Metodi asintotici di verosimiglianza per esperimenti sequenziali

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(23) ! "#$%&' ('$)'*+'. !. ,-. "#$%&' ('$)'*+'. ! !" # $%&$ ' ()*%+,&-$. /'*0-,('+-' 1&&' .'*&/ Xi 0 Yi 1 i = 1, 2, . . . , n1 #0 2*3'*4'#' $*5%*#' 3*663050&7*&7' #0 3'56/570 80' 5/99077' *5509&*7' * 8%0 73*77*+0&7'1 %&/ 8' $/&73/##/1 '# $%' 6*3*+073/ 2033: '&8'$*7/ $/& '# 608'$0 C 0 %&/ 5603'+0&7*#01 '# $%' 6*3*+073/ 5*3: $/&73*8; 8'57'&7/ 8*# 608'$0 E ! .' 5%66/&9* $)0 5'*&/ '&8'60&80&7' $/& 8'573'4%<'/&0 &/3+*#0 $/& 2*3'*&<* &/7*= Xi ∼ N (µC , σ 2 )1 Yi ∼ N (µE , σ 2 )! .' $/&5'803' #*. 203'>$* 8?'6/705' H0 : µE = µC $/&73/ #?*#703&*7'2* 4'#*703*#0 H1 : µE 6= µC 1 $/& 63/4*4'#'7: 8' 033/30 8' @ 7'6/ α 0 6/70&<* 1 − β 603 µE − µC = ±δ ! .0 *8 /9&' 73*77*+0&7/ 5/&/ *5509&*7' $/+6#055'2*+0&70 n 5/99077'1 603 203'>;. $*30 #?'6/705' 5' 6%A %7'#'<<*30 #* 57*7'57'$* 7057 57*&8*38'<<*7* Zep $)0 *23: #* B/3+*. 1. n X. n X. !. Xi C !,D Yi − (2nσ 2 ) i=1 i=1 p $/& 8'573'4%<'/&0 N ((µE −µC ) n/(2σ 2 ), 1)! .' *23: E%'&8' $)0 Zep ∼ N (0, 1) Zep = p. 5/77/ H0 0 '# 7057 3'>%7* H0 50 |Zep | > Φ−1 (1 − α/2)!. @ 30E%'5'7' 8' 6/70&<* 603 '# 7057 $/& #'20##/ α1 $/&5'803*&8/ $)0 5/77/ p #?'6/705' *#703&*7'2* H1 : µE − µC = ±δ 5' )* Zep ∼ N (±δ n/(2σ 2 ), 1)1 5/&/. 5/88'5B*77' 50. p 1 − β = F{|Zep | > Φ−1 (1 − α/2) − δ n/2σ 2 } r r n n α α −1 −1 = F{Zep > Φ (1 − ) − δ } + F {Z < −Φ (1 − } ) − δ ep 2 2σ 2 2 2σ 2 p p # # = 1 − Φ Φ−1 (1 − α/2) − δ n/2σ 2 + Φ −Φ−1 (1 − α/2) − δ n/2σ 2 p p # # = Φ −Φ−1 (1 − α/2) + δ n/2σ 2 + Φ −Φ−1 (1 − α/2) − δ n/2σ 2 .. G*7/ $)0 #* 63/4*4'#'7: 8' 3'>%7*30 H0 : µE − µC = 0 '& B*2/30 8' θ < 0 . # E%*&8/ '# 203/ 2*#/30 H θ = δ H 73*5$%3*4'#01 5' )* $)0 1 − β = Φ −Φ−1 (1 − p α/2) + δ n/2σ 2 ! I56#'$'7*&8/ n '& E%057* 0E%*<'/&0 5' /77'0&0 #* &%+03/5'7: $*+6'/&*3'*.

