Modello di change-point per dati esponenziali
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Testo completo
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(13) 1. 2. 3. 2.0. 3. 4. 2. 3. 4. 5. 6. 8 4 0. 0. 1. Sample quantiles. Exp(1) n = 1000. 0. 2. 4. 6. Weibull(2,1) n = 1000. Weibull(0.5,1) n = 1000. 2. 4. 6. 0. 30. Theorical quantiles. Sample quantiles. Theorical quantiles. 0.0. 0. 2. 4. 6. Theorical quantiles. Gamma(4,1) n = 1000. Gamma(0.2,1) n = 1000. 2. 4. 6. Theorical quantiles. 0. 2. Theorical quantiles. Sample quantiles. 0. 2. Exp(1) n = 250. 8. 0. 1. Theorical quantiles. 2.0. 0. 0. Theorical quantiles. 3. 0. Exp(1) n = 50. 0.0. 0.5. 3.0. Sample quantiles. Exp(1) n = 20. 2. Sample quantiles. Sample quantiles. Sample quantiles. Sample quantiles. ! ! "#$%&%'( )#**+(,,-./0 )%,/$%1-/%"0 # )% %.)%2#.)#.3( !. 0. 2. 4. 6. Theorical quantiles. "#$%&' ( ) *+' ,-&#- .# /0/ 1234 5,31&' 3$+# %'.&3 7#-+- #+.#8'4' 2' .#,0 4	%:#3+- .'22' 6%'2- # .'4# ,3+3 ,'4# $-+-&'4#; +-2 8',3 .-22' <-#9%22 - .-22' ='>>' 7#-+- #+.#8'43 4&' 1'&-+4-,# 1&#>' #2 1'&'>-4&3 .# ?3&>' - 13# 6%-223 .# ,8'2'@.
(14) !"#$%&% '( !))*+$# ,- *+! ".#/! )%&*0#%+,. !. !"#$!%&'!()! *$(%+%!,!"#!-./ */$ #/0*! 1! +##/"+. "#$%&' ()* +' ,('( -'..' $/0/&'1#22'3' /4 '15%0# 5'6# 073/,71# 8706#4/&#'.7 7&' %0' 5/&3' 5'3/$7&#' 4# ,('( 59/ 6# :766707 #05703&'&/ ;%'0< 47 6# 9' ' 59/ ='&/ 570 4'3# 59/ &'::&/6/03'07 3/.:# 4# '33/6'( +' -'..' $/0/&'1#22'3' > %0' ,('( 5703#0%' 1' 5%# =%02#70/ 4# 4/06#3? > 5'&'33/'3' 4' 3&/ :'&'./3&#* β x kβ−1 −( x )β e θ f (x) = Γ(k)θ θ. θ > 0 > #1 :'&'./3&7 4# 65'1'@ ./03&/ k > 0 / β > 0 6707 # :'&'./3&# 4# =7&.' A#1 6%::7&37 4/11' ,('( > x > 0B( 8706#4/&'047 4/11/ :'&3#571'&# &#:'&'./3'2#70#@ 59/ ;%# 070 6:/5#C59#'.7 @ / :'&3#571'&# ,'17&# :/& ;%/63/ 6# '4/ #0 %07 4/# 3&/ 673373#06#/.# A#1 5'67 +7$07&.'1/ > %0 5'67 1#.#3/ :/& k → ∞B 59/ 6# ,/4707 0/11' "#$%&' ()( D713# '13&# :73&/EE/&7 /66/&/ # 67337#06#/.# 4' 5706#4/&'&/@ .' ;%/63# 6707 6#5%&'./03/ # :#F 4#G%6# 0/11/ '::1#5'2#70# / 59/ 6:/66/ ,713/ 6# H570=704707I 570 #1 .74/117 /6:70/02#'1/( J'# $&'C5# #0 "#$%&' (K :766#'.7 #03%#&/ 57./@ '059/ 570 %0 5'.:#70/ 4# 070 .74/63' 0%./&76#3?@ $1# 63&%./03# 4#'$0763#5# 5# :766'07 :7&3'&/ '11L'55/33'2#70/ 4/11L'66%037 4#63&#E%3#,7 /6:7< 0/02#'1/ ;%'047 6#'.7 ,#5#0# '11L#03/&6/2#70/ 3&' $1# #06#/.# -'..' / M/#E%11 / 5# 3&7,#'.7 #0 &/'13? #0 %07 4/# 4%/ 7::%&/ '057&' 'EE#'.7 %0' +7$07&.'1/ 570 %0 :'&'./3&7 4# =7&.' .#07&/ 4# /4 %0 :'&'./3&7 4# 65'1' 3'1# 59/ 1' 67.#$1#'02' 4#63&#E%3#,' 570 1L/6:70/02#'1/ > =7&3/( N766#'.7 4#&/ 59/ 1' :&7E'< E#1#3? 4# 57../33/&/ %0 /&&7&/ 4/1 6/57047 3#:7 #0 ;%/63# 5'6# > .7137 '13'@ .' #0 &/'13? 1L'::&766#.'2#70/ '11L/6:70/02#'1/ > 6/06'3' /4 '059/ /6/.:1#C5'3/ 0/11L#03/&:&/3'2#70/ 4/1 =/07./07(. 23!)1!*/)1/)'+ +L#04#:/04/02' 4/# 3/.:# 4# '33/6' 6# 9' 6/ #1 :&75/667 4# N7#6670 6733763'03/ $74/ '059L/667 4/11' :&7:&#/3? 4# #04#:/04/02'( N766#'.7 %3#1#22'&/ 63&%./03# $&'C5# 57./ #1 57&&/17$&'..' /4 # 1'$<:173@ 3/63 57./ ;%/117 4# +O%0$<P7Q :/& 1' ,/&#C5' 4/11L'66%037 4# #057&&/1'2#70/@ '059/ 6/ 570 3'1# 63&%./03# #0 &/'13? ,/&#C59#'.7 6/ # 4'3# 6707 #057&&/1'3# A/ 57./ 6'::#'.7 1L#04#:/04/02' #.:1#5' 1L#057&&/1'2#70/ .' 070 ,#5/,/&6'B( R6#63707 4#,/&6# ./374# :/& 1' ,/&#C5' 4/1< 1L#04#:/04/02' 6375'63#5' A766#' 1L#04#:/04/02' :#F =7&3/ / $/0/&'1/ 59/ :766#'.7 5705/:#&/B 59/ ;%# 3&'1'65#'.7 &#.'04'047 '11L/6/.:#7 4/11L%13#.7 5':#3717 #0 5%# ,#/0/ %3#1#22'37 %0 3/63 χ2 ( !" #$%%&'"& (!))$%*& +,**$ -$##$ %!.!"$*&//$)$ +& 0!($ &* +&)' 1!2 ! "#$%&&!'() ' &* *&2"' (& )!+)' *+,+#',& -(."&/ ,0. -"+1(./ 2(3 4#2"+#)" 5,+,3 4$1*!++ 56.( !(78.
(15) 3. 4. !. 0. 1. 2. Exp( 1/3 ) Weibull( 0.7 , 1/3 ) Gamma( 0.7 , 1/3 ) Lognorm( 1/3 , 3 ). 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. 0.0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. 0.4. 0.8. 0.0. 0.0. Cumulative density function. Probability density function. !"! #$ %$&'$ ()*+$&' ,*& -$'. *(,/0*01.$#.. "#$%&' ()* +,-%.# -'/# #. -%# ,0 1#/2%4#5.# 6'..5 %. '.1'70.25 89%'/#: 0/;5.0.4#',0(. !" #$ %$&'$ ()*+$&' ,*& -$'. */,01*12.$3. <'2' ,' 1#=-5,2> .0,,?';;,#-'4#5.0 10,,' -'&2' 1# -5.2&5,,5 @60A'&2 ' 1'2# 0/;5B .0.4#',# 15C%2' ',,' D5&20 '/#7702&#' 1# 2',0 1#/2%4#5.0E F /2'2' ;&5;5/2' 1' G0,/5. %.' /07;,#-0 2&'/D5&7'4#5.0 1# ;520.4' -60 -# D' ;'//'&0 1',,' 1#/B 2%4#5.0 0/;5.0.4#',0 ',,' 1#/2%4#5.0 1# H0#3%,,( @0 X ∼ Exp(λ) 01 5;0&#B 1 '75 ,' 2&'/D5&7'4#5.0 X 3.6 E /# 6' -60* 3.6. 1. P [X 3.6 ≤ x] = P [X ≤ x3.6 ] = 1 − e−λx. I' D%.4#5.0 1# 10./#2> F 1%.9%0 J#, /%;;5&25 F 1'25 1' x > 0K*. f (x). = =. 3.6. 3.6λx3.6−1 e−λx 3.6−1 3.6 x 1. λ− 3.6. 1. λ− 3.6. −. e. x − 1 λ 3.6. 3.6.
