ Âø && " ÿ$ø ø $5ÿ+Äÿ+$5ü$# ù β > 0 f X î î (x) = βe βx α > 0 β > 0 f X î (x) = βα Γ (α) xα1 e βx x 0 x 0 c > 0 î τ 1 f X (x) = cτx τ1 e cxτ î x 0 β i > 0 î n i=1 a i = 1 f X (x) = n i=1 (a iβ i e β ix ) î x 0 % ÿ '&"( 'ø úÿ /øïù  Aø Âøúý /ù! ÿ ì ' - ÿ+ Âÿ$ Aù ì " ÿ$ø úý /ý- úý && ì ' Aÿ+ Âý / A c > 0 î 0 < τ < 1 f X (x) = cτx τ1 e cxτ î x 0 µ R σ > 0 f X (x) = 1 2πσx e (log(x)µ)2 /2σ 2î x 0 α > 0 î β > 0 f X (x) = βα (log(x)) α1 x β+1 Γ (α) α > 0 î ν > 0 f X (x) = α ν+x ( ν α > 0 î ν > 0 î τ > 0 î ν+x )α î (ν+x τ ) α+1 f X (x) = ατνα x τ1 x 1 x 0 x 0 Ì ±@1 )2ŒŠ$ )}a C _>,= (*k>,ƒ" C k9a@ Ž Š%=i> @ Ž"Ž" À ) 0YŒ C;% i,ˆ =~ C -DÄŽ @C Š_@ ~, \;%=~Ž" Nƒ" Ë> @ Ž >,ŒŽ @/ )ƒ" ± ) N;%Œƒ" Ë;%=~ @ ƒ"ž"ƒ 7 iƒ" \ ) _> @1Š% G=~ Ž"ƒ" ~ % A '@ ƒ"ž" U>,Ž @ ƒ"}kšc Ç Œ=ªŠ% γ = sup z M X (z) < - =ªŠ%ƒ )ƒ ª ]Š%=~Ž"Œ )ƒ"=~ L=~ Æ ) _ Œ@/ )ƒ"=~ 9AʪD 1 + (1 + θ)µr = M X (R), R < γ, y! '+( v'w' #wz= z #w ƒ Š > @ Ž"Ž" @, - L@ @ Ž )ƒ > @ ŽÆŠ$=~Ž"Œ )ƒ"=~ Ë )=X Œ@/ )ƒ"=~ 9AʪDn,ˆ ƒ Š$ Šr=~ Ž \ =~;r,b >,Ž @ ƒ"}z ƒ Š2 );%ƒ" Œ )ƒ"=~ ŠC - =5B, ª C;Cœ ƒ ȃ Š:Œƒ ) 5@1Š$ _ )=E=~ '@ ƒ" a@ Œ}a C;%ƒ > @ ŽŠ$=~Ž"Œ )ƒ"=~ ªÌ >,= (*a>,ƒ" C R Š@/ )ƒ Š$ Š ) Nƒ" Œ@ Ž"ƒ 7 9 Ó D R < 2θµ µ (2), B C;% µ (2) = EXi 2 9*>, 7 -EŠ$}lŒŠ)Š% C œ 1Ã~Ã~êD' Y, D(z) = 1 + (1 + θ)µz M X (z) Ìr ŒŠ,œ ) X@1 )2ŒŠ$ )}a C G>,= (*k>,ƒ" C R > 0 Š@/ )ƒ Š$ ŠÄ ) l Œ@/ )ƒ"=~ D(R) = 0 =~;) C;r )=a ~, E@XŠ%=~Ž"Œ )ƒ"=~ L=~ _} @C ŒŠ% N ),B )=~ Ä@ Š$=~ =~;%}lœž @

É É )í ð / /ÿ$þ7ûcø Ü7Ú Ù'æ 9ÂÕ~D R j+1 = R j D(R j) D (R j ), B ƒ ) \ ) Gƒ" ƒ )ƒ @ Ž >,=~ ƒ )ƒ"=~ R 0 = 2θµ/µ (2) œ B C;% D (z) = (1 + θ)µ M X (z) ÔU=~;% C=5 ª C;Cœ ƒ" ƒ ƒ Šn=ªŠ)Š$ƒ" Ž" E )=S> @ Ž >,ŒŽ @/ ) Ä ) ) ƒ";) k;@cbí}a=~}k C µ (3) œb E> @ =~ '@ ƒ" À@XŠ% @ ;% C; =~Œ Ë ) @ 9 Ó DN9*Š% C _Ÿ!@ )2 C;Ä@ i: ƒ"ž"ž"}a=1 œy ª ~D R < 12µθ 3µ (2) + 9(µ (2) ) 2 + 24µµ (3) θ, @ ËŒŠ% _ƒ Ä@1Š ) _ƒ" ƒ )ƒ @ Ž >,=~ ƒ )ƒ"=~ Lƒ" <9ÂÕ~D' :F L =1 ) L ) @/ \}a=ªš$ =~ E ) L}k, ) = Šk=~ N Š2 )ƒ"} @/ )ƒ" < ) ;%Œƒ" ;%=~ @ ƒ"ž"ƒ 7 ƒ Š)>,ŒŠ%Š% ]ƒ" N ) n@ C;Æ;% Œƒ";% n=~ Ž G ) n,ˆ ƒ Š2 ) C >, Å=~ ;)Š$ Y 7B= =~; ) ;% C Å}a=~}a C Š =~ ) Ë>,Ž @ ƒ"}qš%ƒ"šc Ë ƒ Š2 );%ƒ" Œ )ƒ"=~ œå@ Š%=~}a \=~ ) C}j@ Ž Š%=À ) i,ˆ ƒ Š$ ) C >, Ë=~ ) }a=~}k C r ~ C C;@/ )ƒ" k Œ >- )ƒ"=~ =5B]œ >,=~ Š$ƒ C;Æ ) Å@ ~ ~;% C @/ ) nž"=ªš)šy;%= >, Š%Š S t B ƒ ) c = 1 5ŸŒ ξ(u) = S τ(u) uœæb C;% τ(u) ƒ Š] ) k )ƒ"}k a )=À;%Œƒ" C Í9$ 5D' ÈÌ =~Ž"Ž"=5B ƒ" Š$ '@/ ) C}a C ;% Š% C Š_@k ~ C C;@ ŽÈ =~;%}lœž @k =~;Ä ) X;%Œƒ" À;%=~ @ ƒ"ž"ƒ 7 Àƒ" ƒ" ƒ ) S )ƒ"}k i9*š% C l ~ -EŠ$}XŒŠ%Š% C œè 1Ã~Ã~êD' :-ÄŠ)Š$Œ}k ± ) @/ l =~;SŠ%=~}a R > 0 ) k;%= >, Š)Š {e RS t } t 0 ƒ Šl@ } @ ;$ )ƒ" @ Ž" l@ S t @ ÏŠC =~ {τ(u) = }. Ìr C e 9 ªD Ru ψ(u) = E(e