(24) !. "#$%&'(' )* +,$-&&% &-'.%/%. /0- ,'11%,2# % .-34%,%&% 1% $'&-56#. # 2 nf,2 = Φ−1 (1 − β) + Φ−1 (1 − α/2) 2σ 2 /δ 2. 7)*)8. /0- 9 %( 54:-.' 1% $#6%-5&% 1# #,,-;5#.- #1 ';545' 1-% 14- &.#&&#:-5&%* <-./%= /':$(-,,%>#:-5&- % ,';;-&&% ',,-.>#&% 5-((?-,$-.%:-5&' ,#.#55' 2nf,2 * @- (?%$'&-,% #(&-.5#&%># 2',,- ,&#&# 45%(#&-.#(-A /'5 ;(% ,&-,,% $#,,#;;% ,% '&&%-5-B. # 2 nf,1 = Φ−1 (1 − β) + Φ−1 (1 − α) 2σ 2 /δ 2 .. 7)*C8. @% 5'&% /0- (' ,&-,,' -,-:$%' $4= -,,-.- .%,/.%&&' %5 &-.:%5% 1% 34#5&%&D 1% >-.',%:%;(%#56# 4&%(%66#51' (# 5'&#6%'5- ,-;4-5&-* @% /'5,%1-.% 45 :'1-((' ,&#&%,&%/' /'5 ;-5-.%/# 2456%'5- 1% 1-5,%&D f (x; ψ) 1'>- %( $#.#:-&.' ψ ∈ Ψ ⊆ Rp A 9 ,411%>%,' %5 14- /':$'5-5&%A ψ = (θ, λ) 45# 1% %5&-.-,,-A θA - 45# 1%,&4.E'Aλ* F# /':$'5-5&- ,4((# 34#(- ,% 1-,%1-.#. 2#.- %52-.-56# 9 θ /0- 0# 1%:-5,%'5- A :-5&.- (# /':$'5-5&- 1% 1%,&4.E' 0# 1%:-5,%'5- p − 1* F# 2456%'5- 1% ('; >-.',%:%;(%#56# 1-( :'1-((' 9 l(ψ)A -. ,- ,% 9 %5&-.-,,#&% # 2#.- %52-.-56# ,4( ,'(' $#.#:-&.' 1% %5&-.-,,- ,% 4&%(%66# (# ('; >-.',%:%;(%#56# $#.6%#(-A ' ('; >-.',%:%;(%#56# $.'G('A lP (θ) = l(ψˆθ ) =. ˆ θ )A 1'>- λ ˆ θ 9 (# ,&%:# 1% :#,,%:# >-.',%:%;(%#56# 1% λ /'5,%1-.#51' θ l(θ, λ G,,#&'* H( >-&&'.- 1-((- 1-.%>#&- $#.6%#(% $.%:- 9 l∗ (ψ)* I% -,,'A (?-(-:-5&' .%2-.%&' ,'(#:-5&- #((# 1-.%>#&# .%,$-&&' # θ ,% 1-5'&# /'5 lθ (ψ) - ,%:%(:-5&$-. ;(% #(&.% $#.#:-&.%* @% 1-G5%,/-A $-. ,-:$(%G/#.- (# 5'&#6%'5-A S(θ) = ˆ θ )A (# !"#$ .%,$-&&' # θ /#(/'(#&# %5 7θ, λ ˆ θ 8* F# :#&.%/- 1% %52'.:#6%'5lθ (θ, λ ',,-.>#&#A ',,%# (# :#&.%/- 1-((- 1-.%>#&- ,-/'51- .%,$-&&' # ψ /#:E%#&- 1% ,-;5'A 9 j(ψ) - 0# 1%:-5,%'5- p × p* J#/-51'5- %( >#('.- #&&-,' ,% '&&%-5- (#. :#&.%/- 1% %52'.:#6%'5- #&&-,# ' 1% J%,0-.B i(ψ)* K-( ,-;4%&' >-..D 4&%(%66#&' %( ,-;4-5&- .%,4(&#&' #,%5&'&%/'A ,'&&' /'51%6%'5% 1% .-;'(#.%&DA. ψˆ∼N ˙ p (ψ, i(ψ)−1 ), 1'>- %( ,%:E'(' ∼ ˙ %51%/# /0- (# 1%,&.%E46%'5- 9 #$$.',,%:#&#A - i(ψ) $4=.

