THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION

Transcription

1 Advce d Alcto Stttc Volue Nue Pge 5-55 Th e vlle ole t htt://hco/oul/dht Puh Pulhg Houe THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION CRISTIANE RODRIGUES GAUSS M CORDEIRO CLARICE G B DEMÉTRIO d EDWIN M M ORTEGA Deteto de Cêc Et Uvedde de São Pulo Pcc SP Bzl e-l: codgue_7@hotlco clce@elu edw@elu Deteto de Ettítc Uvedde Fedel de Peuco e-l: gucodeo@uolco Atct We ooe the Weull egtve ol dtuto tht ute flele odel to lyze otve dt d clude ecl uodel the Weull Weull Poo d Weull geoetc dtuto Soe of t tuctul oete follow fo the fct tht t dety fucto c e eeed tue of Weull dete We ovde elct eeo fo oet geetg fucto e devto Bofeo d Loez cuve utle fucto ellty d etoy The dety of the Weull egtve ol ode tttc c e eeed te of fte le coto of Weull dete We ot two ltetve eeo fo the oet of ode tttc The ethod of u lelhood vetgted fo ettg the odel ete d the oeved foto t clculted We ooe ew egeo odel ed o the logth of the ew Mthetc Suect Clfcto: 6E5 6F 6N 6N Keywod d he: geetg fucto foto t u lelhood oet Weull dtuto Weull egtve ol dtuto Receved Mch

2 6 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA dtuto The uefule of the ew odel llutted thee lcto to el dt Itoducto The Weull dtuto vey oul odel tht h etevely ee ued ove the t decde fo lyzg dt uvvl ly ellty egeeg d flue ly dutl egeeg to eeet ufctug d delvey te etee vlue theoy wethe foectg to dece wd eed dtuto wele coucto d uce to edct the ze of euce cl I hydology t led to etee evet uch ully u oe-dy fll d ve dchge The eed fo eteded fo of the Weull dtuto e y led e ut the eegece of uch eteo the tttc ltetue oly vey ecet Followg de due to Add d Lou [] fo g ocedue of dtuto we defe the Weull egtve ol (WNB) dtuto d tudy evel of t thetcl oete The Weull dtuto eeet oly ecl u-odel of the ew dtuto Let W WZ e do le fo Weull dety fucto wth cle ete > d he ete > ely g ( ω) ω ω e ( fo ω > ) We ue tht the do vle Z h zeo tucted egtve ol dtuto wth olty fucto z z P( z; β) β [( β) ] z N R z d β ( ) Hee Z d W e ued deedet do vle Let X z ( W WZ ) The f ( z; ) z e d the gl olty dety fucto (df) of X wth fou ete educe to f ( ; θ) β [( β) ] e ( βe ) > β ( ) () whee θ ( β) I the euel () efeeed to the WNB dety fucto whch cutoy fo uch e to e gve to odel g v the oeto of cooudg (g) dtuto

3 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 7 By tegtg () led to t cuultve dtuto fucto (cdf) gve y [( β) ( βe ) ] F( ; θ) > [( β) ] The hzd te fucto coeodg to () () ( ) ( βe βe ) τ ; θ (3) [( βe ) ] () () Fgue Plot of the WNB dety fo oe ete vlue () () Fgue Plot of the WNB hzd te fucto fo oe ete vlue