Rξ(u) τ(u) < ). =~; ) l>,ž @1Š%Š%ƒ > @ Žž;%ƒ Š$ Ë}a= CŽY ) S =~;% C ~=~ƒ" Ë@1Š%Š%Œ}a )ƒ"=~ ŠÄ =~Ž À@ R ƒ ŠE@ @1 )2ŒŠ$ )}a C Ä>,= (*k>,ƒ" C µ T ¼1 _» ½ /¹ ž½ ½ 5½ µ É =5B]œ1B @ ;% ~=~ƒ" N )=N;% Š% C @G>,=~Ž"Ž" >- )ƒ"=~ =~ ž @1Š$ƒ >_,ˆ@1>- ;% Š$ŒŽ Š =~ i ) _;%Œƒ" U;%=~ @ ƒ"ž"ƒ 7 Uƒ" ˃" ƒ ) _ )ƒ"}a ~ # way z'y= w4y "{ : C u = 0 ƒ ƒ Šr 5@1Š$ \ )=a=~ '@ ƒ" Ë ) _,ˆ@1>- =~;%}XŒŽ @ ψ(u) = 1 1 + θ. =~;E}a=~;% X, '@ ƒ"ž ŠNŠ% C l ~ ÆÁE;@ CŽ"Ž 9$ 5D' =1 )ƒ >, S ) @/ G ) l =~;%}XŒŽ @ C C Š =~ Ž i=~ θœ;% C @ ;) Ž" Š%Š =~ ) _>,Ž @ ƒ"} Š$ƒ"šC ] ƒ Š$ );%ƒ" Œ )ƒ"=~ # #way z&"y"'lvc{ Ìr,ˆ Ž"ƒ >,ƒ œ 5@1Š2 A )=1 4> @ Ž >,ŒŽ @/ ) N =~;%}lœž @N,ˆ ƒ Š$ ŠÅ =~;Å,ˆ =~ C )ƒ @ Žž>,Ž @ ƒ"}kš,œ @ }k CŽ 9 ~D ψ(u) = 1 1 + θ e θβu 1+θ. y"' 'ayaz& y"'xv,{ nb @1ŠnŠ% =5B k ÁG;@ CŽ"Ž @ \Ç C ~ C;) @ Ž!9$ 5DÈ ) @/ =~; ) _ @ }a} @X>,Ž @ ƒ"}z ƒ Š2 );%ƒ" Œ )ƒ"=~ LB ƒ ) U}a 5@ 1 @ α 1œ 9 ªD ψ(u) = θ(1 R/α)e Ru αθ sin(απ) + I, 1 + (1 + θ)r (1 + θ)(1 R/α) π

B C;% I = 0 ö,ö1ôaè 'Þ ÝÙ'Ú"Þ"è'élè Èô7â Þ é ö1ôaècãâù,ã2þ æ Þ Ú"ò )í x α e (x+1)αu dx [x α (1 + α(1 + θ)(x + 1)) cos(απ)] 2 + sin 2 (απ). Ìr Sƒ" ) C ~;@ Ž I @1ŠÄ )= l> @ Ž >,ŒŽ @/ ) Œ}a C;%ƒ > @ Ž"Ž ªœ Œ EB ƒ ) Š%=~}a l> @ ;% S 5@ ; 0 ) ƒ ŠS> @ =~ ;% >,ƒ Š% CŽ ª Æ:< k =1 )ƒ >, k ) @/ ] ) \@1Š)Š%Œ}a )ƒ"=~ =~ F ) k}a 5@ <ƒ Š =U;% Š$ );%ƒ >- )ƒ"=~ Š%ƒ" >, a =~;S>,Ž @ ƒ"}kš X B ƒ ) @ ;% ƒ );@ ;$ }k 5@ µ B a @C ª ψ X (u) = ψ X/µ (u/µ) -EŠ\ ) @ }a}\@ ƒ Š2 );%ƒ" Œ )ƒ"=~ Òƒ ŠU>,Ž"=ªŠ$ Œ C;LŠ)> @ Ž" F> @ ~ ŠiB =~ '@ ƒ" ψ G(α,β) (u) = ψ G(α,α) (βu/α) @ B G> @ Ë =5B > @ Ž >,ŒŽ @/ ) G ) G,ˆ@1>- ;%Œƒ" ;%=~ @ ƒ"ž"ƒ 7 Ë ƒ @X Œ@/ )ƒ"=~ 9 ªD' w( x # #way úvc{ n =~;Y ) Å>,Ž @ ƒ"}dš$ƒ"šc ƒ Š2 );%ƒ" Œ )ƒ"=~ X Cƒ" G@}kƒ ˆ )Œ;% =~ n,ˆ =~ C )ƒ @ Ž Š±B ƒ ) @ ;@ }a, ) C;)Š β 1 < < β n @ B Cƒ" ~ Š a 1,..., a n œ ŒŠ%ƒ" ) 0 @ Ž @1>, l );@ Š$ =~;%} ƒ" ª C;)Š$ƒ"=~ œ =~ S}\@C U=~ '@ ƒ" @ À,ˆ@1>- N =~;%}lœž @k=~ ) _ =~;%} 9*²EŒ ;% Š$ l@ ÁG C;% C;CœÆ ªD 9 ªD ψ(u) = n C k e rku, k=1 B C;% r 1,..., r n @ ;% n =ªŠ$ƒ )ƒ ª SŠ%=~Ž"Œ )ƒ"=~ Š )=X ) _ Œ@/ )ƒ"=~ (1 + θ)µ = n j=1 a j β j r, B ƒ ) 0 < r 1 = R < β 1 < r 2 < β 2 < < r n < β n Ì ]>,= (*a>,ƒ" C Š C k @ ;% _ ~ƒ ª C U \ ) N =~;%}XŒŽ @ a j C k = 1 n j=1 β j r k µ r k n j=1 a j. (β j r k ) 2 R ) À> @1Š$ =~ G}aƒ ˆ )Œ;% L=~ Ä 7B=<,ˆ =~ C )ƒ @ Ž Š 9 n = 2D±@FŠ$ƒ"}aŽ" À@ @ Ž )ƒ > ;% Š%ŒŽ ƒ Šr ~ƒ ª C 9*Ÿ!@ )2 C;E@ i:íƒ"ž"ž"}a=1 œy ª ~D (10) ψ(u) = B C;% @ 1 (1 + θ)(r 2 r 1 ) {(ϱ r 1) exp(r 1 u) + (r 2 ϱ) exp(r 2 u)}, r 1 = ϱ + θ(β 1 + β 2 ) [{ϱ + θ(β 1 + β 2 )} 2 4β 1 β 2 θ(1 + θ)] 1/2, 2(1 + θ) r 2 = ϱ + θ(β 1 + β 2 ) + [{ϱ + θ(β 1 + β 2 )} 2 4β 1 β 2 θ(1 + θ)] 1/2 2(1 + θ) ϱ = β 1 (1 p) + β 2 p, p = a 1 β1 1 a 1 β1 1 + a 2 β2 1.