(16) !"#$%&% '( !))*+$# ,- *+! ".#/! )%&*0#%+,. !. 1. "#$%& '( &)&(*' +$(,'-.%.%./$ 0& %'1&%&/.*%'22&2'$(. δ = λ− 3.6 .- '( -.3('*')&4 1. X 3.6 ∼ W eibull(3.6, δ) "$). 56! 7 '0 1&%&/.*%$ -' 8$%/& . δ 7 '0 1&%&/.*%$ -' ,+&0&6 90 /$*')$ -' :;.,*& *%&,8$%/&2'$(. 7 -&*$ -&0 8&**$ +<. ;(& -',*%'=;2'$(. -' >.'=;00 +$( 1&%&/.*%$ -' 8$%/& 56! 7 /$0*$ ,'/'0. &00& -',*%'=;2'$(. ($%/&0.6 ?.% ,'/'0. ,' '(*.(-. '0 8&**$ +<. 0. -;. -',*%'=;2'$(' <&(($ 0#'(-'+. -' &,'//.*%'& . -' +;%*$,' ,*&(-&%-'22&*' +<. -'@.%',+$($ -' 1$+$6 9(8&**' -&*$ '0 1&%&/.*%$ -' 8$%/& 1&%' & 56! . -' ,+&0& 1&%' & δ 0& /.-'& . 0& -.)'&2'$(. ,*&(-&%- -.00& >.'=;00 ,$($ %',1.**')&/.(*.4. 1 µ = δΓ 1 + 3.6 s 2 2 σ = − µ2 δ Γ 1+ 3.6 . 9(3(. 0#'(-'+. ,*&(-&%-'22&*$ -' &,'//.*%'& . -' +;%*$,' %',;0*&($ .,,.%.4. γ1 γ2. = =. ! δ3 Γ 1 +. ! δ4 Γ 1 +. 3 3.6. . − 3µσ 2 − µ3. σ3 4 3.6. . − 4γ1 µσ 3 − 6µ2 σ 2 − µ4 σ4. A@.*;&(-$ &0+;(' ,./10'+' 1&,,&BB' ,' 1;C 8&% ).-.%. +<. ,'& γ1 +<. γ2 ($( -'1.(-$($ -& δ 6 D' <& +$,E +<. 0#'(-'+. -' &,'//.*%'& 7 +'%+& 5.6 · 10−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
(17) 1.5. !. 0.0. 0.5. 1.0. Weibull(3.6,1) Normale(0.9,0.28). 0.5. 1.0. 1.5. 2.0. 0.0. 0.5. 1.0. 1.5. 2.0. 0.4. 0.8. 0.0. 0.0. Cumulative density function. Probability density function. !"! #$ %$&'$ ()*+$&' ,*& -$'. *(,/0*01.$#.. "#$%&' ()* +' ,#-.&#/%0#123 ,# 43#/%55 612 7'&'83.&1 ,# 91&8' :(; 3 ,# -6'5' ' 6129&12.1 612 %2' <1&8'53 6=3 =' 73& 83,#' 3 ,3>#'0#123 -.'2,'&, 53 -.3--3 ,355' -%,,3..' 43#/%55(.
(18) !"#$%&% '( !))*+$# ,- *+! ".#/! )%&*0#%+,. 50. 60. !. 30. *. 20. ARL. 40. *. 0. 10. Limiti approssimati Limiti esatti. 0.0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. r. "#$%&' ()* + ,&-./# 01//2345 ,1& %6' 7'&8' 0# 7-68&-//- 9:1;'&8 ,1& /' <10#' 01/ ,&-71==-> ,1& /#<#8# 1='88# =# #68160-6- # /#<#8# 7'/7-/'8# ' ,'&8#&1 0'# ?%'68#/# 01//' @1#A%//B ,1& /#<#8# ',,&-==#<'8# =# 7-6=#01&' #6C171 /2',,&-==#<'D#-61 '//' E-&<'/1> /' ,&-A'A#/#8F 0# 7-<<1881&1 %6 1&&-&1 01/ ,&#<- 8#,- G .=='8' '0 α = 0.025( 4#7-&0#'<- #6.61 7:1 r = H<10#' #6 7-68&-//-IJH<10#' K%-&# 7-68&-//-I(.
(19) !"! #$ %$&'$ ()*+$&' ,*& -$'. *(,/0*01.$#.. L.inf. Media in controllo. !. L.sup. "#$%&' ()* +&, -,#.%// ' 0123&1241* #2 2,&1 #/ 0'51 #2 0124&1//1 , /, '/4&, 6%, 5121 0'5# 6# 6#7#2%8#12, 6,//' 7,6#' 6,/ 9&10,551 #2 0124&1//1( 155,&:'8#12# ;4&'531&7'4,< ,24&1 # /#7#4# 6# 0124&1//1 = #23'44# '6 %2' 6#7#2%8#12, 6,//' 7,6#' 6,/ 9&10,551 01&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
(20) !"#$%&% '( !))*+$# ,- *+! ".#/! )%&*0#%+,. !. " #$%$ &'%(&)*& )&++,()-(.(-/01& #&22($103&)*( )&+ 4(4*&30 #1$-/**(.$ &- 50 ().&%& 6/$)& #1&4*07($)( )&++,()-(.(-/01& 3(2+($103&)*( -&+ 4(4*&30 #1$-/**(.$8. !" #$ %$&'$ ()*)+ ,-& .$'/ -0,12-23/$4/ 5%-26 2/7 9)*1$-/%(03$ 61&.&3&)*& +0 %01*0 :;<;= 6(+0*&10+& >() ?$130 0+2$1(*3(%0@ #&1 -0*( &4#$)&)7(0+(A 066(03$ %5& (+ .0+$1& -&++0 4*0*(4*(%0B %5& 1(+&.0 +& -(3()/7($)( )&++0 3&-(0B )&++,(4*0)*& (C&4(3$ " -0*$ -0D. Ti = min (0, Ti−1 + xi − kT ) =&)*1& E/&++$ -&++0 4*0*(4*(%0 %5& 1(+&.0 2+( 0/3&)*( " -0*$ -0D. Si = max (0, Si−1 + xi − kS ) F$.& kT & kS 4$)$ -&( .0+$1( %5& -&1(.0)$ -0+ <GHI > !"#!$%&'( )*+,',&(&%.'%&+ /!0%@B & 4$)$ -0*( -0 >µ1 " +0 3&-(0 () %$)*1$++$ & µ2 " +0 3&-(0 ?/$1( %$)*1$++$@D µ1 µ2 µ2 kJ = log µ2 − µ1 µ1 <& J = T 0++$10 µ1 > µ2 B 4& J = S 0++$10 µ1 < µ2 8 J(&)& +0)%(0*$ /) 0++013& $2)( E/0+.$+*0D Si > hS ∨ Ti < −hT F$.& hS &- hT 4$)$ ( +(3(*( -( %$)*1$++$ %5& .&)2$)$ %0+%$+0*( /)0 .$+*0 K440*( +,LHM () %$)*1$++$ >%5(03&13$ H(0) +,LHM () %$)*1$++$ %5& 4( 50 )&++0 1(+&.07($)& -( 0/3&)*( )&++0 3&-(0 () %$)*1$++$ &- L(0) E/&++$ )&++& -(3()/7($)(@ & (+ .0+$1& kJ #&1 J = S, T 8 M,LHM 2+$60+& " -0*$ -0++0 4&2/&)*& 0##1$44(307($)& >%5& 4$**$ #01*(%$+01( %$)-(7($)( 1(4/+*0 &44&1& &40**0@D. . ARL0 =. H(0)L(0) H(0) + L(0). G&1 0##1$?$)-(3&)*( & %5(01(3&)*( 4/++0 :;<;= #&1 -0*( &4#$)&)7(0+( 4( 1(30)C -0 0 N0) >OPPQ@8.