(25) ! "#$%&' ('$)'*+'. ,-. ˆ 8$50! 9*$. . :*#7*&; .//.0. /1/2'2%'2* 3*##4'&510+*6'1&. 1//.07*2* j(ψ) 1 j(ψ) <<,; 9*0*=0*51 >!?@! A.# $*/1 /' 3./'3.0'; $1+. &.=#' ./.+B' /.=%.&2'; 5*0. '&5.0.&6* /%# /1#1 B*0*+.201 θ; /' %2'#'66* #4'&510+*6'1&. *22./* B*06'*#.C. V (θ, λ) = iθθ (θ, λ)−1 = iθθ (θ, λ) − {iθλ (θ, λ)}{iλλ (θ, λ)}−1 {iθλ (θ, λ)}T , 317. *3 ./.+B'1; iθ,λ D '# E#1$$1 0.#*2'71 * θ . λ 3.##* +*20'$. i(ψ); +.&20.. iθλ D #4*&*#1=1 E#1$$1 3' i(ψ)−1 ! :' )* $). V (θ, λ) 3'B.&3. /'* 3*# B*0*+.201 3' '&2.0.//. $). 3* F%.##1 3' 3'/2%0E1! G' /.=%'21 7.00H %2'#'66*2* &.##* 510+* ˆ θ )−1 ; 317. *# B*0*+.201 3' 3'/2%0E1 D /2*2* /1/2'2%'2* #* /%* V (θ) = iθθ (θ, λ /2'+* 7'&$1#*2*! I* F%*&2'2H *&*#1=* * V (θ); +* 0'5.0'2* *##4'&510+*6'1&. 1//.07*2* j(ψˆθ ) 7.00H '&3'$*2* $1& I(θ)!. :' )* F%'&3' $). θˆ *70H 3'/20'E%6'1&. */'&212'$* N (θ, V (θ)−1 ); . *&$). F%' ˆ 8/' 7.3* 9*$. . '# 0'/%#2*21 &1& $*+E'* /. * V (θ) 7'.&. /1/2'2%'21 I(θ) 1 I(θ) :*#7*& ,JJ>; 9*0*=0*51 ?!>@! K.&=1&1 '&1#20. 3.L&'2' '# 2./2 #1= 0*BB1021 3' 7.01/'+'=#'*&6*; '# 2./2 !"#$ . '# 2./2 3' M*#3 $). /1&1; '& 103'&.;. ˆ − l(ψ)), W = 2(l(ψ). Wu = lψ (ψ)T i(ψ)−1 lψ (ψ),. We = (ψˆ − ψ)T i(ψ)(ψˆ − ψ) . )*&&1 3'/20'E%6'1&. */'&212'$* &%##* χ2p ! A.# $*/1 '& $%' '# B*0*+.201 ψ /'* /$*#*0.; 2*#' 2./2 7.&=1&1 $1&/'3.0*2' &.##* #101 7.0/'1&. 3'0.6'1&*#.; $'1D. Z = /=&(ψˆ − ψ). q ˆ − l(ψ)), 2(l(ψ). Zu = lψ (ψ)(i(ψ))−1/2 ,. Ze = (ψˆ − ψ)(i(ψ))1/2 , $1& 3'/20'E%6'1&. */'&212'$* &%##* A8<;,@! N& B0./.&6* 3' B*0*+.20' 3' 3'O /2%0E1; +* $1& θ /$*#*0.; 7.00*&&1 $1&/'3.0*2. #. *&*#1=). 7.0/'1&' B01L#1;.

(26) !. "#$%&'(' )* +,$-&&% &-'.%/%. ',,%#. Zp = ,01(θˆ − θ). q. ˆ − lP (θ)), 2(lP (θ). Zup = S(θ)V (θ)−1/2 ,. Zep = (θˆ − θ)V (θ)1/2 , #2-1&% 3%,&.%456%'1- #,%1&'&%/# 15((# 789: ;* <-. 5(&-.%'.% #$$.'='13%>-1&% ,% .%>#13# # <#/- - ?#(2#1 8 @@A: <#.#0.#B C*D - E*A;* F'.1#13' #((G-,->$%' /'1 .%,$',&- #2-1&% 3%,&.%456%'1- 1'.>#(-: ,% $5H '.# 1'&#.- /I- Zep 1-((# 8)* ; J %( &-,& 3% K#(3 1-((# 2-.,%'1- $.'B(' /'1. V (θ) = I(θ) = iθθ (θ)−1 = n/(2σ 2 )* ?'&&' H0 I# 3%,&.%456%'1- -,#&&# N (0, 1) p - ,'&&' H1 I# 3%,&.%456%'1- -,#&&# N (θ V (θ), 1)* ?% 1'&% /I- %( .-L5%,%&' 3% $'&-16# 8)*); $5H -,,-.- .%,/.%&&' %1 &-.>%1% 3% %1='.>#6%'1- #&&-,#: -,,-13' L5-,&# $.'$'.6%'1#(- #((# 15>-.',%&M /#>$%'1#.%#: '&&-1-13'. # 2 V (θ) = Φ−1 (1 − β) + Φ−1 (1 − α/2) /δ 2 ,. /I- .#$$.-,-1&# %( (%2-((' 3% %1='.>#6%'1- >#,,%># 1-/-,,#.%# $-. .#00%510-.% .-L5%,%&% 3% $'&-16#*. ! !. !"#!$%&'( )*+,',&(&%- .'%&+ /!0% "#$%&'. 7-((G#>4%&' 3% $.'4(->% 3% 2-.%B/# 3G%$'&-,% ,->$(%/%: %( ?<NF 3% K#(3 2%-15&%(%66#&' L5#('.# ,% 2'0(%# >%1%>%66#.- %( 15>-.' #&&-,' 3% .%,$',&- 3# ',,-.O 2#.-* P1 L5-,&% /#,% %1=#&&% 51 &-,& # /#>$%'1- B,,' $'&.-44- 1'1 -,,-.- 51# 45'1# ,/-(&#* ?%#1' x1 , . . . , xn : ',,-.2#6%'1% $.'2-1%-1&% 3#((- /'..%,$'13-1O &% 2#.%#4%(% #(-#&'.%- X1 , . . . , Xn : %13%$-13-1&% -3 %3-1&%/#>-1&- 3%,&.%45%&8%*%*3*; /'1 0-1-.%/# =516%'1- 3% 3-1,%&M f (x; ψ) $-. Xi = x* ?% ,5$$'10# 3% 2'(-. 2-.%B/#.- H0 : ψ = ψ0 /'1&.' (G#(&-.1#&%2# H1 : ψ = ψ1 * P( &-,& ?<NF 3-B1%,/- (# .-0%'1- 3% #//-&&#6%'1- - .%B5&' 3-((G%$'&-,% 15((#.