4 8 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA If X do vle wth dety fucto () the we wte X ~ WNB( β) It eted evel dtuto evouly codeed the ltetue d we tudy oe tuctul oete of () I fct the Weull dtuto wth ete d clely ecl u-odel whe d β Fo d ddto to β t yeld the eoetl dtuto The WNB dtuto lo cot the eoetl Poo (EP) (Ku []) d Weull Poo (WP) dtuto (Beet et l [5]) u-odel whe β λ d the ecod ce ddto to Fo euto () educe to the Weull geoetc (WG) dety fucto (Beto-Souz et l [4]) I Fgue we lot the WNB dety fo elected ete vlue Fo ll vlue of the ete the dety fucto () ted to zeo I Fgue we lot the WNB hzd te fucto fo elected ete vlue howg t flelty The WNB dety c wdely e led y e of egeeg d ology The et of the e ogzed follow: I Secto we deotte tht the WNB dety fucto c e eeed tue of Weull dete Th eult ott to ovde oe thetcl oete of the ew odel dectly fo thoe oete of the Weull dtuto A ge of thetcl oete codeed Secto 3 to 6 Thee clude oet ewe d uto utle fucto geetg fucto e devto d Bofeo d Loez cuve I Secto 7 we how tht the dety of the WNB ode tttc le coto of Weull dete Two elct eeo fo the oet of the WNB ode tttc e oted th ecto whle the ellty d the Réy etoy e deved Secto 8 d 9 eectvely Mu lelhood etto d feece e dcued Secto A log-weull egtve ol egeo odel ooed Secto Thee lcto to el dt Secto llutte the otce of the ew odel Cocludg e e ddeed Secto 3 Eo fo the Dety Fucto Euto () d () e tghtfowd to coute ug y tttcl oftwe Howeve we ot eo fo f d F te of fte weghted u of cdf d df of Weull dtuto eectvely Ug the

5 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 9 Lgge eo (Coul d Foye [7 Secto 6]) fo ( βe ) ( ) ( ) {( βe ) [ ( βe )] } [ ] βe (4) euto () c e futhe eded Hece whee f ( ; θ) [( β) ] β ( ) e f ( ; θ) ω g (5) ω β ( )[( β) ] (6) d g ( ) the Weull dety fucto wth cle ete ( ) d he ete Clely ω Euto (5) evel tht the WNB dety fucto tue of Weull dete tht hold fo y ete vlue Th eult ott to ot oe of t thetcl oete fo thoe of the Weull dtuto The foul elted wth the WNB dtuto tu out gele t how the et of th e d wth the ue of ode coute eouce wth lytc d uecl clte y tu to deute tool cog the el of led tttc Eleety tegto of (5) gve the WNB cuultve dtuto F ( ; θ) ω G (7) whee G e deote the Weull cuultve fucto wth cle ete ( ) d he ete

6 3 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA 3 Moet Fo ow o let X ~ WNB( β) The ody cetl vee d fctol oet of the WNB dtuto c e oted fo fte weghted le coto of thoe utte fo Weull dtuto Fo ele the th oet of the Weull dtuto wth ete d τ Γ( ) whee α w Γ α w e dw the g fucto The th geelzed oet of the WNB dtuto edtely coe fo (5) μ Γ E X ω (8) [ ( ) ] Vou cloed fo eeo c e oted fo (8) tcul ce The cetl oet ( μ ) d the cuult ( κ ) of X e ely oted fo the ody oet y μ μ μ d κ μ κμ eectvely Hee κ μ κ μ μ κ 3 μ 3 3μ μ μ κ4 μ 4 4 4μ 3μ 3μ μ μ 6μ etc eectvely The ewe d uto eue c e oted fo the clcl eltoh volvg cuult: 3 Sewe ( X ) κ 3 κ d Kuto ( X ) κ 4 κ ely 3 Sewe ( X ) 3 E( X ) 3 E( X ) E( X ) E ( X ) 3 V ( X ) 3 d Kuto ( X ) 4 3 E( X ) 4E( X ) E( X ) 6E( X ) E ( X ) 3E ( X ) V ( X ) 4 Plot of the ewe d uto of X fucto of d β e gve Fgue 3 d 4 Both ewe d uto cee wth fo fed β d cee wth β fo fed

7 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 3 Flly the th decedg fctol oet of X gve y [ ( ) E X ] E[ X ( X )( X ) ( X ) ] ( ) μ whee ( ) (!) [ D ] e the Stlg ue of the ft d The WNB fctol oet e oted fo (8) () () Fgue 3 The WNB ewe ( fo 3 d 4) fucto of (fo fed β) d fucto of β (fo fed ) () () Fgue 4 The WNB uto ( fo 3 d 4) fucto of (fo fed β) d fucto of β (fo fed )

