International Journal of Pure and Applied Sciences and Technology

Transcription

1 I J Pu Appl Sc Tchol 8 pp 59-7 Iaoal Joual o Pu ad Appld Sccs ad Tchology ISSN 9-67 Avalabl ol a wwwopaasa Rsach Pap Tasmud Quas Ldly Dsbuo: A Galzao o h Quas Ldly Dsbuo I Elbaal ad M Elgahy * Isu o Sascal Suds ad Rsach Dpam o Mahmacal Sascs Cao Uvsy * Cospodg auho -mal: _lbaal@sacudug Rcvd: 7-7-; Accpd: 5-9- Absac: Th Ldly dsbuo s o o h mpoa o sudyg sss--sgh lably modlg Bsds som sachs hav poposd w classs o dsbuos basd o modcaos o h quas Ldly dsbuo I hs pap a w galzd vso o hs dsbuo whch s calld h asmud quas Ldly dsbuo s oducd A comphsv mahmacal am o h TL dsbuo s povdd W dv h h mom ad mom gag uco hs dsbuo Moov w dscuss h las squas wghd las squas ad h mamum lklhood smao o hs dsbuo Kywods: Quas Ldly dsbuo Hazad uco Moms Mamum lklhood smao Ioduco ad Movao Quas Ldly dsbuo wh paams ad s dd by s pobably dsy uco pd g ; > > > I ca asly b s ha a h QLD Equao ducs o h Ldly dsbuo 958 wh pobably dsy uco g ; > >

2 I J Pu Appl Sc Tchol ad a ducs o h gamma dsbuo wh paams Th pd Equao ca b show as a mu o poal ad gamma dsbuos as ollows g pg p g Wh p g ad g Th cumulav dsbuo uco cd o QLD s obad as G > > > Wh s scal paam Ghay al 8a hav dscussd vaous pops o hs dsbuo ad showd ha may ways Equao povds a b modl o som applcaos ha h poal dsbuo A dsc vso o hs dsbuo has b suggsd by Dz ad Oda havg s applcaos cou daa lad o suac Sakaa 97 obad h Ldly mu o Posso dsbuo Ghay al 8b c obad sz-basd ad zo-ucad vso o Posso- Ldly dsbuo ad dscussd h vaous pops ad applcaos Ghay ad Almua 9 dscussd as vaous smao mhods o h dsc Posso- Ldly dsbuo Bakouch al obad a dd Ldly dsbuo ad dscussd s vaous pops ad applcaos Mazuchl ad Achca dscussd h applcaos o Ldly dsbuo o compg sks lm daa Rama ad Msha sudd quas Ldly dsbuo Ghay al dvlopd a wo-paam wghd Ldly dsbuo ad dscussd s applcaos o suvval daa Zakzadah ad Dola obad a galzd Ldly dsbuo ad dscussd s vaous pops ad applcaos Tasmuao Map I hs subsco w dmosa asmud pobably dsbuo L F ad F b h cumulav dsbuo ucos o wo dsbuos wh a commo sampl spac Th gal ak asmuao as gv 7 s dd as G u F F u ad G u F F u R No ha h vs cumulav dsbuo uco also kow as qual uco s dd as F y F y o y R { } [ ] Th ucos G R u ad G R u boh map h u val o sl ad ud suabl assumpos a muual vss ad hy sasy G ad G A quadac Rak Tasmuao Map QRTM s dd as R R I [ ] R G R u u u u Fom whch ollows ha h cd's sasy h laoshp

3 I J Pu Appl Sc Tchol F F F 4 Whch o dao ylds [ F ] 5 Wh ad a h cospodg pds assocad wh cd F ad F spcvly A sv omao abou h quadac ak asmuao map s gv Shaw al 7 Obsv ha a w hav h dsbuo o h bas adom vaabl Th ollowg Lmma povd ha h uco gv 5 sass h popy o pobably dsy uco Lmma: gv 5 s a wll dd pobably dsy uco Poo: Rwg as F w obsv ha s ogav W d o show ha h gao ov h suppo o h adom vaabl s qual o Cosd h cas wh h suppo o s I hs cas w hav [ ] d [ F ] d d F d Smlaly oh cass wh h suppo o h adom vaabl s a pa o al l ollows Hc s a wll dd pobably dsy uco W call h asmud pobably dsy o a adom vaabl wh bas dsy Also o ha wh h Ths povs h qud sul May auhos dalg wh h galzao o som wll- kow dsbuos Ayal ad Tsokos 9 dd h asmud galzd m valu dsbuo ad hy sudd som basc mahmacal chaacscs o asmud Gumbl pobably dsbuo ad has b obsvd ha h asmud Gumbl ca b usd o modl clma daa Also Ayal ad Tsokos psd a w galzao o Wbull dsbuo calld h asmud Wbull dsbuo Rcly Ayal poposd ad sudd h vaous sucual pops o h asmud Log- Logsc dsbuo ad Muhammad kha ad kg oducd h asmud modd Wbull dsbuo whch ds c dvlopm o asmud Wbull dsbuo by Ayal al ad hy sudd h mahmacal pops ad mamum lklhood smao o h ukow paams Elbaal psd Tasmud Modd Ivs Wbull Dsbuo Th s o h pap s ogazd as ollows I Sco w dmosa asmud pobably dsbuo h hazad a ad lably ucos o dsbuo I Sco w sudd h sascal pops clud qual ucos moms mom gag uco Th dsbuo o od sascs s pssd Sco 4 Th las squas ad wghd las squas smaos a oducd Sco 5 Fally I Sco6 w dmosa h mamum lklhood smas o h ukow paams