É %ë, ð / /ÿ$þ7ûcø Ü7Ú Ù'æ µ K»Æ ¹ /¹ ƽ /½¹ µ : C U ) ]>,Ž @ ƒ"}jš$ƒ"šc l ƒ Š2 );%ƒ" Œ )ƒ"=~ Àƒ Š,ˆ =~ C )ƒ @ ŽG9 =~;]>,Ž"=ªŠ$ CŽ ;% CŽ @/ ) D'œÆŠ%ƒ"}aŽ" @ @ Ž )ƒ >a;% Š$ŒŽ Š_ =~;N ) a;%œƒ" ;%=~ @/ ƒ"ž"ƒ 7 ƒ" ƒ" ƒ ) i )ƒ"}a i@ ;% i=ªš)š$ƒ" Ž" <9*Š$ C LÇ >- )ƒ"=~ ~D'?=~;l}a=~;% \ ~ C C;@ Žr>,Ž @ ƒ"} @ }a=~œ ƒ Š$ );%ƒ" Œ )ƒ"=~ Š,œN ~ E 5@C A '@ ƒ"ž" œg ) @ Ž @1>, );@ Š% =~;%} ) > ƒ Œ = Šr =1 rb=~;%ï@ Ë=~ N} @C C ËŠ%=~}a G Š$ )ƒ"} @/ ) ŠC i ) ƒ Š Š% >- )ƒ"=~ ËB N;% Š$ C 12 ƒ" C;% C XB CŽ"ŽÅ =5B R@ =1 ±Š$= B CŽ"ŽÅ =5B @ ;%=5ˆ ƒ"}\@/ )ƒ"=~ Š, È:F \ƒ" );%=1 Œ>, U@ ;%=~}aƒ Š%ƒ" À}k= ƒ" > @/ )ƒ"=~ )=À ) Ë²Ä \³! Ž C;a@ ;%=5ˆ ƒ"} @/ )ƒ"=~ > @ Ž"Ž" ) Ê/ Â}a=~}k C ] @ }a} @i²e a³! Ž C;l@ ;%=5ˆ ƒ"} @/ )ƒ"=~ Œ}a C;%ƒ > @ ŽÅ>,=~}a@ ;%ƒ Š$=~ =~ ) @ ;%=5ˆ ƒ"} @/ )ƒ"=~ Šƒ Šn ~ƒ ª C iƒ" ËÇ >- )ƒ"=~ \Ó :< G@ Ž Š%=S =1 ) Ä ) @/,B @ ;%=5ˆ ƒ"} @/ )ƒ"=~ Š wåy "{ @C ª N;% >, C )Ž i C C Ë;%=~=ªŠ$ ˃" \ ) NŽ"ƒ ) C;@/ )Œ;% 9*Š% C N ~ @Yƒ"} @ œ 1Ã~ê l@ Š CŽAœY 1Ã~à 5D'œ Œ N@1Š ), ËB=~;% Ë =~;EŠ$ >,ƒ" >X>,Ž @1Š%Š% ŠÄ=~ ƒ Š$ );%ƒ" Œ )ƒ"=~ ŠN@ @ ;% *@ ; ;%=~}Z>,=~}aŒ '@/ )ƒ"=~ @ Ž Š%ƒ"}aŽ"ƒ >,ƒ 7 ªœB GB ƒ"ž"žy =1 >,=~ Š%ƒ C; ) C}L µ* µ x)y ',x:(%#u x $py!x, 'ay~w "#{ Ì J =~Ž"Ž"=5B ƒ" j@ ;%=5ˆ ƒ"} @/ )ƒ"=~ =~Ž Š 9$ ~ 5D ψ CL (u) = Ce Ru, B C;% C = M X θµ (R) µ(1 + θ). =~;G ) ±;%= =~ ÅB X;% C C;N )= ÁE;@ CŽ"ŽÄ9$ 5D' ÆÌ a>,ž @1Š)Š%ƒ > @ ŽÅ;@ }C;2 Œ C;% @ ;%=5ˆ ƒ"} @/ )ƒ"=~ ƒ" CŽ Š Œƒ ) F@1>C>,Œ;@/ ) ;% Š$ŒŽ Š =5B, ª C;CœB }XŒŠ2 i;% C}a C}l C; ) @/ ƒ" l=~;) C;È )=GŒŠ% ƒ @ a@1 )2ŒŠ$ )}a C n>,= (*k>,ƒ" C @1Š )=G,ˆ ƒ Š2 œ1 ) C;% C =~;% r}k C;% CŽ ) _Ž"ƒ" ~ % A '@ ƒ"ž" ƒ Š2 );%ƒ" Œ )ƒ"=~ ŠÄ> @ L G '@ ~ C Uƒ" )= >,=~ Š$ƒ C;@/ )ƒ"=~ =~;,ˆ =~ C )ƒ @ Ž >,Ž @ ƒ"} Šr =~;%}lœž @U9$ ~ 5Dn ƒ" CŽ Š ) N,ˆ@1>- ;% Š%ŒŽ µ µ @"# #way Èy!x, 'ay~w "#{ Ìr ƒ ŠE@ ;%=5ˆ ƒ"} @/ )ƒ"=~ B @1Š ;%=~=ªŠ$ @ C;%ƒ ª U Ë²Ä N³ Ž C;]9$ ªD ( ψ E (u) = exp 1 2µθu µ (2) (µ (2) ) 2 + (4/3)θµµ (3) ). µ µ :(%#u x $_y=!x, 'ay1w "#{ y= x, Ìr =~Ž"Ž"=5B ƒ" =~;%}lœž @ œ/> @ Ž"Ž" N ) 'ay1w # œ>,=~}a Šr ;%=~}JÁE;@ CŽ"Ž9 1Ã~Ã~êD ψ L (u) = [ 1 + ) (θu µ(2) 4θµ 2 µ (3) 2µ 3(µ (2) ) 3 ] e 2µθu/µ(2). :(%#u µyµ B'ay"# x-v y!x, 'ay~w "#{ Ì C C } @ =5B C;)Š @ ;%=5ˆ ƒ } @/ )ƒ"=~ UŒŠ% Š ) _ =~Ž"Ž"=5B ƒ" a;% C;% Š$ C '@/ )ƒ"=~ À=~ Æ ) _;%Œƒ" U;%=~ @ ƒ"ž"ƒ 7 ψ(u) = P(M > u) = P(M > 0)P(M > u M > 0). Ìr ƒ 5@ =~ a ) @ ;%=5ˆ ƒ"} @/ )ƒ"=~ ƒ ŠÀ )=Í;% CŽ @1>, ) R>,=~ ƒ )ƒ"=~ @ Ža;%=~ @ ƒ"ž"ƒ 7 1 P(M > u M > 0) B ƒ ) @i @ }a} @i ƒ Š$ );%ƒ" Œ )ƒ"=~ Œ >- )ƒ"=~ G(u) % )ƒ" ) _ ;)Š2 7B=a}a=~}a C Š_9*Š% C láe;@ CŽ"ŽAœ 1Ã~Ã~êD' Ìr ƒ ŠrŽ" 5@1 Šr )= 9$ 5 ~D ψ BB (u) = 1 (1 G(u)), 1 + θ x $

B C;% G ) ]@ ;@ }a, ) C;)Š αœ β =~ G @ ;% _ ~ƒ ª C U α = 1 + (4µµ(3) /3(µ (2) ) 2 1)θ, β = 1 + θ ö,ö1ôaè 'Þ ÝÙ'Ú"Þ"è'élè Èô7â Þ é ö1ôaècãâù,ã2þ æ Þ Ú"ò %ëb 2µθ µ (2) + (4µµ (3) /3µ (2) µ (2) )θ. Ì C C } @ =5B C;)Š@ ;%=5ˆ ƒ"} @/ )ƒ"=~ X ~ƒ ª ŠÈ;@/ ) C;!@1>C>,Œ;@/ ) r;% Š$ŒŽ ŠCœ1ƒ" l ),ˆ =~ C )ƒ @ Žn> @1Š$ Xƒ N >,=~}a ŠE ) l,ˆ@1>- N =~;%}lœž @ _> @ lœš% À =~;N ƒ Š2 );%ƒ" Œ )ƒ"=~ ŠYB ƒ ) ] ƒ ) Å ;)Š2 Æ ) ;% C Å}a=~}a C ŠCœ ƒa ~ 5 =~;Æ,ˆ@ }kž" =~;Æ,ˆ =~ C )ƒ @ ŽAœ @ }k} @ œ Ž"=~ ~ =~;%} @ ŽAœ );%Œ > @/ ) =~;%} @ Ž!@ :F Cƒ" ŒŽ"Ž! ƒ Š$ );%ƒ" Œ )ƒ"=~ Š, =~;EŽ"=~ ~ @ }a} @k ƒ Š$ );%ƒ" Œ )ƒ"=~ B S @C ª l )=iš$, β > 3œ =~;GŸ!