8 3 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA 4 Qutle Meue The WNB utle fucto coeod to the vee of () gve y u Q ( u ) F ( u ; θ) { log[ β ( {[ ( β) ] ( β) } )] } (9) The ug (9) ulto fo the WNB do vte tghtfowd y X V { log[ β ( {[ ( β) ] ( β) } )] } whee V ufoly dtuted do vle ove the tevl We develo ct ug the oftwe R (R Develoet Coe Te 8) to ulte X Aed A () Fgue 5 The Bowley ewe of the WNB dtuto ( fo 3 d 4) fucto of (fo fed β) d fucto of β (fo fed ) We ow coute utle eue fo the ewe d uto The Bowley ewe (Keey d Keeg [9]) ed o utle Q( 3 4) Q( ) Q( 4) B () Q( 3 4) Q( 4) d the Moo uto (Moo [3]) ed o octle Q( 7 8) Q( 5 8) Q( 3 8) Q( 8) M () Q( 6 8) Q( 8) whee Q ( u) clculted fo (9) Thee eue e le etve to outle d ()

9 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 33 they et eve fo dtuto wthout oet Plot of () d () fo elected ete vlue e gve Fgue 5 d 6 eectvely Fo fed β whe cee the Bowley ewe d the Moo uto ft cee to u d the decee The ehvo of oth eue whe β cee deed uo the vlue fed fo () Fgue 6 The Moo uto of the WNB dtuto ( fo 3 d 4) fucto of (fo fed β) d fucto of β (fo fed ) 5 Moet Geetg Fucto The oet geetg fucto (gf) M ( t) E[ e( tx )] follow fo the owe ee eo fo the eoetl fucto d (8) () Γ ω M t t [ ( ) ]! Now we deve two elct eeo fo M ( t) ug Mee G-fucto d Wght geelzed hyegeoetc fucto Let A β [( β) ] Ft we hve () ( )[ β ( )] ( M t A e t e e ) d A β e ( t) e[ ( ) ] d () ()

10 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA 34 Fo the Mee G-fucto defed y Γ Γ Γ Γ π L t dt t t t t G d the eult { } ( ) e g G g fo g ty fucto we c wte [ ] e d t K ( ) e d G t If we ue tht whee d e co-e tege the euto (4) Pudov et l [5 Volue 3] yeld { [ ] } π t G t K Fo () d the lt two euto we ot () [ ] β π β t t M { [ ] } t G (3) The codto (3) ot etctve ce evey el ue c e oted y tol ue A ecod eeetto fo the tegl M c e oted ug the Wght geelzed hyegeoetc fucto gve y! ; β Γ α Γ β β α α Ψ B A B B A A

11 We et tht THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 35 K t! e{ [ t ( ( ) ) ] } d ( ) ( l [ ( ) ] )! Γ Ψ ( ) ( ) t [ ] ; (4) ovded tht > Cog () d (4) we ot the ecod eeetto fo M () t () [ ] β t M t Ψ ( ) ; (5) β [ ( ) ] ovded tht > Clely ecl foul fo the gf of the Weull WG d WP dtuto c edly e deteed fo euto (3) d (5) y uttuto of ow ete 6 Me Devto The e devto out the e ( δ( X ) E( X μ )) d out the ed ( δ ( X ) E( X )) c e wtte δ ( X ) μ F( μ ) T ( μ ) d δ ( X ) μ T (6) eectvely whee μ E( X ) gve y (8) F ( μ ) coe fo () Med ( X ) Q( ) couted fo (9) d T( ) f d Fo (5) d ug the colete g fucto γ( α ) we ot w α e ω dw ω T ( ) γ( ( ) ) (7) [ ( ) ] The the e devto c e clculted fo (7) Futhe we c ot