(21) !"#$%&% ' ! "#$%!!# $& '()*+%,-#&*. ! "!#$%&&#&%'(% )!% *!+,% )% #**!-#. )#*' /( ,/(*' )% -0'$*# τ !) /( 1#+,%'(! )% #+,%!&&# 2--# ,#"% #) (. ,'--'(' !--!"! +')!$$#*! 1'+! -!3/!4 ( Yi ∼ Exp(µ1 ) i = 1, . . . , τ − 1 Yi ∼ Exp(µ2 ) i = τ, . . . , n µ1 > 0,. µ2 > 0,. µ1 6= µ2 ,. τ ∈ {2, . . . , n + 1}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τ ,/B 0#"%#"! C(!$$;%(-%!+! )!% 0#$'"% %(*!"%D )# E 2(' #) ( -! %$ 1#+:%#+!(*' 8 '11'"-' ! -! ('( #::%#+' (!--/( 1#+:%#+!(*' #$$'"# τ = n + 1 16! 8 !=/%0#$!(*! #$$# -1"%**/"# τ > n ,'%168 -%#+' %( A#-! 5< @! -%#+' %( A#-! 55 %(0!1! ! #--/+%#+' 16! %$ ,"'1!--' 8 %( 1'(*"'$$' C'--%# -%#+' -'**' H0 D $;%(-%!+! )% )!2(%&%'(! )% τ ('( 8 ,%F $%+%*#*' ,!"168 ( ('( 8 ,%F 1'-*#(*!< 5$ +')!$$' )% 16#(3!7,'%(*. 1'+! 0!)"!+' #0#(*%. 8 %+,%!3#*' '$*"! 16! ,!" $# 1'-*"/&%'(! )!$$# -*#*%-*%1# *!-* #(16! 1'+! -*"/+!(*' ,!" $! -*%+! )% µ1 , µ2 ! τ< 5('$*"! -! µ1 > µ2 #$$'"# #0"!+' /( ,!33%'"#+!(*' )!$ -%-*!+# ,"')/**%0' ,'%168 %$ *!+,' +!)%' )% #**!-# *"# )%9!**% 1'(-!1/*%0% 8 )%0!(*#*' ,%F ,%11'$' +!(*"! -! µ1 < µ2 #0"!+' /( +%3$%'"#+!(*' )!$ -%-*!+# ,"')/**%0'< EG.
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(23) !"! #$%&' ()$*+%,'- *$((.*' /$, 0+%+1$#%, 2 *'#'. !. "# ∧. =. Tmax,n. log. ∧. ∧. LH1 (µ1 , µ2 , τ ) ∧. LH0 (µ1 ) ∧. ∧. ∧. ∧. = lH1 (µ1 , µ2 , τ ) − lH0 (µ1 ) $% &'()*++( " ,- )(-.'(%%( /01-2( +,13( +(..( H0 4 1%.',3*-., +(..( H1 5 6(3* +, -(.1 /01-2( +,13( +(..( H0 %70-,)( &1'13*.'( 21 +.,31'* " µ1 4 &(,)8" %1 %(9:;*'(+,3,9%,1-<1 ,- /0*+.( )1+( 2,&*-2* +(%( 21 µ1 5 =, 81 ,-(%.'* )8* %* +.,3* 2, 31++,31 ;*'+(3,9%,1-<1 2*, &1'13*.', >1 +*)(-21 2*%%7,&(.*+, +(..( %1 /01%* ), .'(;,13(? 21.( 0- )13&,(-* )1+01%* +*3&%,)* y = ( y1 y2 . . . yn ) +(-( %* +*90*-.,# ∧. µ1H. 1. = y1. > 5@?. 1. = y2. > 5 ?. 0. = y. > 5!?. ∧. µ2H ∧. µ1H A(;*#. n. y=. 1X yi , n i=1. t−1. y1 =. 1 X yi , t − 1 i=1. n. y2 =. X 1 yi n − t + 1 i=t. ∧. . -(- " 1%.'( )8* τ %1 +.,31 2, 31++,31 ;*'(+,3,9%,1-<1 2, τ 4 ( 3*9%,( # ∧. ∧. ∧. ∧. lH1 (µ1 , µ2 , t) = max lH1 (µ1j , µ2j , j) 2≤j≤n. > 5B?. A1%%* > 5@?4 > 5 ? * > 5!? +, 81 )8*# ∧. ∧. ∧. lH1 (µ1 , µ2 , τ ) ∧. lH0 (µ1 ). = −(t − 1) log y 1 − (n − t + 1) log y 2 − n = −n log y − n. > 5C? > 5D?. =, -(., ,% ;,-)(%( %,-*1'*# ∧. ∧. (t − 1) µ1H1 + (n − t + 1) µ2H1 µ1H = 0 n ∧. > 5E?. ! "#$%&'()' )*++* ,-./0 *("1' !+ $#2!3# )'++* (#2*4!#(' )'++* 52*2!52!"* 2'52 Tmax,n 6 !(7*22!8 Tmax,n = max Tj,n 2≤j≤n. ∧. ∧. ∧. "#( Tj,n = lH1 (µ1j , µ2j , j) − lH0 (µ1 )..
(24) !"#$%&% '( #& )%*+&&% *# ,!-.+/"%#-$. !. "#$$% & '()* & '+) % & ',) -. /00.%1%2. Tmax,n. = −(t − 1) log y 1 − (n − t + 1) log y 2 + n log y − t−1 n−t+1 n n y1 y1 + (n − t + 1) y (t − 1) y2 2 & '3) = n log n. 2.0 1.5 1.0 0.0. 0.5. Valore atteso della statistica T (appr.). 2.5. 3.0. 4% 5%61.#7/ r = y 1 8y 2 #$$/9# $# -0#0.-0.:# Tmax,n 7.-;9# .1 ;1 :%90/ -%1-/ $# <5.-0#1=#> 5. r 5# ?@ -% .1A#00. .$ 9%#$% 9#BB/90/ 5%$$% 7%5.% C ? &D;#15/ -.#7/ -/00/ H0 ) :. #-B%00.#7/ :E% $# -0#0.-0.:# 0%-0 Tmax,n #--;7# F#$/9. F.:.1. #$$/ G'. 0. 1. 2. 3. 4. 5. 6. Rapporto tra le medie. H.I;9# '?2 J9#6:/ 5. :/7% #BB9/--.7#0.F#7%10% $# -0#0.-0.:# 0%-0 Tmax,n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max,n 0+%# "/( 1(& n → ∞ $#$ *, /+ *#%%# '<,1#%(*, $)''+ '+ 0,*%&,@)9,#$( +*,$%#%,"+ χ21 8)$ *#'# 6>0>'> 1#,"/2 ;+&(44# ,$;(&($9+ *#'# *) τ :A ,$ B+*( C 1#,"/2 '# *%+%# ,$ "#$%&#''# -,($( ,1#%,99+%# *#'+4($%( $(' "+41,#$( + 0,*1#*,9,#$( (0 ,$ B+*( CC (**($0# '<+$+',*, *(3)($9,+'( 8+-&(4# + "/( ;+&( "#$ 0,*%&,@)9,#$, "#$0,9,#$+%( 0, Tmax,n "#4( -(0&(4# +-+$%,:>. !"!#. $ %&'&(& )& *+,(-+%%+. D(@@($(7 "#4( "/,+&,%# 1&,4+7 0#@@,+4# %($(& "#$%# 0(''( 1(")',+&,%E 0(''+ "+&5 %+ 0, "#$%&#''# FGH 1#**,+4# *;&)%%+&( "#4)$3)( 6', *%&)4($%, %(#&,", 0(''<,$5 ;(&($9+ "'+**,"+ 1(& 4(6',# "+&+%%(&,99+&'+> I(& 1#%(& #%%($(&( , $#*%&, ',4,%, 0, "#$%&#''# 0#@@,+4# 1&,4+ .**+&( '+ 1&#@+@,',%E 0(''<(&&#&( 0(' 1&,4# %,1# α> C$ B+*( C 8"#$ ,' "+41,#$( 0, +41,(99+ .