@ ;%, )=i ƒ Š$ );%ƒ" Œ )ƒ"=~ α > 3œ @ =~;N ÅŒ;%; ƒ Š$ );%ƒ" Œ )ƒ"=~ ατ > 3 µ µ #@ ny=!x, 'ay1w "#{ Ìr C ƒæ@ ;%=5ˆ ƒ"} @/ )ƒ"=~ Uƒ Š @1Š% Ë=~ U@l>,Ž @1Š$ Š%ƒ > @ Žž;% Š%ŒŽ _@ =~Œ p A ) ƒ" ƒ" œ5 C ƒaîïšä ) C=~;% C} 9*Š$ C káe;@ CŽ"ŽAœÆ 1Ã~Ã~êD' G} @C @ Ž Š%=± _ C;%ƒ ª U ;%=~} 9$ 5 ~DÅB C ib N;% CŽ @1>, N ) N @ }a}\@s ƒ Š$ );%ƒ" Œ )ƒ"=~ L Œ >- )ƒ"=~ G B ƒ ) L@ U,ˆ =~ C )ƒ @ ŽÆ=~ ~œ}\@/ > ƒ" a=~ Ž ) ] ;)Š2 }a=~}a C Ä>,=~ŒŽ Ë _;%, @ ;) i@1š@]š%ƒ"}až"ƒ" a ª C;)Š%ƒ"=~ \=~ ) E C C } @ =5B C;)Šr@ ;%=5ˆ ƒ"} @/ )ƒ"=~ Ë@ ƒ Š ~ƒ ª C U µ µ ψ R (u) = 1 e2µθu/µ(2) (1+θ). 1 + θ xxy=!x, 'ay1w "#{ Ìr ]ƒ 5@a=~ È ) ƒ ŠG@ ;%=5ˆ ƒ"} @/ )ƒ"=~ Àƒ Š )= ;%, Ž @1>, ) ;%ƒ Š%Ẍ;%= >, Š%ŠB ƒ ) ± ) =~ rb ƒ ) θ = θœ λ = λ @ a,ˆ =~ C )ƒ @ Ž>,Ž @ ƒ"}kš B ƒ ) Ë@ ;@ }k, ) C; βœ % )ƒ" ± ) N ;)Š$ ) ;% C _}k=~}a C Š_9*²E N³! Ž C;CœÆ ªD', β = 3µ(2) 9λµ(2)3, λ =, θ = 2µµ(3) θ. µ (3) 2µ (3)2 3µ (2)2 Ìr C U²E N³! Ž C;CÎÏŠÄ@ ;%=5ˆ ƒ"} @/ )ƒ"=~ Lƒ Šr ~ƒ ª C L ψ DV (u) = 1 1 + θ eθβu/(1+θ). ƒ"=~œš%ž ªœ ƒ" a ) Ä,ˆ =~ C )ƒ @ Ž > @1Š$ ) }a, ) = k ~ƒ ª Š ),ˆ@1>- ;% Š%ŒŽ %=~; =1 ) C;]>,Ž @ ƒ"} ƒ Š$ );%ƒ" Œ )ƒ"=~ Š]ƒ" =~;) C;N )=L@ Ž À ) k@ ;%=5ˆ ƒ"} @/ )ƒ"=~ œ Š$ƒ"}kƒ"Ž @ ;%Ž )= ) ] C C } @ =5B C;)Š @ ;%=5ˆ ƒ"}\@/ )ƒ"=~ œ ) _ ;)Š$ r ) ;% C _}k=~}a C Š @C ª N )=±,ˆ ƒ Š$ µµ ' ' #w $y"' 'ay x\y!x, 'ay~w "#{ -Ä C;% ab kƒ" );%= Œ>, Ë@,BJ@ ;%=5ˆ ƒ"} @/ )ƒ"=~ @1Š$ =~ ²E i³ Ž C;CÎÏŠ±ƒ 5@ )= ;% CŽ @1>, Ë ) U;%ƒ Š% ;%= >, Š)Š B ƒ ) <@ =1 ) C;_=~ ±B C;% ± ) ±,ˆ@1>- ];%Œƒ" F;%=~ @ ƒ"ž"ƒ 7 ƒ ŠN =5B ÆÌ ƒ ŠG )ƒ"}a XB Š$B ƒ > )=i @ }a} @ ƒ Š2 );%ƒ" Œ ) F>,Ž @ ƒ"}kše % )ƒ" i ) X ;)Š$ G ) ;% C X}k=~}a C ŠG@ ŒŠ% _ ) _,ˆ@1>- E =~;%}lœž @U9 ªD =~; ) ];%Œƒ" L;%=~ @ ƒ"ž"ƒ 7 Uƒ" U ) ] @ }a} @X> @1Š% ~ -Ä=5Br, ª C;Cœ ) _;%ƒ Š%ï;%= >, Š)Š B ƒ ) L @ }a} @X>,Ž @ ƒ"} Š ƒ Š, ) C;%}kƒ" U Ë =~Œ; @ ;@ }a, ) C;)Š (λ, θ, µ, µ (2) )œ~š%=g =5B =~ @1 ƒ )ƒ"=~ @ Ž Œ@/ )ƒ"=~ ±ƒ ŠÈ C Š$ ) 5@1 ±=~ ) r =~Œ;$ ) ;@CBÒ}k=~}a C œ B N;%=~=ªŠ$ µ = µ Ç ƒ" >, ES t = θλµt, ES 2 t = λµ (2) t + (θλµt) 2, ES 3 t = λµ (3) t 3(λµ (2) t)(θλµt) (θλµt) 2,

,, %ëcö ð / /ÿ$þ7ûcø Ü7Ú Ù'æ @ iƒ" ) G @ }a}\@]> @1Š% µ (3) = µ(2) µ (2µ(2) µ 2 )œ ) G@ ;@ }a, ) C;)Š (λ, θ, µ, µ (2) ) }lœš$ Š@/ )ƒ Š$ µ = µ, θλµ = θλµ, λµ (2) = λµ (2), λµ (3) = λ µ(2) µ (2µ(2) µ 2 ) @ ib N ~, ) _Š%=~Ž"Œ )ƒ"=~ λ = 2λ(µ (2) ) 2 µ(µ (3) + µ (2) µ), θ = θµ(µ(3) + µ (2) µ) 2(µ (2) ) 2, µ = µ, µ (2) = µ(µ(3) + µ (2) µ) 2µ (2). žƒ" @ Ž"Ž ªœB N=~ '@ ƒ" U ) S@ ;%=5ˆ ƒ"} @/ )ƒ"=~ B C;% θ(1 R/α)e βru/α ψ 3MGDV (u) = 1 + (1 + θ)r (1 + θ)(1 R/α) I = 0 + αθ sin(απ) π x α e (x+1)βu dx [x α (1 + α(1 + θ)(x + 1)) cos(απ)] 2 + sin 2 (απ), @ α = µ 2 /(µ (2) µ 2 )œ β = µ/(µ (2) µ 2 ) À ) X,ˆ =~ C )ƒ @ Ž@ @ }a} @ > @1Š$ X ) ƒ ŠN}a, ) = À ~ƒ ª ŠE ) ±,ˆ@1>- _;% Š$ŒŽ =~;r=1 ) C; >,Ž @ ƒ"}p ƒ Š2 );%ƒ" Œ )ƒ"=~ Š ƒ" Ë=~;) C;r )=a@ Ž \ ) ]@ ;%=5ˆ ƒ"} @/ )ƒ"=~ œš$ƒ"}kƒ"ž @ ;%Ž )=k ) l²e _³ Ž C;G@ ;%=5ˆ ƒ"} @/ )ƒ"=~ œ ) S ;)Š$ Ä ) ;% C l}a=~}a C Š @C ª ] )=,ˆ ƒ Š$ :F B ƒ"ž"žš% =5B ) @/ n ) }a, ) = aƒ ŠÅ@NŠ%Ž"ƒ" ~ Ń"}a;%=5 ª C}k C n=~ a ) ²E ³ Ž C;Å@ ;%=5ˆ ƒ"} @/ )ƒ"=~ B ƒ > Br@1Š]Š)@ ƒ <ÁE;@ CŽ"ŽE9 1Ã~Ã~êDE )=Ë ± ) a Š$ l@ }a=~ \Ð2Š$ƒ"}kŽ" -Ñ @ ;%=5ˆ ƒ"} @/ )ƒ"=~ ŠC µ µ y"blw*x)y ziy= x, 'ay1w #{ Ìr a ) C;%} Ð7 5@C );@ *a>'ñk>,=~}a Š_ ;%=~} Œ CŒƒ" G ) C=~;$ ª l;%ƒ Š% _ ) C=~;$ lƒ!}a 5@ Š ) @/!=~ l ) @C ª C;@ ~ ) ;% C}kƒ"Œ}kŠž,ˆ >, C a=~ Ž aš%ž"ƒ" ~ )Ž X ),ˆ >- ) >,Ž @ ƒ"}kš, Ìr ƒ Šƒ"}aŽ"ƒ" Š! ) @/ ) ÄŠ@, 7 ±Ž"=@1 ƒ" θ ƒ Š =ªŠ%ƒ )ƒ ª @ XŠ$} @ Ž"ŽA "-EŠ%}lŒŠ%Š% C Ë9 1Ã~Ã~êDÆŠ%Œ ~ ~ Š$ Šž ) =~Ž"Ž"=5B ƒ" _@ ;%=5ˆ ƒ"} @/ )ƒ"=~ ( ψ HT (u) = exp 2θµu µ (2) Ìr ƒ Šl}a, ) = ;% Œƒ";% Šl ) i,ˆ ƒ Š$ ) C >, Ë=~ r ) i ;)Š2 X 7B=À}a=~}a C ŠS=~ ) Ë>,Ž @ ƒ"} Š%ƒ"šC ] ƒ Š$ );%ƒ" Œ )ƒ"=~ œš%=xb ]@1Š%Š%Œ}a β > 2 =~;r ) _Ž"=~ ~ @ }a} @X> @1Š% ~œ α > 2 =~;r ) Ÿ!