12 36 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA Bofeo d Loez cuve fo (7) Thee cuve hve lcto ecooc ellty deoghy uce d edce d e defed y B T( ) ( μ ) d L T( ) μ eectvely whee Q clculted y (9) fo gve olty 7 Ode Stttc The dety fucto f : of the th ode tttc fo coeodg to the do vle X X followg the WNB dtuto c e wtte { } f : f F B whee f the df () F the cdf () d B( ) Γ ( ) [ Γ( ) Γ ] the et fucto Settg u e( ) we c wte fo (5) d (7) f : ω B ( ) u u ω We ue thoughout euto of Gdhtey d Ryzh [8 Secto 34] fo owe ee ed to otve tege y c y (8) whoe coeffcet euto c ( fo ) e ely oted fo the ecuece c ( ) ( ) c (9) whee c Hece the coeffcet c coe dectly fo c c

13 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 37 d theefoe fo Ug (8) t follow tht ω : u B f u c u Hee the cott c e deteed fo euto (6) d (9) [ ] ω ω l l l c l l c whee c ω Hece c c e clculted fo c c d theefoe fo ω ω By cog te we ot ω : B c f u Settg δ the ove eeo we ot δ η : g f () whoe coeffcet η e ely oted fo ω η B c Euto () how tht the dety fucto of the WNB ode tttc c e eeed fte weghted le coto of Weull dete We c deve oe thetcl eue of the WNB ode tttc dectly fo thoe utte of the Weull dtuto

14 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA 38 Ug the le coto eeetto () the oet of the WNB ode tttc c e wtte dectly te of the Weull oet δ η Γ : X E () whee δ d η e gve Secto 7 Altetvely we ot othe cloed fo eeo fo thee oet ug eult due to Bt d Adelde [3] led to the deedet d detclly dtuted (d) ce I X E : () whee { } d F I The tegl c e wtte { ) ω e d c d u I whee ω ω c c d c ω Settg y we ot Γ c I Fo euto () we hve Γ c X E : (3)

15 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 39 We c coute the oet of the WNB ode tttc y thee dffeet wy: () fo euto () tht volve two fte u d oe fte u ut o colcted fucto; () fo euto (3) tht volve oly two u oe fte d othe fte; o () y dect uecl tegto The oet of the WNB ode tttc lted Tle e geeet ug the thee ethod Tle Moet of the WNB ode tttc fo 5 4 β 8 d 3 Ode tttc : 5 X X : 5 X 3: 5 X 4: 5 X 5: Vce Sewe Kuto Rellty Hee we deve the ellty R P( X < X ) whe X d X hve deedet WNB ( β) d WNB ( β) dtuto wth the e he ete The dety of X d the cdf of X e oted fo euto (5) d (7) f ( ) ω u d F ω u (4) whee u e ( ) fo d the cott ω d ω e gve y (8) wth the coeodg ete of the dtuto of X d X eectvely We hve R f F d

16 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA 4 d the ω ω d u u R { [ ] } ω ω e d By the lcto of μ μ e d we ot [ ] ω ω R 9 Etoy The etoy of do vle X wth dety f eue of vto of the ucetty Réy etoy defed y log d f IR (5) whee f the df of X > d Fo do vle X wth WNB dtuto we hve [ ] β β β d e e d f Ug the Lgge eo (4) the lt euto we ot β ν d f (6) whee [ ] β Γ ν (7)

17 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 4 Suttutg (6) d (7) to (5) we c wte I R ( ) log() ( )[ log( ) log] log[ Γ( ) ] log[( ) ] log β β ( ) Etto d Ifeece Hee we code the etto of the odel ete of the WNB dtuto y the ethod of u lelhood Suoe X X do le fo () d θ ( β) T the ete vecto The log-lelhood (LL) fucto y log L log{ L( β) } fo the fou ete log { L( β) } log( ) log ( ) log log ( β) log( ) ( ) log( βe ) log[( β) ] (8) It follow tht the u lelhood etto (MLE) e the ulteou oluto of the euto: ( ) βe βe log ( ) log( ) ( ) log( ) βe βe ( β) ( β) ( βe log log ) [( β) ]