(27) ! "#$%&' ('$)'*+'. ,-. .%##* /*.0 1' 1%0 .23#'0 A 0 B 0 10# 4*552462 1' 7042.'+'3#'*&8*. γn =. n Y. f (Xi ; ψ1 )/f (Xi ; ψ0 ).. i=1. 1270 f (Xi ; ψ1 ) 0 f (Xi ; ψ0 ) '&1'$*&2 #* 9%&8'2&0 1' 10&.'6: .2662 #;'5260.' *#604&*6'7* 0 .2662 #;'5260.' &%##* 4'.5066'7*+0&60! <0 γn ≤ A *##24* .' *$$066* H0 = .0 γn ≥ B .' *$$066* H1 0 .0 A < γn < B. .' 5*..* *1 *&*#'88*40 %&* 4'.52.6* .%$$0..'7* 52'$)> &2& .' 704'?$* #* 4032#* 1' *440.62!. @*#0 10.$4'8'2&0 7*#0 *&$)0 &0# $*.2 1' 2..047*8'2&' &2& '!'!1!= /*.6* $2&A .'104*40 γn $2+0 '# 4*552462 1' 7042.'+'3#'*&8* 10##0 1'.64'/%8'2&' $2&3'%&60 1' (X1 , . . . , Xn ) .2662 H0 0 .2662 H1 ! <' 10?&'.$* Pi (Qi ) #* 542/*/'#'6: .2662 ψi $)0 .' 704'?$)' Qi B 504 C%*#.'*.' '&.'0+0 1' 2..047*8'2&' '&1'50&10&6' (x1 , . . . , xn ) 6*#0 $)0 γn .'* &0##* 82&* 1' Q # *$$066*8'2&0 1' H1 = .' )* $)0 ni=1 f (Xi ; ψ1 )/f (Xi ; ψ0 ) ≥ B = 504$'D. P1 (Q1 ) ≥ BP0 (Q1 ).. E !FG. H& +212 *&*#232= 504 C%*#.'*.' '&.'0+0 1' 2..047*8'2&' 6*#0 $)0 γn .'* &0##* Q # 82&* 1' *$$066*8'2&0 1' H0 = .' )* $)0 ni=1 f (Xi ; ψ1 )/f (Xi ; ψ0 ) ≤ A= 504$'D. P1 (Q0 ) ≤ AP0 (Q0 ).. E !IG. J0 542/*/'#'6: 1' 044240 1' H 0 HH 6'52 52..2&2 0..040 10?&'60 $2+0 α = P0 (Q1 ) 0 β = P1 (Q0 ) 01 0..0&12 Q0 0 Q1 1'.3'%&6'= .' )*K P0 (Q0 ) = 1 − α 0 P1 (Q1 ) =. 1 − β ! L4*= 1*##0 10?&'8'2&' 1' 044240 1' H 0 HH 6'52 1*60 0 1*##0 0C%*8'2&' E !FG 0 E !IG 52..'*+2 104'7*40 1%0 '+5246*&6' 1'.%3%*3#'*&80 $)0 1072&2 0..040 .211'.9*660 1*' 7*#24' A 0 B. A ≥ β/(1 − α) B ≤ (1 − β)/α..