@ ;%, )=k> @1Š% ~œ@ ατ > 2 =~;r ) ] Œ;%; > @1Š% ~ µ µ :$waw*x)y zfy!x, 'ay~w "#{ -ÄŠi =~;i 5@C );@ *k>1œ ) À ) C;%}fÐ7Ž"ƒ" ~ );@ *k>'ñ >,=~}a Š ;%=~} Œ CŒƒ" ] ) C=~;$ ªœ Œ Å @1Š@ a=~ ƒ"=~œšåƒ" ) C;%;%, '@/ )ƒ"=~ Ë@ Ž Š%=]ƒ" ;%ƒ Š% k ) C=~;$ ªœ @ }k CŽ ªœ=~ \ ) ]@C ª C;@ ~ ~œ ) N;% C}aƒ"Œ}kŠr@ ;% N}XŒ> iž @ ;% ~ C; ) @ Ë ),ˆ >- ) Ë>,Ž @ ƒ"}kš, Ìr ƒ ŠÅƒ"}kŽ"ƒ" Šn ) @/ ) GŠ@, 7 Ž"=@1 ƒ" θ ƒ ŠÅ=ªŠ$ƒ )ƒ ª ]@ Ž @ ;% ~ ~ ). I,

, É x :< N} @C \=~ '@ ƒ" U ) N =~Ž"Ž"=5B ƒ" @1Š$ }a )=1 )ƒ >N =~;%}lœž @ ö,ö1ôaè 'Þ ÝÙ'Ú"Þ"è'élè Èô7â Þ é ö1ôaècãâù,ã2þ æ Þ Ú"ò %ë ψ LT (u) = 1 (1 + θ)µ u F X (x) dx. a;%ƒ Š% l ) C=~;$ ± 5@C ± );@ *a>rƒ Šn}a=ªŠ$ =~ ) C \@ ;% ~Œ ± )=] r ) 7 ƒ > @ Ž> @1Š$ ;@/ ) C; ) @ Ž"ƒ" ~ Ä );@ *a>1 - =5B, ª C;CœŽ"ƒ" ~ Ä );@ *a>]ƒ Š =~!Š$=~}k _ƒ" ) C;% Š$ N@1ŠE@a>,=~}aŽ" C}a C )=\ 5@C U );@ *k> ƒ Gƒ ŠE@ Ž Š%=\ C =~;Ä ) lƒ" ) C;%=~Ž @/ )ƒ"=~ F@ ;%=5ˆ ƒ"} @/ )ƒ"=~ À )=\ Š$ )Œ ƒ" Uƒ" i ) _,ˆ ÄŠ%Œ Š$ >- )ƒ"=~ µ* Yµ y"b) $w_w*x)y z y=!x, 'ky1w "#{ Ì Ëƒ 5@Àƒ Šl )=F>,=~}l ƒ" Ë 5@C @ ËŽ"ƒ" ~ E@ ;%=5ˆ ƒ"} @/ )ƒ"=~ Š ψ HLT (u) = θ 1 + θ ψ LT ( ) θu + 1 + θ 1 (1 + θ) 2 ψ HT(u) 9*Š% C -ÄŠ%}lŒŠ)Š$ C œ 1Ã~Ã~êD' ÆÌr ±@ ;$ )ƒ >,ŒŽ @ ;_ 5@/ )Œ;% a=~ ) ƒ ŠS@ ;%=5ˆ ƒ"} @/ )ƒ"=~ Fƒ ŠE ) @/ ƒ Gƒ ŠÄ,ˆ@1>- G =~;Ä ) l,ˆ =~ C )ƒ @ Ž! ƒ Š$ );%ƒ" Œ )ƒ"=~ F@ @1Š2 }a )=1 )ƒ > @ Ž"Ž À>,=~;%;% >- N =1 ) ƒ" UŽ"ƒ" ~ E@ Ë 5@C \ );@ *k>1 µ* µ y"b7w4y"4 _z&"y"'lv y=!x 'ay1w "#{ žƒ";)š2 œ Ž", ÈŒŠ ƒ" );%= Œ>, ) >,Ž @1Š%Š wy { S =~ žš%œ,ˆ =~ C )ƒ @ Žž ƒ Š$ );%ƒ" Œ )ƒ"=~ Šl9*Š% C _ ~ }l ;% > Š œæ ~D S = { F F : lim 2 (x) = 2 x F (x) } { F F : lim n (x) = n, n 2 x F (x) Ì ƒ ŠU>,Ž @1Š%ŠL>,=~ '@ ƒ" ŠUŽ"=~ ~ =~;%} @ ŽS@ Í:< Cƒ" ŒŽ"Ž] ƒ Š$ );%ƒ" Œ )ƒ"=~ Š 9 =~; τ < 1D' ÔL=~;% C=5 ª C;CœE@ Ž"Ž] ƒ Š2 );%ƒ" Œ )ƒ"=~ Š B ƒ ) @ ;% C ~ŒŽ @ ;%Ž ~@ ;$ ƒ" '@ ƒ"ž 9 ~ Ž"=~ ~ @ }a} @ œ Ÿ!@ ;%, )=Ä@ _ Œ;%;Æ ƒ Š2 );%ƒ" Œ )ƒ"=~ Š'DÆ@ ;% nš$œ,ˆ =~ C )ƒ @ ŽA =~;YŠ%Œ,ˆ =~ C )ƒ @ Ž ƒ Š$ );%ƒ Œ )ƒ"=~ Š B n> @ S =~;%}XŒŽ @/ ) n ) Å =~Ž"Ž"=5B ƒ" G@ ;%=5ˆ ƒ"}\@/ )ƒ"=~ l=~ ) Å;%Œƒ" l;%=~ @ ƒ"ž"ƒ 7 ª F Sœ ) C Ë ) S@1Š2 }k )=1 )ƒ >N =~;%}lœž @X =~;rž @ ;% ~ u ƒ Šr ~ƒ ª C U 9$ ʪD ψ HTC (u) = 1 ( u ) µ F (x) dx θµ 9*Š% C -EŠ%}lŒŠ%Š% C œä 1Ã~Ã~êD' Ìr ƒ Ši}k, ) = > @ FŒŠ$ =~;Ë:F Cƒ" ŒŽ"ŽAœ Ž"=~ ~ =~;%} @ ŽAœ Ž"=~ ~ @ }a}\@ œÿè@ ;%, )=\@ U Œ;%; ƒ Š$ );%ƒ" Œ )ƒ"=~ Š, µ* µ "' (w xy= x, 'ay1w # AyEw "yªz!#z!# x' ( "yª{ Ìr ƒ Š )ƒ"}a ÄB ÄŒŠ% Ä ) E;% C;% Š% C '@/ )ƒ"=~ 9 ~D=~ ) G;%Œƒ" \;%=~ @ ƒ"ž"ƒ 7 Ë@ ) G >,=~}a=1 Š%ƒ )ƒ"=~ =~ n ) ±} @/ˆ ƒ"}lœ} M @1Š]@\Š%Œ} =~ nž @1 C;_ Cƒ" ~ ŠC 8Y, L 1 l ) X ~@ Ž"Œ ) @/ _ ) ±;%= >, Š)Š {S t } ;% 5@1> ŠN =~;N ) a ;)Š2 _ )ƒ"}a a@ =5 ª a ) ±šc C;%=UŽ", ª CŽA,ˆ œ Ž", L 2 X ) l ~@ Ž"Œ XB ƒ > ƒ ŠG=~ '@ ƒ" =~;E ) X ;)Š$ N )ƒ"}a a@ =5 ª X ) XŽ", ª CŽ "y!! L 1 L 3 $w*v, L 4,... @ ;% S C ƒ" L ) SŠ@ }a _B @C ª Ìr _ ~@ Ž"Œ Š L k @ ;% l> @ Ž"Ž" U ) Ç ƒ" >, ] ) S;%= >, Š)Š {S t } @1ŠÄŠ$ '@/ )ƒ"=~ @ ;$ @ Lƒ" C C C Nƒ" >,;% C}a C ŠCœ {L k } k=1 ƒ ŠG@ Š$ Œ C >, ±=~ ƒ" C C C S@ ƒ C )ƒ > @ Ž"Ž ƒ Š$ );%ƒ" Œ ) ~@ ;%ƒ @ Ž" ŠC S> @ LŠ% =5BÒ ) @/ Ä ) ] Œ}l C; K =~ žž @1 C;Ä Cƒ" ~ Š Œ )ƒ"ž ;%Œƒ" U )ƒ"}a ]ƒ Š ~ƒ ª C a@e ~ C=~}a, );%ƒ > ƒ Š2 );%ƒ" Œ )ƒ"=~ kb ƒ ) a@ ;@ }a, ) C;)Š p = 1/(1+θ) @ q = θ/(1+θ) 0 }.