**+%+ +0 $: ,$;+%%, 3)(*%+ 2 0+%+ *(41',"(4($%( 0+A PH0 [Tmax,n > hn,α ] = α 0#-( hn,α 2 ,' $#*%&# ',4,%( "#$ ,' 3)+'( *%+@,',+4# ,$ @+*( +' -+'#&( #**(&-+%# 0(''+ *%+%,*%,"+ Tmax,n *( '<,$*,(4( 0, 0+%, 0, "), 0,*1#$,+4# 1&#-,($( 0+ )$ 1&#"(**# ,$ "#$%&#''# ( $(' 3)+' "+*# *%,4,+4# ,' -+'#&( 0, µ1 > J(''+ B+*( CC '( "#*( "+4@,+$#> I#%&(44# ,$,9,+&( +0 (K(%%)+&( ,' %(*% 0+''+ %(&9+ #**(&-+9,#$( 0, "), 0,*1#$,+4#7 4+ 3)(*%# 2 ,$ &(+'%E 4#'%# +99+&0+%# 1#,"/2 "#$ *#'( %&( #**(&-+9,#$, $#$ *,+4# +K+%%# ,$ 6&+0# 0, 1#%(& -(&,."+&( *( , 0+%, 1&#-($6#$# 0+ )$+ 0,*%&,@)9,#$( (*1#$($9,+'( ( '+ '#&# ,$0,1($0($9+> I+&( 1(&%+$%# 1,L #11#&%)$# +"")4)'+&( )$ "(&%# $)4(&# 0, #**(&-+9,#$, 1&,4+ 0, 0+&( ,' -,+ +''<+$+',*, *(3)($9,+'( 0(, 0+%,7 (0 M+NO,$* 1#$( "#4( $)4(&#*,%E 4,$,4+ 0, 1+&%($9+ $ 1+&, + PQ> C$ B+*( CC '+ 1&#@+@',%E 0(''<(&&#&( 0(' 1&,4# %,1# 2 0+%+ 0+A. PH0 [Tmax,n > hn,α | Tmax,j ≤ hj,α , j < n] = α C$ 1+&#'( α 2 R'+ 1&#@+@,',%E 0, '+$",+&( )$ ;+'*# +''+&4( 0+%# "/( 1&,4+ $#$ 2 *%+%# 4+, '+$",+%#S7 #**,+ 0#@@,+4# "#$0,9,#$+&", +' ;+%%# "/( ,$ %)%%, , 1&("(5 0($%, %(*% $#$ *%+%, "#44(**, (&&#&, 0(' 1&,4# %,1#> D, $#%+ 0)$3)( "/( '+ 0,*%&,@)9,#$( 0(''+ &)$ '($6%/ *( ,' 1&#"(**# 2 ,$ "#$%&#''# 2 6(#4(%&,"+7 ,$;+%%, "#4( $(''+ D/(N+&% ARL0 = 1Tα U (3),-+'($%(4($%( 1#**,+4# 0,&( "/( '+ ;)$5 9,#$( 0, &,*"/,# 0(''+ &)$ '($6%/ 2 "#*%+$%( ( 1+&, +0 α > V@@,+4# #%%($)%# -,+ W#$%( X+&'# 8*, -(0+ '+ Y+@(''+ >P: 0(''( *%,4( 1&(5 ',4,$+&, 0(, ',4,%, 0, "#$%&#''#> M+NO,$* $(''# *%,4+&( , ',4,%, 0, "#$%&#''# $(' "+*# $#&4+'( "#$ "+4@,+4($%, $(''+ 4(0,+ 0(' 1&#"(**# 1+&%( 0+ PZ 4,',#$, 0,. ! "#!# $%&'%()*+! $&,% ,%--! .)&.)*%(/ ,* '!0+!01! ,* '%'&)*! %, 2! 30! 4301*&0% ,* )*5+2*& +&5(!0(%# 604!((*7 PH0 [RL = n|RL ≥ n] = α.
(26) !"#$%&% '( #& )%*+&&% *# ,!-.+/"%#-$. !. α " &# && & &' &) &% &! &* &( &+ # & ' ) % ! * ( + '# '& ' '' ') '% '! '* '( '+ )# )& ) )' )) )% )! )* )( )+ %#. #$#% '$(&! '$''+ '$&' '$# % $+! $+ ! $+#& $(+! $((' $(*% $(!) $(!' $(%( $(!& $(!% $(! $(*' $(%! $(%* $(!' $(%* $(% $()) $(%+ $(!* $(! $(%) $(*) $(!) $(%( $(!' $(%% $(%! $(!& $(*+ $(%! $(* $(*& $((% $(!* $(('. #$# % )$%!* )$#*+ '$((% '$*+* '$*)* '$* ( '$!+( '$!+ '$!+) '$!!) '$!!( '$!!+ '$!* '$!* '$!!% '$!* '$!%! '$!* '$!!! '$!%% '$!*% '$!*( '$!%% '$!** '$!* '$!( '$!*! '$!*& '$!!( '$!+& '$!*' '$!!* '$!+' '$!(' '$!+% '$!* '$*#( '$*#% '$!(+ '$!( '$!!+. α #$##% !$&+( %$**& %$%() %$%'! %$%#+ %$%#( %$)+) %$%#% %$)(% %$%#' %$)( %$%# %$%&* %$%#% %$%#! %$% ) %$% ( %$% %$%#& %$%#% %$% & %$%&+ %$% %$%)' %$% ( %$%% %$% %$%!' %$%)& %$%'! %$%% %$%'* %$%) %$%' %$%%% %$%'( %$%'& %$%)! %$%&& %$%)% %$%!+. #$##& *$()! *$)!& *$'#& *$' & *$ & *$ &* *$ '% *$ ) *$ #( *$&(! *$ )! *$ '' *$ ' *$ !* *$ ! *$ !+ *$ )( *$ '! *$ )! *$ *) *$'%& *$ (% *$ &% *$ *! *$'# *$ (& *$ + *$ %+ *$'#* *$ ++ *$' ' *$'%+ *$ +! *$ +) *$ (* *$ (+ *$''( *$'** *$') *$'#( *$'*(. " %& % %' %) %% %! %* %( %+ !# !& ! !' !) !% !! !* !( !+ *# *& * *' *) *% *! ** *( *+ (# (& ( (' () (% (! (* (( (+ +#. #$#% $(*' $(*& $(%! $(!* $((& $((% $(%% $((& $((' $((' $(*( $(!' $(! $(+% $(!( $(+& $+&* $(*& $+#! $(( $+&( $(*) $(( $(%+ $(!+ $+# $(+ $(%+ $(' $(%) $+#& $(() $((! $(') $('' $+ * $(%% $(' $((+ $+ &. #$# % '$!(+ '$!(& '$*#' '$*& '$* '$!** '$!!+ '$!+& '$!!+ '$* '$!(' '$* '$!(! '$!(! '$*#' '$!!+ '$!+' '$*#! '$!!% '$!(% '$!+( '$!(* '$!( '$!(! '$*#+ '$*#% '$*#( '$*&! '$*#+ '$* ! '$*#! '$!+' '$* '$*#( '$!+& '$!!( '$!(+ '$!*& '$*& '$*&'. #$##% %$%%' %$%!& %$%) %$%') %$%) %$%%& %$%*! %$%! %$%!+ %$%%% %$%'( %$%! %$%+ %$%%% %$%* %$%*% %$%!& %$%!( %$%!) %$%'( %$%% %$%*& %$%%' %$%!( %$%( %$%* %$%+& %$%() %$!#' %$%+! %$%!% %$%*( %$%*) %$%** %$%%+ %$%*( %$%%( %$%*+ %$%*) %$%%). #$##& *$)#% *$'(& *$' % *$'') *$' *$''( *$'( *$'%' *$'*( *$' * *$') *$''+ *$'&' *$'!! *$) *$'%& *$'(' *$'*) *$'+( *$')% *$)#% *$'(! *$'+* *$) & *$'*! *$)#+ *$)) *$'+& *$'!! *$'*) *$')* *$'!& *$'( *$''' *$''( *$' + *$'%! *$')+ *$'' *$). ,-./00- $&1 2345/ 64- 74580-94:"/ ;/4 045434 ;4 <:"3=:00: ;-3: 8" 046/00: ;4 <:">;/"9- α / "85/=:743? "$.