18 4 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA d ( ) e βe ( β ) [( β) ] β Fo tevl etto d hyothe tet of the odel ete we eue the foto t The 4 4 oeved foto t K K( θ) { κ } β whoe eleet e gve Aed B Ude codto tht e fulflled fo the ete θ the teo of the ete ce ut ot o the oudy the ytotc dtuto of ( θ ˆ θ) ( N K θ ) We c ue the ytotc 4 ultvte ol N 4 ( K( θ) ) dtuto of θˆ to cotuct ote cofdece ego fo oe ete d fo the hzd d uvvl fucto I fct ( γ) ytotc cofdece tevl fo ech ete θ gve y θ θ κ ( ) ( ˆ ˆ θ AIC θ ˆ ˆ θ θ θ γ θ z κ θ z κ ) γ whee ˆ eeet the ( ) dgol eleet of K( θ) fo 3 4 d z γ the utle γ of the tdd ol dtuto The ytotc olty lo ueful fo tetg goode of ft of the WNB dtuto d fo cog th dtuto wth oe of t ecl u-odel ug lelhood to (LR) tttc We code the tto θ ( θ T θ T ) whee θ uet of ete of teet of the WNB dtuto d θ uet of the eg ete The LR tttc fo tetg the ull hyothe H : θ θ veu the ltetve hyothe H : θ θ gve y ω ~ {( l θˆ θ) } whee ~ θ d θˆ deote the MLE ude the ull d ltetve γ hyothee eectvely The tttc ω ytotclly ( ) dtuted χ whee the deo of the uet θ of teet Fo ele we c coe the WG odel gt the WNB odel y tetg H : veu H :

19 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 43 The log-weull Negtve Bol Model Let X e do vle hvg the WNB dety fucto () The do vle Y log( X ) h log-weull egtve ol (LWNB) dtuto whoe dety fucto (etezed te of λ log ( λ)) f ( y) β [ ] y μ y μ e e σ β σ σ σ d μ e e y μ β (9) σ whee < y < σ > d < μ < We efe to (9) the (ew) LWNB dtuto y Y ~ LWNB( β σ μ) whee μ the locto ete σ the deo ete d d β e he ete So f X ~ WNB( β ) the Y log( T ) ~ LWNB( β σ μ) The lot of (9) Fgue 7 fo elected ete vlue how get flelty of the dety fucto te of the he ete d The uvvl fucto coeodg to (9) ecoe ( ) e e y μ β β σ S y (3) [( β) ] () () (c) Fgue 7 Plot of the LWNB dety fo oe ete vlue: () μ σ () μ σ d (c) μ

20 44 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA We defe the tddzed do vle Z ( Y μ) σ wth dety fucto π( z; ) β { }[ { }] ( e z e z β e e z ) z R [( β) ] (3) The ecl ce coeod to the (ew) log-weull geoetc (LWG) dtuto The th ody oet of the tddzed dtuto (3) gve y μ E( Z ) β { }[ { }] ( ) z e z e z β e e z dz [( β) ] By edg the ol te d ettg e( z) we ot μ β [( β) ] ( ) β log e[ ( ) ] d The ove tegl c e clculted fo Pudov et l [5 Volue euto 6] d the ( ) [( I ) Γ ( )] μ β [( β) ] ( ) β I( ) I y ctcl lcto the lfete e ffected y eltoy vle uch the choleteol level lood eue weght d evel othe Petc egeo odel to ette uvte uvvl fucto fo ceoed dt e wdely ued A etc odel tht ovde good ft to lfete dt ted to yeld oe ece ette of the utte of teet Bed o the LWNB dety fucto we ooe le locto-cle egeo odel fo ceoed dt lg the eoe vle y d the eltoy vle vecto