x œ %ë ð / /ÿ$þ7ûcø Ü7Ú Ù'æ Ìr ŒŠ,œ ) _;@ =~}P ~@ ;%ƒ @ Ž" M }\@C \ N,ˆ ;% Š)Š$ M = K L i. Ìr ƒ Š]ƒ"}kŽ"ƒ" Š] ) @/ M @1ŠX@L>,=~}a=~Œ ~ C=~}a, );%ƒ >\ ƒ Š2 );%ƒ" Œ )ƒ"=~ R ~ƒ ª C F ) ƒ Š$ );%ƒ" Œ )ƒ"=~ L Œ >- )ƒ"=~ F M (x) = i=1 θ 1 + θ G n (x), B C;% G n ƒ Šn ) n ) i>,=~ ª=~Ž"Œ )ƒ"=~ U=~ Y ) N ƒ Š2 );%ƒ" Œ )ƒ"=~ ËB ƒ ) \ ) N C >- )ƒ ª _ C Š%ƒ 7 @ i ) ] C Š%ƒ 7 9$,Ó D z g(x) = n=0 1 µ(1 + θ) F X(x) = 1 1 + θ b 0(x), b 0 (x) = F X(x). µ Ì k@ =5 ª ± *@1>- N )=~ ~, ) C;GB ƒ ) À ) ±;% C;% Š$ C '@/ )ƒ"=~ Í9 ~DÄŽ" 5@1 ŠG )=\ )!#z!# x='0( y =~;r ) N;%Œƒ" L;%=~ @ ƒ"ž"ƒ 7 9$ 5Õ~D ψ(u) = P(M > u) = θ 1 + θ n=0 ( 1 1 + θ ) n B n 0 (u), B C;% B 0 ƒ ŠÈ ) r '@ ƒ"ž=~ ) ƒ Š$ );%ƒ" Œ )ƒ"=~ k Œ >- )ƒ"=~ >,=~;%;% Š%=~ ƒ" ] )=G ) C Š$ƒ 7 b 0 @ B0 0(u) I {u 0} À> @ ŒŠ% L ) ƒ Šk =~;%}lœž @ )= C;%ƒ ª,ˆ Ž"ƒ >,ƒ ËŠ$=~Ž"Œ )ƒ"=~ Š\ =~;i@ Œ}l C; =~ >,Ž @ ƒ"} @ }a=~œ ƒ Š$ );%ƒ" Œ )ƒ"=~ ŠN9*Š$ C Ä ~ -EŠ$}lŒŠ)Š% C œ 1Ã~Ã~ÃN=~;ÅŸ!@ )2 C;@ ±: ƒ"ž"ž"}a=1 œ ª ~D' ) @/ Nƒ ŠG =1 N=ªŠ)Š$ƒ" Ž" ~œ ) X =~;%}lœž @\> @ a ƒ";% >- )Ž @ Ž"ƒ" À )=Ë> @ Ž >,Œ Ž @/ )ƒ" ) \;%Œƒ" ;%=~ @ ƒ"ž"ƒ 7 ª - =5B, ª C;Cœ ƒ lƒ" ª=~Ž ª Ša@ ƒ" ƒ ) ËŠ$Œ}Uœ C >, \B ŒŠ% E ) ]ÔU=~ ) @ ;%Ž"=±}a, ) = )=±Š%ƒ"}lŒŽ @/ ) E ) G;@ =~} ~@ ;%ƒ @ Ž" M @ i> @ Ž >,Œ Ž @/ ) ] ) _;%Œƒ" L;%=~ @ ƒ"ž"ƒ 7 ª ;%=~}p9$ 5Õ~D ) ];%Œƒ" L;%=~ @ ƒ"ž"ƒ 7 ψ(u) = EZœB C;% Z = 1(M > u)œ} @C \ _ ~ C C;@/ ) @1Šr =~Ž"Ž"=5BÄŠC 9$ 5D ÁE C C;@/ ) _@S;@ =~} ~@ ;%ƒ @ Ž" K ;%=~} ) G ~ C=~}a, );%ƒ >N ƒ Š2 );%ƒ" Œ )ƒ"=~ ËB ƒ ) @ ;@ }a, ) C;)Š p = 1/(1 + θ) @ q = θ/(1 + θ) 9 ~D ÁE C C;@/ ) _;@ =~}P ~@ ;%ƒ @ Ž" Š X 1,..., X K ;%=~}P ) _ C Š%ƒ 7 b 0 (x) 9AʪD @ Ž >,ŒŽ @/ ) M = X 1 + + X K 9 Ó D M > uœž", Z = 1œ=1 ) C;$B ƒ Š$ ]Ž", Z = 0 Ì } @ ƒ" ;%=~ Ž" C} Š% C C}kŠa )=< ÀŠ%ƒ"}lŒŽ @/ )ƒ" ;@ =~}f ~@ ;%ƒ @ Ž" Š B ƒ ) ) C Š%ƒ 7 b 0 (x) w 9 9 B2765249:3 Ê!; "v-w*x=âu(w #vsz"#v #v= w #y"' + b 0 (x) y vlyiz& v x' y "# 1x "( x0

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%ë ð / /ÿ$þ7ûcø Ü7Ú Ù'æ ø/ð~ö ð ÿèÿ+$5ü1ý /ÿ Âø þ -ùaÿ Yø β = 0.01 î θ = 5 u 1000 ø/ð5ð ÿèÿ+$5ü1ý /ÿ Âø þ -ùaÿ Yø β = 0.01 î θ = 1 u 1000,ˆ );% C}a CŽ UB CŽ"ŽAœ ) XŸ =~Ž"Ž @1>,šC C ª «N ƒ" > ƒ" =~;%}XŒŽ @a =~Ž ŠG@± ~= = @1>C>,Œ;@1>- À@ ) _;% Š2 Ä=~ Æ ) _}a, ) = Š =a =1 B=~;% \Š$=±B CŽ"ŽA -Ä=5B, ª C;CœŽ"= =~ ƒ" \@/ r ) y 7@/ˆ ƒ Š Š)> @ Ž" ]9 Ž"ƒ"}aƒ ŠÈ ;%=~} Ã Ò )=GÓ ad B }\@C SŠ)@C S ) @/ Å@ Ž"Ž}a, ) = ŠÈ,ˆ>, C ) Ž"ƒ" ~ );@ *k>g ~ƒ ª l@1>c>,œ;@/ ) _;% Š%ŒŽ ŠC :Í C θ = 5 9*Š$ C 0žƒ" ~Œ;% ± ~D'œ ) XŸž=~Ž"Ž @1>,šC C ª «G ƒ" > ƒ" =~;%}XŒŽ @±B=~;% Š ª C;$ B CŽ"ŽA ªÌr r 5@C ÂŽ"ƒ" ~ n );@ *a> @ ;%=5ˆ ƒ"} @/ )ƒ"=~ >,=~}a ŠŠ$ >,=~ Ì r=1 ) C;Å@ ;%=5ˆ ƒ"} @/ )ƒ"=~ ŠÄ@ ;% ]>,Ž" 5@ ;%Ž iœ @1>C>, C '@ Ž" ~ L ) l> @1Š% θ = 1 9*Š% C 0 ƒ" ~Œ;% ±ÊªD'œ ) XŸ =~Ž"Ž @1>,šC C ª «N ƒ" > ƒ" =~;%}lœž @±B=~;% Š,ˆ );% C}a CŽ ib CŽ"ŽAœ ) N=1 ) C;E@ ;%=5ˆ ƒ"} @/ )ƒ"=~ Š =~Ž ËŽ"ƒ % )Ž" S@1>C>,Œ;@1>- ª - C;% 5@ ) C;kƒ" R ) ƒ ŠaŠ$ >- )ƒ"=~ B ib ƒ"ž"žrœš%œ@ Ž"Ž @1Š)Š%Œ}a i ) @/ k ) Ë}a 5@ =~ Ä ) >,Ž @ ƒ"}z ƒ Š$ );%ƒ" Œ )ƒ"=~ ƒ Šr Œ@ Ž )= 1 @ θ = 0.1 Ò ) @ }k} @ > @1Š% B > @ d=~ '@ ƒ",ˆ@1>- ~@ Ž"Œ Š ƒ @ =~;%}lœž @d9 ªD @ ŒŠ% ) C} ƒ" =~;) C; )= >,=~}a@ ;% @ Ž"Žl}a, ) = ŠÀ,ˆ>, C ) 5@C A '@ ƒ"ž" @

ö,ö1ôaè 'Þ ÝÙ'Ú"Þ"è'élè Èô7â Þ é ö1ôaècãâù,ã2þ æ Þ Ú"ò %ë ø/ð /ð ÿ nþ 'ùaÿ Yø α = 0.01 î β = 0.01 î θ = 0.1 u 1000 ø ð í5ð ÿ nþ -ùaÿ Yø α = 0.001 î β = 0.001 î θ = 0.1 u 1000 Ê/ Â}a=~}k C G @ }a} @ ²E l³ Ž C;_@ ;%=5ˆ ƒ"} @/ )ƒ"=~ ŠCœ B ƒ > À ƒ" CŽ À ) X,ˆ@1>- _;% Š$ŒŽ : C α = 0.01 @ β = 0.01 9*Š% C žƒ" ~Œ;% SÓ D'œ @ Ž"ŽÈ@ ;%=5ˆ ƒ"} @/ )ƒ"=~ ŠE,ˆ>, C E ) 5@C ªœŽ"ƒ" ~ ±9* ƒ Š)@1Š$ );%=~ŒŠE;% Š$ŒŽ Š'DÄ@ L 5@C ÂŽ"ƒ" ~ E );@ *k>1œ ~ƒ ª X@±;% CŽ @/ )ƒ ª l C;%;%=~; =~ Ž" Š%ŠÄ ) @ 3% : C LB Sƒ" >,;% 5@1Š$ l ) S ~@ ;%ƒ @ >, U9 α = 0.001 @ β = 0.001œ Š% C ƒ" ~Œ;% ËÕ~D'œY ) kš%ƒ )Œ@/ )ƒ"=~ ƒ Š]Š%ƒ"}aƒ"Ž @ ;CœY Œ S ) ƒ ŠN )ƒ"}a a ) a 5@C ÂŽ"ƒ" ~ l );@ *k> @ ;%=5ˆ ƒ"} @/ )ƒ"=~ XŽ"= =~ Šž}lŒ> S, % ) C;!@ ] ) Ÿž=~Ž"Ž @1>,šC C ª «N ƒ" > ƒ" ± =~;%}lœž @ >,=~}k Š ;)Š$ B ƒ ) L@ Ž"}a=ªŠ$,ˆ@1>- ;% Š%ŒŽ Š, :Í C ± ) >,Ž @ ƒ"}w ƒ Š2 );%ƒ" Œ )ƒ"=~ ƒ Šn@E}kƒ ˆ )Œ;% r=~ ) ;% C,ˆ =~ C )ƒ @ Ž ŠG9*Š% C žƒ" ~Œ ;% ªD'œ ) ;@ }C;2 YŒ C;% œæ²e l³ Ž C;CœÆÊ/ Â}a=~}a C _ @ }a} @ ²E l³ Ž C;]@,ˆ =~ C )ƒ @ ŽY@ ;%=5ˆ ƒ"} @/ )ƒ"=~ Š ~ƒ ª Œƒ ) G@1>C>,Œ;@/ ) E;% Š%ŒŽ Š,œ B ƒ"ž" ) E Å C C }\@ =5B C;)Š @ YŒ C;% \@ ;%=5ˆ ƒ"} @/ )ƒ"=~ ŠÄ@ ;% )2ŒŠ$ E@1>C>, C '@ Ž" ~ Ç ƒ" >, S ) C;% a@ ;% l =\,ˆ@1>- N}a, ) = ŠE =~;E=1 ) C;N ƒ Š2 );%ƒ" Œ )ƒ"=~ Š_>,=~ Š%ƒ C;% œ B @ ;% ~=~ƒ" )=R> @ Ž >,ŒŽ @/ ) ) ;% CŽ @/ )ƒ ª C;%;%=~;)ŠiB ƒ ) ;% Š% >- Ë )= ) }a=ªš$ U@1>C>,Œ ;@/ ) }a, ) = D;%=~} žƒ" ~Œ;% Š ' F ) =ªŠ)Š$ƒ" Ž" F> @ ƒ @/ ) ŠU@ ;% À ) ;@ }C;2

- ð / /ÿ$þ7ûcø Ü7Ú Ù'æ ø/ð ë5ð î ÿ ø $" - / AÿNý Aÿ$ÿ_ÿ $5ü1ý /ÿ+ Âø úù Æø î î β 1 = 0.014631 β 2 = 0.190206 β 3 = 5.514588 ÿ2ø î î î a 1 = 0.0039793 a 2 = 0.1078392 a 3 = 0.8881815 θ = 0.1 u 1000 ýü ' AøúùÂý ]ý ÿžÿ " /ÿ$ $î rý ÿ žÿ" /ÿ$ Åìý 'þ$ÿ$û ø /þ ø 'ü/ü/ Aý%$ ø ' Âøúý /ù /ÿ$ ÿ ø $ - / Aÿ!ý Aÿ$ÿÈÿ+$ ü~ý /ÿ Âø úù%ð ÿ ' Âø &,ÿèÿ2 A Aý, AùYø ð θ u ψ(u) E PK E DV E 3MGDV 5ð,í % 5ð ( 5ð B )í 5ð Ö' -ð ë- -ë,ö 5ð % % 5ð 5ð,Ö B í ðcö ( Ö ð ë 5ð )í % 5ð -Ö Ö 5ð,Ö' ë5ð B 5ð %,Ö 5ð Ö' % 5ð ë-ëb 5ð - -í 5ð,í 5ð,í 5ð Ö-í % 5ð ë- $ 5ð- % 5ð í'ëb ) 5ð -ë-ë 5ð í B /ð,í -ð - % 5ð Ö'ë 5ð,í'ë-ë ) 5ð ë-ë Ö ð - 5ð,í % - 5ð 5ð % % - 5ð í 5ð B ' -ð )Ö-í 5ð ë-,ö 5ð )í % - 5ðCÖ 5ð,í - -ðb 5ð CÖ 5ð Ö' % - 5ð Cí,í 5ð B Ö ð )Ö' -ð - 5ð Ö-í % - 5ð Ö,ë 5ð,Ö 5ð -ð % B 5ð- % - 5ð Ö,ë 5ð,Ö'ë 5ð -ë,ö-ð -ð - % - 5ð -Ö 5ð Ö' ð Ö' -ë 5ð 5ð,í % - - 5ð 5ð B )í 5ð - 5ð - 5ð % % - - 5ð,Ö ) 5ð Ö' 5ð CÖ 5ð B - 5ð )í % - - 5ð -,í 5ð -í 5ð ' - - 5ð Ö 5ð Ö' % - - 5ð - B 5ð í',í'ë 5ð, - - -ðb%ë 5ð Ö-í % - - 5ð - - ( 5ð- % - - 5ð - - ð 5ð ' - 5ð - - -ð - % - - 5ð - -,