(28) !. "#$%&'(' ) *+$,&&% &,'-%.%. /#(% 0%+,12#1(%#34, 0%5,3&#3' #$$-'++%6#&#6,3&, 0,((, 212#1(%#34, 3,( .#+' %3 .2% (, +'1(%, A , B 5,31'3' +2$,-#&, 0% $'.' 0# γn 72#30' &,-6%3# %( .#68 $%'3#6,3&') 9, 3,( &,+& 0:%$'&,+% +% 0,+%0,-# ;#-, %3 6'0' .<, (, $-'=#=%(%&> 0% ,--'-, 0% $-%6' , +,.'30' &%$' 3'3 +%#3' +2$,-%'-% #0 α , β -%+$,&&%5#6,3&,? #(('-# (, +'1(%, #$$-'++%6#&, 2&%(%44#&, +'3' A = β/(1 − α) , B = (1 − β)/α) @( &,+& 9AB/ C 23 &,+& '&&%6'? '++%#? +, (, '++,-5#4%'3% +'3' %)%)0) 6%3%8. 6%44# (# 326,-'+%&> .#6$%'3#-%# #&&,+# D. !"#$%" &$'()" *+',"#? *9EF $,-. ψ = ψ0 , ψ = ψ1 &-# &2&&% % &,+& .'3 $-'=#=%(%&> 0% ,--'-, 0% $-%6' , +,.'30' &%$' 3'3 +2$,-%'-% #0 α , β ) @3 5#-%, +%&2#4%'3% $-#&%.<, +% $'&-,==, ,++,-, %3&,-,++#&% # 5,-%G.#-, 23 %$'&,+% .'6$'+%&# 0,( &%$' H0 : ψ ≤ ψ0 .'3&-' H1 : ψ ≥ ψ1 $'3,30' %( 5%3.'('. α(ψ) ≤ α $,- &2&&% % ψ ≤ ψ0 , β(ψ) ≤ β $,- &2&&% % ψ ≥ ψ1 ) 9, (# 0%+&-%=24%'3, .'31%23&# 0% (X1 , . . . , Xn ) <# 23 -#$$'-&' 0% 5,-'+%6%1(%#34# 6'3'&'3' $,'13% n? #(('-# +% $2H 0%6'+&-#-, .<, %( &,+& 9AB/ -,+&# #$$-'$-%#&'? 6#. 3'3 +'00%+;# 3,++23# $-'$-%,&> 0% '&&%6#(%&> 3,% .#+% %3 .2% ψ 6= ψ0 , ψ1 ) @3 '13% .#+'? +$,++' -%+2(&# .<, (:*9E 0,( 9AB/ C 6%3'-, 0,((# 326,-'+%&> .#6$%'3#-%# 3,.,++#-%# $,- 23 &,+& .'--%+$'30,3&, # .#6$%'3, G++' $,- '13%. ψ ) A,- 0%6'+&-#4%'3% , #$$-';'30%6,3&% +% 5,0#3' I#(0 DJKLMF , N'&4 D !!O? $$) PO!P8POJLF). ! !" #$%&'$()*$+,- .+,/$),&0 1$ ),0 %)..-%%$+,- 1$ %&02 &$%&$.3- 4-' '$%4+%&- .+, 1$%&'$()*$+,- ,+'506- 70'$0,*0 ,+&0 9% +2$$'31# '-# 0% 2&%(%44#-, (' +&,++' 6'0,((' 0,( A#-#1-#;' ) )J $,- .'38 02--, %( 6,0,+%6' ,+$,-%6,3&' 6# %3 6'0' +,72,34%#(,) Q:'=%,&&%5' C +,6$-, 72,((' 0% 5,-%G.#-, +, (:212#1(%#34# 0,% 02, &-#&&#6,3&% C $(#2+%=%(, '$$2-, 3') Q# +&#&%+&%.# 2&%(%44#&#? Zep,k ? #5-> (# ;'-6# D )JF? 6# +% #5-> # .<, ;#-, .'3 23# +2..,++%'3, (Zep,1 , . . . , Zep,K ) 0'5, #0 '13% #3#(%+% +% 2&%(%44#3' % 5#8 ('-% 0%+$'3%=%(% #((:#3#(%+% $-,.,0,3&, , (, 32'5, '++,-5#4%'3%) *( $'+&' 0% n +% 2&%(%44# mk $,-.<R %( 326,-' 0% '++,-5#4%'3% 5#-%# %3 =#+, #( 1-2$$' .<,.