(27) !"! #$%&' ()$*+%,'- *$((.*' /$, 0+%+1$#%, 2 *'#' " #$ ## # #( #% #' #* #! #+ #) $ # ( % ' * ! + ) ($ (# ( (( (% (' (* (! (+ () %$ %# % %( %% %' %* %! %+ %) '$ '# ' '( '% '' '* '! '+ '). hn,0.025 %&''( %&$+ (&+++ (&!)# (&!% (&!## (&*)' (&*+% (&*!+ (&*!% (&*! (&*!# (&*!# (&*!# (&*!# (&*!# (&*! (&*!( (&*!( (&*!( (&*!' (&*!* (&*!* (&*!! (&*!+ (&*+ (&*!) (&*+ (&*+ (&*+ (&*+ (&*+ (&*+( (&*+( (&*+( (&*+% (&*+! (&*+* (&*+* (&*+' (&*+* (&*+* (&*++ (&*+* (&*+! (&*+! (&*) (&*) (&*) (&*). eperc. $&$) $&$*) $&$* $&$') $&$* $&$*' $&$*' $&$*+ $&$! $&$*+ $&$*' $&$*+ $&$*+ $&$*% $&$! $&$!' $&$!! $&$+ $&$+ $&$) $&$!( $&$!! $&$) $&$+) $&$+( $&$+) $&$)( $&$) $&$)' $&$)! $&$)! $&$) $&#$( $&# $&#$) $&$)( $&$+! $&$)! $&# $&$)) $&#$ $&##% $&#$! $&# # $&##' $&### $&# % $&##! $&# $&# +. " *$ *# * *( *% *' ** *! *+ *) !$ !# ! !( !% !' !* !! !+ !) +$ +# + +( +% +' +* +! ++ +) )$ )# ) )( )% )' )* )! )+ )) #$$ #$# #$ #$( #$% #$' #$* #$! #$+ #$). hn,0.025 (&*) (&*) (&*)# (&*) (&*)# (&*)# (&*) (&*) (&*)( (&*) (&*)# (&*) (&*)# (&*) (&*)# (&*)( (&*)( (&*) (&*) (&*)% (&*)# (&*)* (&*) (&*)' (&*)( (&*) (&*) (&*)% (&*)' (&*)' (&*)+ (&*)% (&*)% (&*)* (&*)+ (&*)+ (&*)! (&*)% (&*)! (&*)' (&*)* (&*)# (&*)* (&*)' (&*)+ (&*)! (&*)% (&*)! (&*)) (&*)+. eperc. $&# % $&##! $&#(' $&# ' $&#(+ $&#(+ $&#( $&#(( $&# * $&#%( $&#%+ $&#( $&#%% $&#*' $&#'( $&#* $&#'# $&#!! $&#%% $&#%+ $&#%% $&#!* $&#'* $&#'' $&#'+ $&#' $&#!! $&#+( $&#*) $&#*) $&#*( $& $# $&#+! $&#+' $&#+ $& $! $&#+ $&#) $& $) $& #) $&#)( $&#+) $& $( $&#)% $& $! $& #' $& ) $& %' $& $' $& #'. " ##$ ### ## ##( ##% ##' ##* ##! ##+ ##) # $ # # # # ( # % # ' # * # ! # + # ) #($ #(# #( #(( #(% #(' #(* #(! #(+ #() #%$ #%# #% #%( #%% #%' #%* #%! #%+ #%) #'$ #'# #' #'( #'% #'' #'* #'! #'+ #'). hn,0.025 (&*)* (&*)+ (&*)+ (&*)* (&*)% (&*)# (&*)% (&*)+ (&*)* (&*)' (&*)+ (&*)% (&*)' (&*)* (&*)! (&*)% (&!$# (&*)( (&*)+ (&*)* (&*)* (&*)+ (&! (&*)! (&*)+ (&!$# (&! (&*)+ (&*)' (&*)+ (&*)' (&*)( (&*)' (&!$# (&! (&*)+ (&*)* (&!$( (&!$ (&*)! (&!$( (&*)* (&!$# (&!$* (&*)' (&! (&! (&!$' (&*)( (&*)). eperc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α = 0.025@ 499/"A94 <-00- ;/<6- =-;864"-36- <6 #$$ :96;/@ / 3/0-9674 /3343/ 8/3=/"9A-0/&. hn,0.025 (&!$+ (&!$! (&!$ (&! (&!$# (&!$( (&!$' (&*)+ (&!$# (&*+) (&*)( (&!$% (&*)' (&!$# (&*)! (&*)# (&*) (&*)) (&*++ (&! (&!$# (&*)* (&*)' (&*) (&*)) (&!## (&!#( (&!$! (&! (&!$ (&! (&!$# (&!$) (&!$! (&!$) (&!#* (&!$+ (&*)! (&! (&!$ (&*)+. eperc. $&%%# $&%'( $&%$( $&%%# $&%!# $&%** $&('+ $&%') $&%+ $&%!) $&%*% $&%+ $&%*+ $&' ( $&%)* $&%%) $&'(( $&' ( $&'%' $&'' $&'(* $&%)* $&''( $&'%+ $&''% $&'+' $&*!+ $&')! $&'' $&')% $&') $&*%! $&')( $&*+' $&**# $&**) $&**( $&**) $&*'' $&!( $&!*(.
(28) !"#$%&% '( #& )%*+&&% *# ,!-.+/"%#-$. 4.2 4.0 3.8. Limiti di controllo. 4.4. !. 0. 50. 100. 150. n. "#$%&' ( ) *&'+,- ./# 0#1#2# .# ,-32&-00- -&2'2# #3 5'6/00' ( 7/&8- 3(. -88/&7'9#-3#: 0/ 3-82&/ 82#1/ #37/,/ 8-3- -22/3%2/ 4'&2/3.- .' ;<< 1#0' -88/&= 7'9#-3#( ># 8#'1- ,-8? ,-3,/32&'2# 8% %3 4'&2#,-0'&/ 0#7/00- .# ,-3+./39': -88#' α = 0.025: / 4'&2/3.- 8/14&/ .' ;<< 1#0' -88/&7'9#-3/ '66#'1- /2%2- 0@/8/= ,%9#-3/ ./00@'0$-- 4/& A<< 7-02/ B8# 7/.' 0' 5'6/00' ( C 84#3$/3.-,# +3- '. n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α 4-#,HF 4#K F $&'3./ 4#K 0' 7'&#'6#0#2J 8'&J 1'$$#-&/ ' 4'J .# 3CL 4/& n = 200 '66#'1- #3E'22# ,#&,' 500000 0.975200 ≈ 3160 -88/&7'9#-3# 4/& 82#1'&/ #0 0#1#2/ .# ,-32&-00-: .# E'22- %3 3%1/&- 3-3 1-02- '02-( M3-02&/ 3/00' 5'6/00' ( 8/ '7/88#1- 7-0%2&#.%&&/ .# %3 %02/&#-&/ -&.#3/ .# $&'3./99' 0@/&&-&/ 4/&,/32%'0/ '7&/11- .-7%= 2- /2/&/ 4/& A<<<< 7-02/ 0@/8/,%9#-3/ ./00@'0$--: .'2- ,H/ 0' ./7#'9#-3/ √ 82'3.'&. F σB,α N m B./' .'0 E'22- ,H/ E',,#'1- 0' 1/.#' ,'14#-3'&#' .# m 7'0-&#L σB,α F 0' ./7#'9#-3/ 82'3.'&. ./00- 82#1'2-&/ ./0 I%'32#0/= α ,'0,-0'2- 8% O -88/&7'9#-3#C: 4/& I%/82- 1-2#7- ,# 8#'1- E/&1'2# '. m = 100( >-1/ 8# 7/./ .'00' "#$%&' ( # 0#1#2# .# ,-32&-00- '0 .#7/&$/&/ .# 3 ,-37/&$-37/&8- %3 7'0-&/ ,H/ 8/16&' /88/&/ 3-3 1-02- 4#K $&'3./ .# P(Q / ,H/ 4%R /8= 8/&/ %2#0#99'2- ,-1/ 7'0-&/ '44&-88#1'2- ./# 0#1#2# ' 4'&2#&/ ,#&,' .' n = 100(.
(29) !"! #$%&' ()$*+%,'- *$((.*' /$, 0+%+1$#%, 2 *'#'. !. 0.015 0.010 0.000. 0.005. RL0 probability density function. 0.020. 0.025. "#$%#& ' (')'*' +' ,-$*.-((- /( +'0&.1&.& +' $ 2' 2*/3'('44/$-5 6 ,-$7&.)/ +' %#/$*- +&**- 8.&,&$+&$*&)&$*& 0&+'/)- ,9& (:&..-.& 8&.,&$*#/(& /#)&$*/ ,-$ $; ,'< 2' .'=&**& '$ >'1#./ 5 ,-$ '( )/11'-. ?.#)-.&@ ,9& 2' 9/ 0&.2- $ 1./$+'5. 0. 50. 100. 150. 200. 250. 300. n. >'1#./ 5AB C& ('$&& 0&.*',/(' $&.& ./88.&2&$*/$- (/ 7#$4'-$& +' 8.-3/3'('*D 2*')/E */ +&((/ .#$ (&$1*9 '$ ,-$*.-((-F (/ ('$&/ ,-$*'$#/ '$ .-22- ./88.&2&$*/ 7#$4'-$& +' 8.-3/3'('*D +' #$/ 05/5 1&-)&*.',/ ,-$ p = 0.025 G(/ ,-$*'$#'*D +&((/ ('$&/ .-22/ $-$ 9/ 2&$2- )/ .&$+& 0'2'3'()&$*& 8'H 7/,'(& (:'$*&.8.&*/4'-$&I5 J' ,9'&+'/)- -./ %#/$*- (& 2*')& +&' (')'*' +' ,-$*.-((- 2'/$- 3#-$&; $-$ 2-(- '$ *&.)'$' +' &..-.& 8&.,&$*#/(&; & 2& (:/88-.-22')/4'-$& +&' (')'*' +' ,-$*.-((- /( 0/(-.& A5K 8&. n ≥ 100 2'/ 2&$2/*/ -88#.& $-5 L&. .'28-$+&.& 8-22'/)- 2*')/.& (/ 7#$4'-$& +' 8.-3/3'('*D +&((/ .#$ (&$1*9 '$ ,-$*.-((- & 0&+&.& 2& M 1&-)&*.',/ -88#.& 2& ,' 2-$- ,9'/.' +'2,-2*/)&$*' +/ %#&2*/ /22#$4'-$&5 633'/)- -8&./*'$ */( )-+- / 8/.*'.& +/ n = 10 ,-$ NN )'(/ -22&.0/4'-$' & ,-)& 2' 0&+& /$,9& +/((/ >'1#./ 5A G/$,9& .'8&*&$+- 8'H 0-(*& (/ 2')#(/4'-$&I (/ +'2*.'3#4'-$& +&((/ .#$ (&$1*9 ,-$ %#&' (')'*' +' ,-$*.-((- 2*')/*' .'2#(*/ &22&.& /88.-22')/*'0/)&$*& #$/ 05/5 1&-)&*.',/ & $-*'/)- '$-(*.& ,-)& 8&. n ≥ 100 /88.-22')/$+- ' (')'*'.
(30) !"#$%&% '( #& )%*+&&% *# ,!-.+/"%#-$. !. " #$ %& '()*(+,")-.,( /-&&" +0. &-.1,2 3 '2%"+")-.,- %/-.,%'( " 40-&&( /% 0." 5#"# 1-()-,+%'"#. !"!. #$%&'%()*+$ ,$--) +)%.) ,/ +'*.%'--'. 6()- "77%")( 5%8,( &" +0. &-.1,2 %. '(.,+(&&( 8-10- 0." /%8,+%709%(.- 1-()-,+%: '"# ;0(+% '(.,+(&&( %.5-'- &" /%8,+%709%(.- /-&&" +0. &-.1,2 .(. 3 *%< 1-()-,+%'"= (88%" &" >0.9%(.- /% +%8'2%( /%5-.," *%< 1-.-+%'")-.,-?. 1 − βn,τ,r =. PH1 [RL = n] , n≥τ PH1 [RL ≥ n]. &" *+(7"7%&%,@ /% '())-,,-+- 0. -++(+- /-& AA ,%*( /%*-./- /0.40- (&,+- '2/" .= ".'2- /"& *0.,( /% 85(&," τ - /"& 5-+( +"**(+,( ,+" &- )-/%- r = µ1 Bµ2 # C0**(.%")( (+" '2- /(*( %& *0.,( /% 85(&," τ '% 8%" 0. '")7%")-.,( .-&&" )-/%" /" µ1 " µ2 D%& '0% +"**(+,( '()- /-E.%,( *+%)" 3 rF - '"&'(&%")( %& *"+")-,+( /% .(. '-.,+"&%,@?. EH1 [Tmax,n | τ ]. = EH1 [−(τ − 1) log Y 1 − (n − τ + 1) log Y 2 + n log Y ] = −EH1 [(τ − 1) log Y 1 ] − EH1 [(n − τ + 1) log Y 2 ] + + EH1 [n log Y ]. G(%'23 .(. 3 *(88%7%&- -8*+%)-+- %. >(+)" '2%08" %& '"&'(&( /-% ,+- 5"&(+% ",,-8%= 8% -H-,,0".( ,+- 85%&0**% /% I"J&(+ (1.0.( '-.,+",( +%8*-,,%5")-.,- %. µ1 = µ2 - &K0&,%)( %. ((τ − 1)µ1 + (n − τ + 1)µ2 )Bn D'2- .(. 8(.( "&,+( '2- % 5"&(+% ",,-8% +%8*-,,%5")-.,- /% Y 1 = Y 2 - /% Y F# 6(8% "77%")( '2- %& *"+")-,+( /% .(. '-.,+"&%,@ +%80&," -88-+- "**+(88%)",%5")-.,-?. (τ − 1)(r − 1) . EH1 [Tmax,n | τ ] = −(τ − 1) log r + n log 1 + n. DL#MF. N-.,+- 8(,,( H0 &" 8,",%8,%'" ,-8, Tmax,n /%*-./- 8(&( /" .= 8(,,( H1 /%*-./(&,+- '2- /" . ".'2- /" τ -/ r# A& *"+")-,+( /% .(. '-.,+"&%,@= "& /%5-+1-+- /% n= +"11%0.1- 0. &%)%,- '2- "*: *+(88%)",%5")-.,- +%80&," -88-+-?. (τ − 1)(r − log r − 1). DL#O!F. P0-8,( 8011-+%8'- '2- &" >0.9%(.- /% +%8'2%( /-&&" +0. &-.1,2 >0(+% '(.,+(&&( %.: '+-)-.," /(*( '2- %& '")7%")-.,( 3 (''(+8(= *-+ *(% 8,"7%&%99"+8% 80 0. 40"&'2&%5-&&(# 6()- 8% (88-+5" /"&&" ;%10+" L#Q &" >0.9%(.- /% +%8'2%( D8,%)"," " *"+: ,%+- /" L!! )%&" (88-+5"9%(.% '(. τ = 10 - r = 2F +"11%0.1- 0. )"88%)( *-+ *(% /-'+-8'-+- - 8,"7%&%99"+8%? %.,0%")( *-+'%R '2- &- '(/- /-&&" /%8,+%709%(./-&&" +0. &-.1,2 >0(+% '(.,+(&&( .(. 8% /%8'(8,".( )(&,( /" 40-&&- /% 0." /%8: ,+%709%(.- 1-()-,+%'"# SKTUS +"**+-8-.," 0. 8-.8",( '+%,-+%( /% '(.>+(.,( '(. "&,+- '"+,- /% '(.,+(&&( 40"&% "/ -8-)*%( &" 6VCVN D8- &" >0.9%(.- /% +%8'2%( >(88- '(8,".,- &" /%8,+%709%(.- /-&&" +0. &-.1,2 8"+-77- '()*&-,")-.,- '"+",:.
(31) 0.03. 0.04. hazard. 0.05. 0.06. !"! #$%&' ()$*+%,'- *$((.*' /$, 0+%+1$#%, 2 *'#'. 0. 20. 40. 60. 80. 100. n. "#$%&' ()*+ ,' -%./#0.1 2# ř#0 2166' &%. 61.$75 -%0&# 40.7&0660 837#9'7':). !.