/ÿ$ ìý 'þ$ÿ$û! ø /þ ø ö,ö1ôaè 'Þ ÝÙ'Ú"Þ"è'élè Èô7â Þ é ö1ôaècãâù,ã2þ æ Þ Ú"ò ý rü ' AøúùAý Uý žÿ /ÿ2 $î rý ÿ && žÿ 'ü/ü/ Aý%$5ø& ' Âøúý /ù /ÿ$ ÿ úý /ý- & /øúù  Aø ÂøÏý ð ÿ ' Âø &,ÿèÿ$ A Aý- AùYø ð θ u ψ(u) E PK E DV E 3MGDV 5ð,í ) - 5ð í-í' 5ð B,Ö Ö' 5ð ëb )í /ð Ö,ë 5ð % ) - 5ð Cí $ $ 5ð )í ) - 5ð Ö,í 5ð,Ö-íCí /ð Ö-Ö /ð ë, 5ð Ö' ) - 5ð - 5ð 5ð 5ð,,í-Ö 5ð Ö-í ) - 5ð 5ð Ö ð,í Ö5ð,ë,Ö-Ö 5ð- ) - 5ð % 'ëcí 5ð Ö,ë 5ð í - - /ð Ö-ë- -ë -ð - ) - 5ð,Ö-í Cí 5ð "-ð -Ö 5ð ë -Ö-ë 5ð,í %, - 5ð Ö ð ë,í - í,í ð B í 5ð 'ë 5ð % %, - 5ð B % /ð,öb Cí ð í,ö ð,ë 5ð )í %, - 5ð -,í 5ð,Ö 5ð ë-ë- ð ' - 5ð Ö' %, - 5ð - $ $ 5ð Ö-í %, - 5ð -,Ö 5ð Ö B í5ð %ë,ö /ð í ( 5ð- %, - 5ð -,Ö, 5ð, íb-ðb ë5ð í Cí -ð - %, - 5ð - - -ë, ë5ð-, - 5ð,Ö,ë5ð -Ö 5ð,í % -, - 5ð - - - $ $ 5ð % % -, - 5ð - - - 5ð í' - - %, 5ð - - - % - /ð, - - 5ð )í % -, - 5ð - - - CÖ Ö ð -, - %, 5ð - - - % - /ð, - - 5ð Ö' % -, - 5ð - - - CÖ )í ð -, - %, 5ð - - - % - /ð, - - 5ð Ö-í % -, - 5ð - - - í' 5ð -, - %, 5ð - - - % - /ð, - - 5ð- % -, - 5ð - - - Ö-Ö ð -, - %, 5ð - - - % - /ð, - - -ð - % -, - 5ð - - -, YŒ C;% œ²e ³ Ž C;CœÊ/ Â}k=~}a C @ }a} @_²E ³ Ž C;r@ \Ÿž=~Ž"Ž @1>,šC C ª «G ƒ" > ƒ" @ ;%=5ˆ ƒ"} @/ )ƒ"=~ ŠC - =5B, ª C;Cœ ) Å;@ }C;2 YŒ C;% @ ;%=5ˆ ƒ"}\@/ )ƒ"=~ ËB=~;% Šr=~ Ž =~;Ž"ƒ" ~ % A '@ ƒ"ž" k ƒ Š2 );%ƒ" Œ )ƒ"=~ ŠCœ C >, rb @C ª r )=]> = =ªŠ$, 7B C C ± ) ²E ³ Ž C; @ UŸž=~Ž"Ž @1>,šC C ª «N ƒ" > ƒ" @ ;%=5ˆ ƒ"} @/ )ƒ"=~ ŠC ÌÆ=À ) ƒ Š± C B \ '@ ~ ˃" )=F>,=~ Š%ƒ C;@/ )ƒ"=~ @ }aƒ ˆ )Œ;% Ë=~ ) ;% C Ë,ˆ =~ C )ƒ @ Ž ƒ Š$ );%ƒ" Œ )ƒ"=~ Šr@ k@nž"=~ ~ =~;%} @ Ž ƒ Š$ );%ƒ" Œ )ƒ"=~ =~;ÅŠ$ƒ"}aŽ" E@ @ Ž )ƒ > ;% Š$ŒŽ ŠÅƒ" a ) =~;%}a C;> @1Š% GŠ$ C _Ç >- )ƒ"=~ L k ) EŽ @/ % ) C;> @1Š% ~œ B ƒ ) i@]> =~ƒ >, E=~ ÆŠ$ >,ƒ" >Ä@ ;@ }a, ) C;)ŠCœ,ˆ@1>- ~@ Ž"Œ Šr=~ ) _;%Œƒ" Ë;%=~ @ ƒ"ž"ƒ 7 L> @ Ë ]>,=~}aœ ) ËŒŠ$ƒ" a Œ}a C;%ƒ > @ Ž ƒ" ª C;)Š%ƒ"=~ k=~ ) @ Ž @1>, );@ Š% =~;%} 9*Š% C r: ƒ" Š2 '@1 œ 1œ @ kì =~;%ƒ" @ X: ƒ" Š$ '@1 œ ~D'!Y,!ŒŠÈ =5B >,=~}k@ ;% ) ) ;% C @ ;%=5ˆ ƒ"}\@/ )ƒ"=~ Š, ±ÌÈ@ Ž" ŠÊG@ XÓ ) r,ˆ@1>- Å@ a@ ;%=5ˆ ƒ"} @/ ) ~@ Ž"Œ ŠCœ @ l;% CŽ @/ )ƒ ª C;%;%=~;)Šn@ ;% rš% =5B XB ƒ ) ±;% Š$ >- )= u @ θ B ƒ ) F ) \ ƒ Š2 );%ƒ" Œ )ƒ"=~ @ ;@ }a, ) C;)ŠX@1Š_ƒ" ÁE;@ CŽ"Ž@ Ç C ~ C;) @ Ž 9$ 5D >, 7 ƒ" ~Œ;% Š a@ \ƒ Šk 5@1Š$ R )= =1 )ƒ >, À ) @/ i²e L³ Ž C;i@ ÍÊ/ Â}a=~}k C @ }a} @F²E U³ Ž C; @ ;% i =À} @/ > =~;l ) iÿž=~ž"ž @1>,šC C ª «G ƒ" > ƒ" Í@ ;%=5ˆ ƒ"} @/ )ƒ"=~ 9*Š% C \ ) \ =~Ž *@1>,

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