21 T THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 45 v ( v v ) follow: y T v γ σz (3) whee the do eo z h dety fucto (3) γ ( γ ) σ > T γ > d > e uow ete The ete μ γ the locto v T of y The locto ete vecto μ ( μ μ ) T eeeted y le odel μ Vγ whee V ( v v ) T ow odel t The LWNB egeo odel (3) oe ew olte fo fttg y dffeet tye of ceoed dt It eteo of cceleted flue te odel ug the WNB dtuto fo ceoed dt Code le ( y v ) ( y v ) of deedet oevto whee ech do eoe defed y y { log( ) log( c )} We ue ofotve ceog uch tht the oeved lfete d ceog te e deedet Let F d C e the et of dvdul fo whch y the log-lfete o log-ceog eectvely Covetol lelhood etto techue c e led hee The log-lelhood fucto fo the vecto of ete τ ( β σ γ T ) T c fo odel (3) h the fo l ( τ ) l ( τ) l τ whee l ( τ) F C c ( log [ f ( y v )] l ) ( τ) log[ S( y v )] f ( y v ) the dety (9) d S( v ) the uvvl fucto (3) of Y The totl log-lelhood fucto fo τ educe to β l( τ ) log [ ] [ ] z e z σ β F ( ) log[ β e{ e}] C F z ( β) [ β e{ e( z )}] log (33) [( β) ] whee z ( γ) σ d the ue of uceoed oevto (flue) y v T y

22 46 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA The MLE τˆ of the vecto of uow ete c e clculted y zg the log-lelhood (33) We ue the uoute NLMed SAS to ot ˆτ Itl vlue fo β σ d γ e te fo the ft of the LWG egeo odel wth The ft of the LWNB odel yeld the etted uvvl fucto fo y gve y ( ˆ ˆ ) [ ˆ ˆ e{ e( ˆ )}] ( ˆ T β β z S y ; ˆ βˆ σˆ ) γ (34) [( ˆ ˆ β) ] whee z ˆ ( γˆ ) σˆ y v T The ytotc dtuto of ( τˆ τ) ultvte ol N 3( K( τ) ) whee K ( τ) the foto t The ytotc covce t K ( τ) of τˆ c e oted y the vee of the ( 3 ) ( 3) oeved foto t L ( τ) The ote ultvte ol dtuto N 3 ( L ( τ) ) fo τˆ c e ued the clcl wy to cotuct cofdece ego fo oe ete τ We c ue LR tttc fo cog the LWNB odel wth oe ecl u-odel Alcto I th ecto we coe the eult of fttg the WNB d LWNB dtuto to thee el dt et Clofo dt A tudy coducted Chle y D Fed Cvee (Uvety of Vlío) etlhed tht clofo lely cue cogetl lfoto whch c e voded y folc cd The eoe vle w the fetl heght of oue ( llete) (Blh et l []; d Lev et l []) We ftted the WNB WG d Weull dtuto to the dt ug the u lelhood ethod fo ete etto The coutto wee efoed ug the uoute NLMed SAS The covegece w cheved ug the eetezto λ d β e ( β) [ e( β) ] to gutee the ette of β ( )

23 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 47 Tle MLE of the odel ete fo the clofo dt the coeodg SE (gve ethee) d oe tttcl eue Model λ β AIC CAIC BIC WNB (34) (939) (68) (3498) WG (6) (679) - (8898) Weull (8) (956) - - Tle lt the MLE (the coeodg tdd eo e ethee) of the ete fo the ftted WNB WG d Weull odel d the vlue of the followg tttc: AIC (Ae Ifoto Cteo) BIC (Bye Ifoto Cteo) d CAIC (Cotet Ae Ifoto Cteo) Thee eult dcte tht the WNB d WG odel hve the lowet vlue fo the AIC d BIC tttc eectvely og the ftted odel d theefoe they could e choe the et odel I ode to e f the odel ote we lot Fgue 8() the ecl d etted uvvl fucto of the WNB WG d Weull dtuto Fgue 8() gve the htog of the dt d the ftted WNB WG d Weull dtuto The lot dcte tht the ft two dtuto ovde good ft fo thee dt () () Fgue 8 () Etted uvvl fucto fo the ftted WNB WG d Weull dtuto d the ecl uvvl fo clofo dt () Etted dete of the WNB WG d Weull odel fo clofo dt