(32) !"#$%&% '( #& )%*+&&% *# ,!-.+/"%#-$. !. "#$%&&'"' ('))*+,-./ 011#$2%'34 %54)"$# 67# #1%1"# 85' 9:5#1"$' (% 4;;4$"85%"<= 5#>)% %1"'5"% 1866#11%2% ') 6'3?%'3#5"4 %5 68% @ ;%A ;$4?'?%)# 67# )4 1"#114 2#5>' $%)#2'"4B 4 6% '664$>%'34 C8'1% 18?%"4 (#) 6'3?%'3#5"4 4 @ ;%A (%D6%)# E3#54 ;$4?'?%)#. '664$>#$1#5# (4;4/ F5 G'?#))' !/ @ $%;4$"'"4 %) ;$4:)4 (#))*+,-B 85' H'""4 34)"4 #2%(#5"# @ 643# )*+,- (%3%58%16# '))*'83#5"'$# (% τ I 411%' ;%A @ >$'5(# )' 1"4$%' (#) ;$46#114 %5 645"$4))4 # ;%A $';%('3#5"# 1'$#34 %5 >$'(4 (% $%)#2'$#I 85' 24)"' 67# %) ;$4J 6#114 2' H84$% 645"$4))4I %) 6'3?%'3#5"4K %5H'""% '))*'83#5"'$# (% τ '83#5"' %) ;'$'3#"$4 (% 545 6#5"$')%"</. τ OL !M ML OLL !LL. L/!M N/N M/! P/S P/N P/Q. L/M ! /O OM/Q O!/Q OO/ OL/N. L/NM M/O O/Q !S/R !Q/ !M/O. O PL/O R/R R/R PL/P S/!. O/M N/Q !/! !N/M ! /S !O/!. µ2 µ1. ! !R/N OS/ O / OO/Q OL/P. !/M !O/R OL/M S/L N/O Q/N. OM/N N/L M/N M/ M/L. P S/M P/ /S /Q /M. N / !/P !/! !/! !/!. O !/L O/Q O/Q O/Q O/M. G'?#))' !/ B +,- %5 645"$4))4 # H84$% 645"$4))4 C8'5(4 )' ;$4?'?%)%"< (% 6433#""#$# 85 #$$4$# (#) F "%;4 @ :11'"' '( α = 0.025/ T#))' 64)455' %5 68% %) $';;4$"4 "$' )# 3#(%# @ ;'$% '( OI 411#$2%'34 )*+,- %5 645"$4))4 67# 1';;%'34 #11#$# ;'$% '( 1U0.025 = 40I ;4%67@ $%64$(%'34 67# )' (%1"$%?8&%45# (#))' $85 )#5>"7 %5 645"$4))4 @ >#43#"$%6'/.
(33) !"! #$#%&'() *'$+$,-' '. /.+ %'.'#-+ *' 0+-1(.#. !" #$%&'()* +($,$-.( (/ 0/, &(/(%., +( 1,.2)/%. !"#$%&&#'()* +" ,#-#.(- ,%./*"%0%$( "($$# $%0'('%# !!" ,% 1 .*--* %$ "*)( ,% #!$%2 %" 3+% .*"* '%/*'-#-( $( ,#-( ,% 454 (./$*.%*"% %" +"# )%"%('# ,% 3#'0*"( %" *6"+"# ,($$# 7+#$% 3% .*"* .-#-( 48 * /%9 )*'-% .+$ /*.-*2 ,#$ 4: ;#'&* 4<:4 #$ == ;#'&* 45>=?. date 1 1851.203 2 1851.632 3 1851.969 4 1851.975 5 1852.314 6 1852.347 7 1852.358 8 1852.385 9 1852.977 10 1853.196 . . .. . . .. 185 1947.619 186 1947.638 187 1947.687 188 1951.405 189 1957.883 190 1960.489 191 1962.220. @#33%#)* +" 6'#A3* ,($ "+)('* -*-#$( ,% ,%.#.-'% *33*'.* %" *6"% #""* ,#$ 4<:4 #$ 45>=? .% /+B *..('C#'( 3D( ,#$$E#""* 4588 3%'3# %" /*% %$ "+)('* ,% ,%.#.-'% #""+% )(,%#)("-( F ,%)%"+%-* ( .% "*-# #"3D( +"# )%"*' C#'%#0%$%-G H3*)( F.
(34) !"#$%&% '( #& )%*+&&% *# ,!-.+/"%#-$. 3 0. 1. 2. Numero di esplosioni. 4. 5. 6. !. 1860. 1880. 1900. 1920. 1940. 1960. Anno. "#$%&' ()*+ ,-.'/0-12 -03 10/42 .03 -%/0&2 .# .#5'51&# '--%#). 32$#62 '54011'&5# 62- .'1# .# 72#552-8) 70& 4210& '443#6'&0 30 6'&10 .# 62-1&2332 4&050-1'10 9- :%# .2;;#'/2 '<'&0 .'3 .'1'501 # 10/4# 6=0 #-10&62&&2-2 1&' .%0 .#5'51&# 62-506%1#<# 6=0 #- 121'30 5'&'--2 .%-:%0 >?@+. T BD 1 0.4298 2 0.3368 3 0.0055 4 0.3395 5 0.0329 6 0.0110 7 0.0274 8 0.5914 9 0.2190 10 . . . 185 186 187. 0.0329 . . . 0.0192 0.0493 3.7180. 188 6.4778 189 2.6064 190 1.7303. 70& <0`'&0 3A'55%-12 .#51&#;%1#<2 0 3A#-.#40-.0-B' 5%442-#'/2 .# .#542&&0.
(35) !"! #$#%&'() *'$+$,-' '. /.+ %'.'#-+ *' 0+-1(.#. !. 1.0 0.0. 0.5. Sample quantiles. 1.5. 2.0. Exponential Q−Q Plot. 0. 1. 2. 3. 4. Theorical quantiles. "#$%&' ()*+ ,-, ./01 .2& /2 .  !4 0552&6'7#08#) 92//2 .  !4 0552&6'7#08#: 5%//2 ;%'/# <# ='52&230 '8<>2 .2& %8' 51#3' .&2/#3#8'&2 9# µ1 ?<032 5# 629&@ #8 52$%#10 #8 ;%2512 .  !4 0552&6'7#08# #/ .&0<2550 %/12&@ 2552&2 #8 <081&0//0A) B'/ ,-, ./01 0552&6#'30 <>2 808 523=&'80 2552&<# 5<051'3281# '//'&3'81# 9'//' &211' <>2 #812&.0/' #/ . 92<#/2 29 #/ 12&70 ;%'&1#/2: 52 808 #8 .'&12 82//' 708' <281&'/2 2 C8'/2) D80/1&2 <08 #/ 1251 9# 25.08287#'/#1@ 9# E>'.#&0-F#/G: C55'10 α = 0.05: 5# >' <>2 ?#/ 1251 H =#/'12&'/2A+. q50,0.025 = 0.0122 < tS.W.oss. = 0.0136 < q50,0.975 = 0.0340 # 9'1# 5080 <08I0&3# .2110 '//J#.0125# 8%//' 9# 25.08287#'/#1@: '8<>2 52 066#'32812 808 .055#'30 9#&2 <>2 .&0628$080 9' %8' 6)') 25.08287#'/2) 1. 2. 3. y. 0.0. 0.5. 1.0 y. 1.5. 2.0. 0. lag 2. 0.0. 0.5. 1.0. y. y. 1.5. 2.0. lag 1. lag 4. y. 0.0. 0.5. 1.0 y. 1.5. 2.0. lag 3. lag 5 0. lag 6 1. 2. 3. "#$%&' ()K+ L'$-./01 92# 9'1# C80 ' '&90 *) M2& ;%'810 &#$%'&9' /' 62&#C<' 92//J'55%810 9# #89#.28927' # 9'1# %/1'80 2552&2.
(36) !"#$%&% '( #& )%*+&&% *# ,!-.+/"%#-$. TBD. 0.0. 1.0. 2.0. !. 10. 20. 30. 40. 50. 0.0 −1.0. ACF. 1.0. 0. 0. 5. 10. 15. Lag. p−value. 0.0 0.4 0.8. Box−Ljung test. 5. 10. 15. lag. "#$%&' ()*+ ,'--.'-/0 12&30 #- 4'330+ &'55&2326/'7#062 $&'89' :2# :'/#; 90&&2-0$&'<<'; 5=1'-%2 :2- /23/ :# >?%6$=@0A ' 1'&# &#/'&:#). #690&&2-'/# 90<2 3# 60/' :'- 90&&2-0$&'<<' 2 :'# 5=1'-%2 :2- /23/ :# @0A=>?%6$ B#6C'//# -2 '%/092&&2-'7#06# 3060 5&033#<2 '--0 D 2: # 5=1'-%2 3060 /%//# <0-/0 $&'6:# 52& 0550&/%6# &#/'&:#E) ,' 32$6'-'&2 F -' 5&23267' :# %6.'55'&26/2 3/'= $#06'-#/G 9H2 3# 21#692 62--.'6:'<26/0 :2# :'/# 62- /2<50 B5&#<0 5'662--0 :# "#$%&' ()IE 2: '69H2 62# J92&9H#K :2# -'$=5-0/ B"#$%&' ()*E) L0-26:0 12Y'&2 -.2126/%'-2 :#526:267' :2# :'/# :' <232 ' <232 2 :' $#0&60 ' $#0&60M #:26/#89H#= '<0 ' 5'&/#&2 :'--' 5'&/2 C&'7#06'&#' :2--2 :'/2 # $#0&6# 2: # <23#) N'-2 '55&099#0 F '69H2 32$%#/0 :' >%9'3 BOP*QE 62--.'6'-#3# :# %6 C'<030 :'/'32/ :2--2 <0&/# 3%-'10&0) R44#'<0 :%6S%2 -2 32$%26/# /'42--2 :# C&2S%267'+ T232. U) V35-03#06# OW (D (D O OW OD OI O! OD O! OI (W . = 11.7435 − αoss. = 0.3832. !""#$% &!''(#$% )#(*% +,($-! )#..$% $/."% 0/.-$% +.%12% 3!22!4'(! 522%'(! 6%7!4'(! 8$9!4'(! .. χ2oss..