(29) ! "#$%&' ('$)'*+'. ,. -' -.* $/&-'012*&0/! 3/+1 4'-./ 512 ' .1-. * &%+12/-'.6 $*+5'/&*2'* 7--*.*8 p 4*#1 '# 2'-%#.*./ 1-*../ Zep,k ∼ N (θ Vk (θ), 1)8 512$'98 0*./ $)1 /:&' Zep,k ; $/+<'&*='/&1 #'&1*21 0' &/2+*#' '&0'51&01&.'8 #* 0'-.2'<%='/&1 $/&:'%&.*. 01##1 -.*.'-.'$)1 (Zep,1 , . . . , Zep,K ) ; %&* &/2+*#1 +%#.'4*2'*.*! (1-.* 0* $*#> $/#*21 #* $/4*2'*&=* .2* #1 -.*.'-.'$)1 01' 4*2' :2%55'! ?12 @*2#/ -' %.'#'==* '# @*../ $)1 #* 0'-.2'<%='/&1 01##/ -.'+*./21 0' +*--'+* 412/-'+':#'*&=* 512 '# 5*2*+1.2/ 0' '&.121--1 ; θˆ(k) ∼ N (θ, Vk (θ)−1 ) A'& B%1-./ $*-/ ; %& 2'-%#.*./ p P Pmk 1-*../C8 1 $)1 Zep,k = Vk (θ)θˆ(k) 8 0/41 /2* θˆ(k) = ( mk i=1 Yi − i=1 Xi )/mk. 1 Vk (θ) = mk/2σ 2 8 2*5521-1&.*&/ #/ -.'+*./21 0' +*--'+* 412/-'+':#'*&>. =* 1 #D'&@/2+*='/&1 *..1-* 5*2='*#1 512 θ %.'#'==*&0/ #1 /--124*='/&' 7&/ *##*. k >1-'+* *&*#'-'! E' )* $)18 -/../ H0 8 p p 3/4(Zep,k1 , Zep,k2 ) = 3/4( Vk1 (θ)θˆ(k1) , Vk2 (θ)θˆ(k2) ) p p = Vk1 (θ) Vk2 (θ) 3/4(θˆ(k1) , θˆ(k2) ) p p = Vk1 (θ) Vk2 (θ) F*2(θˆ(k2) ) p 512 1 ≤ k1 ≤ k2 ≤ K. = Vk1 (θ)/Vk2 (θ),. A !GC. HD%:%*:#'*&=* 3/4(θˆ(k1) , θˆ(k2) ) = F*2(θˆ(k2) ) -@2%..* '# @*../ $)1 θˆ(k1) ; $*#$/#*./ %.'#'==*&0/ %& -/../'&-'1+1 01##1 /--124*='/&' $/& #1 B%*#' -' $*#$/#* θˆ(k2) /--'*8 1-5#'$'.*&0/8. θˆ(k1) = θˆ(k2) =. mk mk1 X 1 #X1 Yi − Xi mk1 i=1 i=1. mk mk1 mk2 mk2 X X X 1 #X1 Yi − Xi + Yi − Xi mk2 i=1 i=1 i=1+mk i=1+mk 1. =. mk2 X. 1 # mk1 θˆ(k1 ) + (Yi − Xi ) mk2 i=1+mk 1. 1.

(30) !"#$%&'& ( )*#+%%$ %+&,$-$ ,$*.'%". # 1 k1 !&/(θˆ(k1 ) , θˆ(k2 ) ) = !&/ θˆ(k1 ) , θˆ(k1 ) + k2 mk2 2. 2σ k1 0",(θˆ(k1 ) ) + 0 = k2 mk2 = 0",(θˆ(k2 ) ).. mk2 X. i=1+mk1. (Yi − Xi ). . =. 12 "'%+,2"%$/" "''" *%"%$*%$-" 3$ 4"'3 *$ #&%,+55+,& .%$'$66",+ '" !"#$7 & '" *%"%$*%$-" '&8 ,"##&,%& 3$ /+,&*$9$8'$"26"( 12 ,$:+,$9+2%& "'';+*+9#$& 3+' <"= 2 ¯ ,"8,":& ( (>7 *$ ?" -?+ *&%%& H0 7 Sk (0) = mk(Y¯ − X)/2σ = θˆ(k) Vk (θ) + p Zp,k = θˆ(k) Vk (θ)( 1 ,$*.'%"%$ ,$#&,%"%$ 3$ *+8.$%& *&2& @.$23$ +*"%%$( p <+, .2 8+2+,$-& θ *$ ?" Sk (θ) ∼ N (θVk (θ), Vk (θ)) + Zp,k ∼ N (θ Vk (θ), 1)( A" -&/",$"26" %," '+ /",$+ -&9#&2+2%$ #+, Sk (θ) B #",$ "C. !&/(Sk1 , Sk1+k2 ) = !&/(Sk1 , Sk1 + (Sk1+k2 − Sk1 )). = 0",(Sk1 ). D (EF. 9+2%,+ #+, Zp,k /"'+ '& *%+**& ,$*.'%"%& D (GF( H$ -&2*$3+,$ $2&'%,+ '" *.-= -+**$&2+ 3+''+ *%$9+ 3$ 9"**$9" /+,&*$9$8'$"26" 3+' #","9+%,& 3$ $2%+,+**+ (θˆ(1) , . . . , θˆ(K) )( !&2 3"%$ #,&/+2$+2%$ 3" .2" 3$*%,$5.6$&2+ 2&,9"'+ N (θ, σ 2 )7 -&2 /",$"26" 2&%"7 *$ ?" -?+ %.%%+ '+ *.--+**$&2$ -&2*$3+,"%+ ?"22& 3$*%,$5.= 6$&2+ -&28$.2%" 2&,9"'+ 9.'%$/",$"%" -&2. θˆ(k) ∼ N (θ, Vk (θ)−1 ) + !&/(θˆ(k1 ) , θˆ(k2 ) ) = Vk2 (θ)−1 , p p Zep,k ∼ N (θ Vk (θ), 1) + !&/(Zk1 , Zk2 ) = (Vk1 (θ)/Vk2 (θ)), p p Zp,k ∼ N (θ Vk (θ), 1) + !&/(Zk1 , Zk2 ) = (Vk1 (θ)/Vk2 (θ)), Sk (θ) ∼ N (θVk (θ), Vk (θ)) + !&/(Sk1 , Sk2 ) = Vk1 (θ),. #+, k = 1, . . . , K + 1 ≤ k1 ≤ k2 ≤ K ( I*#,$9+,+ .2 %+*% *+@.+26$"'+ " 8,.##$ -&2 5",,$+,+ ,$:+,$%+ " θˆ(k) 7 & " .2" 3+''+ "'%,+ *%"%$*%$-?+ 2&2 "'%+," '+ *.+ #,&#,$+%J +7 2+' -"*& 3$ ,$*#&*%+ 3$/+,*+ 3" @.+''+ 2&,9"'$ " /",$"26" 2&%"7 3$ *&'$%& *$ *-+8'$+ $2 :"/&,+ 3+' 9+%&3& -&9#.%"6$&2"'9+2%+ 9+2& &2+,&*&(.