24 48 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA Aoe dt The dt eeetg e te ( h) fo oe coucto tceve wee ft lyzed y Vo Alve [6] ug two-ete log-ol dtuto Thee dt wee elyzed y Chh d Fol [6] ug twoete vee Gu dtuto d y Koutouvel et l [] ug the vee Gu dtuto wth thee ete We ftted the WNB WG d Weull dtuto to the dt ug the u lelhood ethod fo ete etto The coutto wee efoed ug the uoute NLMed SAS Tle 3 lt the MLE (the coeodg tdd eo e ethee) of the ete fo the ftted WNB WG d Weull odel d the vlue of the AIC BIC d CAIC tttc Thee eult dcte tht the WNB odel yeld the lowet vlue fo thee tttc og the ftted odel d the t could e choe the et odel Tle 3 MLE of the odel ete fo the oe dt the coeodg SE (gve ethee) d the AIC eue Model λ β AIC CAIC BIC WNB (3) (464) (95) (6466) WG (468) (84) - (685) Weull (55) (96) - - The LR tttc fo tetg the hyothee H : veu H : H ot tue e to coe the WNB d WG egeo odel w { ( 98 8)} 4 5 ( - vlue 339) It dcte tht the ooed odel ueo to the WG odel te of odel fttg I ode to e f the odel ote the ecl d etted uvvl fucto of the WNB WG d Weull dtuto e lotted Fgue 9() I Fgue 9() we lot the htog of the dt d the ftted WNB WG d Weull dtuto We coclude tht the ew dtuto ovde good ft fo thee dt

25 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 49 () Fgue 9 () Etted uvvl fucto fo the ftted WNB WG d Weull dtuto d the ecl uvvl fo oe dt () Etted dete of the WNB WG d Weull odel fo oe dt Multly ceoed ely dt A lcto of the LWNB egeo odel we code the dt gve Tle 4 lyzed y Nelo ([4 6]) d coceg to tet dt o oducto ely (thoud of cycle) The flue te of oevto w defed the thoud of cycle d v deote the thee level of oducto (6 6 d 8 ) The oectve to coe the level of oducto elto to thoud of cycle The odel codeed the ly deced y () y γ γv σz whee the do vle y log follow (9) fo 35 Tle 4 Dt o oducto ely Poducto Thoud of cycle 6 : : : The yol dcte ceog

26 5 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA Tle 5 lt the MLE of the ete fo the LWNB d LWG egeo odel ftted to thee dt ug the NLMed ocedue SAS The covegece w cheved ug the e-etezto β e( β ) [ e( β )] to gutee the ette of β ( ) A tl vlue fo the ete γ d σ the tetve zto oce of the log-lelhood fucto 33 we ued the ftted vlue oted wth the log-weull egeo odel Thee eult dcte tht the ew egeo odel h the lowet vlue fo the AIC CAIC d BIC tttc og the ftted odel d the t could e choe the et odel Fo the ftted LWNB egeo odel v gfct t % d the thee gfct dffeece og the level of the oducto fo thoud of cycle The LR tttc fo tetg the hyothee H : veu H : H ot tue e to coe the LWNB d LWG egeo odel w { 95 ( 5)} 5 3 ( -vlue 3) wht dcte tht the LWNB egeo odel ueo to the LWG egeo odel te of odel fttg Tle 5 MLE of the ete fo the LWNB egeo odel ftted to the ely dt the coeodg SE (gve ethee) -vlue [] d the AIC eue Model β σ v v AIC CAIC BIC LWNB (688) (6898) (5) (3) () [<] [<] LWG (7556) (689) (39) (47) [<] [<] A ghcl coo etwee the LWNB d LWG odel gve Fgue () d () Thee lot ovde the ecl uvvl fucto d the etted uvvl fucto gve y (34) Bed o thee lot t evdet tht the LWNB odel ft well to thee dt Fo Fgue we ote tht thee dffeece of the level 6 fo the level 6 d 8