(37) !"! #$#%&'() *'$+$,-' '. /.+ %'.'#-+ *' 0+-1(.# "#$%&$. !. '( )*+,$*#$&# -. /0. !"#$% &'()#$% &#(*+,#$% -.+/#$% 0#"#($% 1'2')+ 3+4#".*' .. ! -1 . = 0.002186. χ2oss. = 20.5759 − αoss. 234#5&63 65,# 6573,,3 83%#9:;#5<$ #, *3=>3&63 *#*63<5 4# #+$63*#?. ( H0 : Indipendenza stocastica da mese a mese H1 : Esiste una qualche relazione da mese a mese ! "#!#$"#$%!&#'"# %() *' !#$+! ,$"#*$-./$()' ). ! (##').#! !"$)#(#$%!0')#' "()(1. 12 X. . 2. ∧. fi − f ∧. f. i=1. .. ∼ χ211. H0. ∧ ,(+'. fi. "()( ' 2*'3.')/' (""'*+!#' ',. f. ' 2*'3.')/' !##'"'4 5' %!"( ,'$ 6$(*)$ "$. 7!))( .) !)! (6( "$"#'0! ,$ $8(#'"$ ' "#!#$"#$%! #'"# 9(++$!0')#' !+*'0( : 6*!,$ ,$. $-'*#; $)+'%' %7' <<=4 5'. %!"( ,'$ 0'"$. $),$8'),')/!= *$". #! '""'*' <?<@<A "'*' <?<@B. ≈ 27.294. ! 2*'3.')/! !##'"! 9"(##(. ≈ 15.92 0')#*' )'. >$8(#'"$ ,$. %!"( ,'$ 6$(*)$ *$". #! '"&. C +! (*$ ,'$ 8&+! .' %$ ".66'*$"%()( %7' 0')#*' "'0-*'*'--'. '""'*%$ $),$8'),')/! ,! 0'"' ! 0'"' %$D )() E !)%(*! +! $,( ,! 6$(*)( ! 6$(*)(4 C)2!##$. ' '"8 ("$()$ "$ %()%')#*!)( 0!66$(*0')#' ,')#*(. ! ,(0')$%! ', $ ,(+.#( !. .)',F "()( 0')( 2*'3.')#$4. 8!*#$%( !*' #$8( ,$ (*!*$(. ! "'##$0!)! 0')#*'. G.'"#( 2!##( 8(#*'--' '""'*'. !+(*!#$+( (""'*+!#( ,.*!)#'. ! "'##$0!)!. $) 0$)$'*!H #')'),( %()#( !)%7' ,'$ %!0-$!0')#$ %7' 8(""()( '""'*' !++').#$ ,!. <IJ< !. <?AA 9"!*'--' $)2!##$ (88(*#.)( !,,')#*!*%$ )'. #*! 0$)'*$! ,$ %!*-()' ' %!8$*' %(0' "$ "()( '+( .#' 0(,( ,! %7$!*$*' $)'! ,'. #.##( $. ! ,$0$)./$()' ,'. ! "#(*$! ,' ! )("&. ' 0$".*' ,$ "$%.*'//! $). ).0'*( 0',$( ,$ '"8 ("$()$ !)).'=4. C). ."#*!#$+! #*!"%.*$!0( 3.'"#! ,$8'),')/! ,! 6$(*)( ! 6$(*)(. !)%7' 8'*%7E %(0' "$ +',*;. ' )("#*' %!*#' ,$ %()#*( ( *$."%$*!))( %(0.)3.'. )' %!"( $) '"!0' !, !""( +'*' $. (*( %(08$#( ,$ "'6)! !/$()' ,' %!0-$!0')#(4. ∧. K! "#$0! ,$ %.$ ,$"8()$!0( 8'* ! 0',$! $) %()#*( ( E )'. ,$"'6)( ,'. µ1 = 0.333H $),$"8')"!-$ '. ' %!*#' L7'M!*# ' NOLOP4 G.'"#! "'%(),! %!*#! "!*; ,$"'6)!#!. $) 0!)$'*! #! ' ,! '""'*' (##$0! ' )' ! "'6)! !/$()' ,$ !.0')#$ 8!*$ ! <4J +( #' ! 0',$! $) %()#*(. ( ' ,$0$)./$()$ 8!*$ ! Q4: +( #'. ! 0',$! $) %()#*( (4 O)(. ,'$ +!)#!66$ %7' %$ 2(*)$"%' ! %!*#! RKS E %7' )() "$ 7! -$"(6)( ,$ ! %.)! "#$0! 8*' $0$)!*' ,$. µ1. 98'* #.##' ' #*'. 8'*. ! ".! !88 $%!/$()'4. K>TSK $) %()#*( ( E U""!#( ! AQQ. ' %!*#'=4. V""'*+!),( $ 6*!U%$ ,'. ' #*' %!*#' ,$ %()#*( ( 8(""$!0( !W'*0!*' %7' )'. %!"( $) '"!0' "$ %(08(*#!)( #.##' ' #*' -')'4. T+'+!0( (""'*+!#( )' >!),!&.
(38) !. !"#$%&% '( #& )%*+&&% *# ,!-.+/"%#-$. "#$%&' ()*+ ,' -'&.' /# -01.&0220 3,4 56# 701#.0&' #2 8&0-9660 ' 8'&.#&9 /'22' /9-#7' 0669&:';#019<= # 1%79&# 608&' # 8%1.# 6010 29 6.#79 /# 7'66#> 7' :9&06#$7#$2#'1;' /# τ 19# :'&# #6.'1.# 6%--966#:# '22' 69$1'2';#019 /92 ?%0&# -01.&0220). "#$%&' ()@A+ ,' -'&.' /# -01.&0220 BC9D'&. 5-01 2#7#.# 96'..#<).
(39) !"! #$#%&'() *'$+$,-' '. /.+ %'.'#-+ *' 0+-1(.#. "#$%&' ()**+ ,' -'&.' /# -01.&0220 3%4%5 6#2'.7&'27). !.
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τ 2 KLM "#$%*# .#* µ1 # µ2 *,-)(%+$& #--#*# *,-.#%%,8+"#$%# !5NK # K5!OKP (+ "#',+ ,$ /&$%*&((& ', 7+%%& 2 %*,.(,/+%+5 Q+( 9*+:/& '#((1RED .#* (+ /+*%+ ;0#<+*% @S,9)*+ K5MF (,$#+ %*+%%#9,+%+ *&--+I -,+"& $#((+ 3&$+ ,$ /), ,( /&".&*%+"#$%& '#((+ /+*%+ 2 "&(%& A)&$& @,$7+%%, . r =KTN # B)#-%& -.,#9+ (+ /#(#*,%J '#((+ /+*%+ $#( -#9$+(+*# (1+)"#$%& '#((+ "#H ',+U '+ $&%+*# ,( 7+(-& +((+*"# /0# 8,#$# (+$/,+%& +((+ B),$',/#-,"+ &--#*8+3,&$#5 V( /+"A,+"#$%& /0# &--#*8,+"& 2 ', 7+%%& )$ /+"A,+"#$%& 9*+$'# #' 2 ',WH /,(# *,)-/,*# + *,(#8+*# '+ B)#-%& )$,/& #-#".,& B)+(# '#((# %*# /+*%# 0+ )$+ .*#-%+3,&$# ",9(,&*#5.
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
(42) !. !"#$%&% '( #& )%*+&&% *# ,!-.+/"%#-$.
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