(31) !" #$%&''$ ()*&+,& *$,-+'&. ". .) *$/) 01& 0$*2)%&,+,& '& 3+,)& +*+')2) & 4') )*/&,3+'') /,+ %) &22& )* 5+2& +. Vk (θ) $ + mk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ep,k ? 7/)')99+/$ *&' /&2/ E$0$0FA )' /&2/ Sk (0) ? 7/)')99+/$ %+ G1)/&1&+% *&' /&2/ /,)+*4$'+,&! E&, Zp,k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i A i = 1A A! ! ! A )*%)6&*%&*/) & *$,@ -+'-&*/& %)2/,)57)/& 0$* 3+'$,& +//&2$ ')*&+,-&*/& %)6&*%&*/& %+' 3&//$,& %) 6+,+-&/,) β = (β1 , . . . , βp )T ! = %+/) 2$*$ ,+00$'/) &% &2+-)*+/) )* -$@ %$ 2&;7&*9)+'& $4*) 3$'/+ 01& 2) $22&,3+*$ m $22&,3+9)$*) +44)7*/)3& 0$* 7* -+22)-$ %) K +*+')2)! D% $4*) +*+')2) ) %+/) %)26$*)5)') 2$*$ mk ! .)+. X (k) = (X1 , . . . , Xmk )T )' 3&//$,& %&''& $22&,3+9)$*) %)26$*)5)') +''+ k @&2)-+.

(32) !. "#$%&'(' ) *+$,&&% &,'-%.%. #/#(%+%) 0% #++12, .3,. X (k) ∼ N (D(k) β, 4mk σ 2 ), 5'6, D(k) 5,/'&# (# 2#&-%., 5,( 5%+,7/' $,- (, $-%2, mk '++,-6#8%'/% , 4mk 9 (# 2#&-%., 5% %5,/&%&: mk × mk ) ;' +&%2#&'-, 5,% 2%/%2% <1#5-#&% 5% β 9 .#(.'(#=%(, #5 '7/% #/#(%+% , $,- % $-%2% k 7-1$$% 5% '++,-6#8%'/% 9 T T βˆ(k) = (D(k) D(k) )−1 D(k) X (k). ,5 9 /'&' .3, 9 /'-2#(2,/&, 5%+&%=1%&' .'/ 6#('-, #&&,+' $#-% # β , 2#&-%., T 5% .'6#-%#/8# >#-(βˆ(k) ) = (D(k) D(k) )−1 σ 2 ) ;?%/@'-2#8%'/, 5% A%+3,- ik (β) 9 (?%/6,-+# 5% &#(, 2#&-%.,B ik (β) = >#-(βˆ(k) )−1 ) C'%.3D '7/% βˆ(k) 9 .'2=%/#8%'/, (%/,#-, 5% /'-2#(% %/5%$,/5,/&%B 2#-7%/#(2,/&, βˆ(k) 3# 5%+&-%=18%'/, /'-2#(, , %( 6,&&'-, (βˆ(1) , . . . , βˆ(k) ) 9 1/# /'-2#(, 21(&%6#-%#&# 5,((# <1#(, +% .'/'+., (# 5%+&-%=18%'/, 5% βˆ(k) , +% 5,6, .#(.'(#-, (# .'6#-%#/8# &-# % βˆ(k) ) 0% 3# T T .3, >#-(βˆ(k) ) = (D(k) D(k) )−1 σ 2 .'/ (D(k) D(k) ) .3, /'/ 5%$,/5, 5# β ) C,-. k1 < k2 % 5#&% 5,((?#/#/(%+% k1 +'/' 1/ +'&&'%/+%,2, 5% <1,((% 5,((?#/#(%+% k2 B $,-.%E +% $1E +.-%6,-, (k2 ) βˆ(k1 ) = AT X .. F#&' .3, βˆ(k1 ) 9 1/' +&%2#&'-, /'/ 5%+&'-&' $,- β B +% 3# .3,. E(AT X '++%# AT D. (k2 ). (k2 ). ) = AT D. (k2 ). β=β. $,- '7/% β. = 4p ) 0% '&&%,/, <1%/5%. T T (k2 ) "'6(βˆ(k1 ) , βˆ(k2 ) ) = "'6(AT X , (D(k2 ) D(k2 ) )−1 D(k2 ) X (k2 ) ) T. = AT >#-(X (k2 ) )D(k2 ) (D(k2 ) D(k2 ) )−1 T. = (D(k2 ) D(k2 ) )−1 σ 2 = >#-(βˆ(k2 ) ). G%#++12,/5'B (# 5%+&-%=18%'/, 5,((# +1..,++%'/, 5,((, +&%2, 5,% $#-#2,&-% 9.