27 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 5 () () Fgue Etted uvvl fucto d the ecl uvvl: () LWNB egeo odel veu K-M () LWG egeo odel veu K-M 3 Cocluo Fo the ft te we toduce the Weull egtve ol (WNB) dtuto d tudy oe of t tuctul oete The ew dtuto geelze oe dtuto tuded ecetly the ltetue It ott odel fo ly of lfete dt ecue of the wde uge of the Weull dtuto d the fct tht the cuet geelzto ovde e of t cotuou eteo to tll oe cole tuto We ovde coeheve decto of oe tuctul oete of the ooed dtuto wth the hoe tht t wll ttct wde lcto evel feld The WNB dety fucto c e eeed tue of Weull dety fucto Th eult llow u to deve oe eo fo the ody fctol d vee oet d oet geetg fucto The dety fucto of the WNB ode tttc c e wtte te of fte le coto of Weull dety fucto We clculte e devto Bofeo d Loez cuve ellty d Rév etoy d ot two eeetto fo the oet of ode tttc The etto of ete oched y the ethod of u lelhood d the oeved foto t deved We ooe ew egeo odel ed o the logth of the WNB dtuto The uefule of the ew odel llutted thee el dt et ug clcl cteo The ooed odel ovde the flele ech fo fttg wde ectu of otve el dt et

28 5 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA Aed A R ct to ulte dt fo the WNB dtuto Ft ulto (ltl(lltrue)) eteed(45) 5 5 et v < uf ( ) < log( et ( ( ( et )) v ( et) )) f < dety ht( fefalse yl f() l ) le(f col ed ) Secod ulto eteed(45) 9 4 et 3 v < uf ( ) < log( et ( ( ( et )) v ( et) )) f < dety ht( fefalse yl f() l ) le(f col ed )

29 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 53 Aed B The eleet of the oeved foto t [ ] β κ e ( βe ) κ ( ) log log( ) βe ( βe ) ( βe ) κ β e ( βe ) κβ ( ) e β ( βe ) κ κ [ log( )] ( ) [ log( )] β e [ βe ] ( βe ) β log e ( βe ) log κβ ( ) e β ( βe ) κ [ log( β) ] ( β) [( β) ]

30 54 RODRIGUES CORDEIRO DEMÉTRIO d ORTEGA κ κ β β e ( βe ) ( β) [ log( β) ][( β) ] [( β) log( β)] [( β) ] β β β β ( ) ( e ) ( βe ) ( β) [ ( ) ( β) ] [( β) ] Refeece [] K Add d S Lou A lfe te dtuto wth deceg flue te Sttt Po Lett 39 (998) 35-4 [] N Blh V Lev A Shuez d E Ce Mtue vee Gu dtuto d t tfoto oet d lcto Stttc 43 (9) 9-4 [3] H M Bt d Y H Adelde Coutg the oet of ode tttc fo odetcl do vle Stt Method Al 3 (4) 5-6 [4] W Beto-Souz A L de Mo d G M Codeo The Weull-geoetc dtuto J Stt Cout Sul 8(5) () [5] E M P Beet F Louzd-Neto d M A P Fco A dtução Weull- Poo SINAPE Sóo Ncol de Poldde e Ettítc 9 São Pedo- SP [6] R S Chh d J L Fol The vee Gu dtuto lfete odel Techoetc 9 (977) [7] P C Coul d F Foye Lgg Polty Dtuto Bhäue Boto 6 [8] I S Gdhtey d I M Ryzh Tle of Itegl See d Poduct Acdec Pe S Dego [9] J F Keey d E S Keeg Mthetc of Stttc Pt 3d ed - Pceto New Jeey 96 [] I A Koutouvel G C Cvo d S G Met Etto the theeete vee Gu dtuto Cout Sttt Dt Al 49 (5) 3-47

31 THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION 55 [] C Ku A ew lfete dtuto Cout Sttt Dt Al 5 (7) [] V Lev M Bo d G A Pul Geelzed Bu-Sude Model Ug R XI Ecol de Modelo de Regeão Recfe Bzl 9 [3] J J A Moo A utle ltetve fo uto J Roy Sttt Soc Se D The Stttc 37 (998) 5-3 [4] W B Nelo Acceleted Tetg Stttcl Model Tet Pl d Dt Aly Joh Wley & So New Jeey 4 [5] A P Pudov Y A Bychov d O I Mchev Itegl d See Volue d 3 Godo d Bech Scece Pulhe Ated 986 [6] W H Vo Alve Rellty Egeeg y ARINC Petce-Hll Ic Eglewood Clff NJ 964