(33) !" #$%&''$ ()*&+,& *$,-+'&. .. /*+ %)01,)2/3)$*& *$,-+'& -/'1)4+,)+1+ 5$*. βˆ(k) ∼ Np (β, ik (β)−1 ) 1 ≤ k ≤ K,. 6$4(βˆ(k1 ) , βˆ(k2 ) ) = 7+,(βˆ(k2 ) ),. 1 ≤ k1 ≤ k2 , ≤ K.. 8 !9:. ;) 0/<<$*=+ 5>& )' <+,+-&1,$ %) )*1&,&00& )* /* &0<&,)-&*1$ 0)+ +% &0&-<)$. θ = β1 ? 5)$@ )' <,)-$ &'&-&*1$ %&' 4&11$,& β ? & 5>& 0) 4$=')+ 4&,)A5+,& H0 : θ = 0 5$*1,$ /*B+'1&,*+1)4+ 2)'+1&,+'&! C* 1+'& &0<&,)-&*1$? θ ,+<<,&0&*1+ 'B&D&11$ %&' 1,+11+-&*1$? -&*1,& =') +'1,) &'&-&*1) %) β 0$*$ 1,+11+1) 5$-& <+,+-&1,) %) %)01/,2$ ,&'+1)4) + 5$4+,)+1& 5>& )*E/&*3+*$ '+ ,)0<$01+ -+ *$* 0$*$ %) <,)-+,)$ )*1&,&00&! C' 1&01 0&F/&*3)+'& + =,/<<) <&, H0 : θ = 0 0) 2+0+ 0/''+ 0/55&00)$*& %) 01)-& (θˆ(1) , . . . , θˆ(K) )! (+ 4+,)+*3+ %) θˆ(k) +''+ k G&0)-+. ˆ (k) = +*+')0) @ 'B&'&-&*1$ %) <$01$ 8H? H: %&''+ -+1,)5& %) 5$4+,)+*3+ 7+, (β) T. (D(k) D(k) )−1 σ 2 ? &% @ )' ,&5)<,$5$ %&''B)*I$,-+3)$*& <+,3)+'& %) J)0>&, <&, θK Vk (θ) = 7+,(θˆ(k) )−1 = [{(D(k)T D(k) )−1 }11 σ 2 ]−1 ,. k = 1, . . . , K.. (+ 01+1)01)5+ Zep,k <&, 4&,)A5+,& H0 @ F/)*%). p θˆ(k) Zep,k = q = θˆ(k) Vk (θ) k = 1, . . . , K. ˆ (k) 7+,(θ) L+1$ 5>& )' 4&11$,& (βˆ(1) , . . . , βˆ(k) ) @ /*+ *$,-+'& -/'1)4+,)+1+? *& 0&=/& 5>& (θˆ(1) , . . . , θˆ(K) ) @ +*5>B&00$ *$,-+'& -/'1)4+,)+1+ 5$*. E(θˆ(k) ) = θ,. 1 ≤ k ≤ K,. 6$4(θˆ(k1 ) , θˆ(k2 ) ) = 7+,(θˆ(k2 ) ),. 1 ≤ k1 ≤ k2 , ≤ K.. M&,5)N 0) >+. (Zep,1 , . . . , Zep,K ) *$,-+'& -/'1)4+,)+1+? p E(Zep,k ) = θ Vk (θ) 1 ≤ k ≤ K, p 6$4(Zep,k1 , Zep,k2 ) = Vk1 (θ)/Vk2 (θ), 1 ≤ k1 ≤ k2 ≤ K.. 8 !O:.