4 I J Pu Appl Sc Tchol Tasmud Quas Ldly Dsbuo I hs sco w sudd h asmud quas Ldly dsbuo Now usg ad w hav h cd o asmud Ldly dsbuo F Wh scal paam ad s h asmud paam Th scos quao o h valus o ad a always h sam Th pobably dsy uco pd o h asmud Ldly dsbuo s gv by Th lably uco RF o h asmud quas Ldly dsbuo s dod by R also kow as h suvvo uco ad s dd as F R I s mpoa o o ha R F O o h chaacsc lably aalyss s h hazad a uco HF dd by [ ] [ ] [ ][ ] 4 F h I s mpoa o o ha h us o h s h pobably o alu p u o m dsac o cycls Ths alu as a dd wh d chocs o paams Th cumulav hazad uco o h asmud quas Ldly dsbuo s dod by H ad s dd as

5 I J Pu Appl Sc Tchol H l 5 I s mpoa o o ha h us o H s h cumulav pobably o alu p u o m dsac o cycls w ca show ha Fo all choc o paams h dsbuo has h dcasg pas o cumulav saaous alu as Sascal Pops Ths sco s dvod o sudyg sascal pops o h dsbuo spccally qual uco moms ad mom gag uco Qual Fuco Th qh qual q o h asmud quas Ldly dsbuo ca b obad om as 4q W smula h dsbuo by solvg h ola quao Wh u has h uom U dsbuo Moms 4u I hs subsco w dscuss h h mom o dsbuo Moms a cssay ad mpoa ay sascal aalyss spcally applcaos I ca b usd o sudy h mos mpoa aus ad chaacscs o a dsbuo g dcy dspso skwss ad kuoss Thom : I X has h h h mom o X s gv by h ollowg Poo: L X b a adom vaabl wh dsy uco Th dsbuo s gv by h oday mom o h

6 I J Pu Appl Sc Tchol { II I d d d X E Wh 4 d d d I Smlaly 5 d II Subsug om 4 ad 5 o w g Whch compls h poo Basd o h s ou moms o h dsbuo h masus o skwss Φ A ad kuoss Φ k o h dsbuo ca obad as [ ] 6 Φ A ad

7 I J Pu Appl Sc Tchol [ ] Φ k Mom Gag Fuco I hs subsco w dvd h mom gag uco o dsbuo Thom : I X has dsbuo h h mom gag uco M X has h ollowg om 8 M X Poo: W sa wh h wll kow do o h mom gag uco gv by 9 II I d d E M X Wh d I Ad d d d II Subsug om ad o 9 w g M X Whch compls h poo

8 I J Pu Appl Sc Tchol Dsbuo o h Od Sascs I hs sco w dv closd om pssos o h pds o h h od sasc o h dsbuo also h masus o skwss ad kuoss o h dsbuo o h h od sasc a sampl o sz o d chocs o ; a psd hs sco L X X X b a smpl adom sampl om dsbuo wh pd ad cd gv by ad spcvly L X X X do h od sascs obad om hs sampl W ow gv h pobably dsy uco o X : say : ad h moms o X : Tho h masus o skwss ad kuoss o h dsbuo o h X : a psd Th pobably dsy uco o X : s gv by : B [ F ] [ F ] 4 Wh F ad a h cd ad pd o h dsbuo gv by spcvly ad B s h ba uco sc < F < o by usg h bomal ss paso o [ ] W hav F gv by [ F ] [ F ] 4 : B [ F Φ ] 4 Subsug om ad o 4 w ca pss h k h oday mom o h h od sascs X : say E X k : as a l combao o h k h moms o h dsbuo wh d shap paams Tho h masus o skwss ad kuoss o h dsbuo o X : ca b calculad 5 Las Squas ad Wghd Las Squas Esmaos I hs sco w povd h gsso basd mhod smaos o h ukow paams o h asmud quas Ldly dsbuo whch was ogally suggsd by Swa Vkaama ad Wlso 988 o sma h paams o ba dsbuos I ca b usd som oh cass also Suppos Y Y s a adom sampl o sz om a dsbuo uco G ad suppos Y ; dos h odd sampl Th poposd mhod uss h dsbuo o G Y Fo a sampl o sz w hav

9 I J Pu Appl Sc Tchol ;o ad k k G Y G Y Cov G Y V E G Y k < S Johso Koz ad Balaksha 995 Usg h pcaos ad h vaacs wo vaas o h las squas mhods ca b usd Mhod Las Squas Esmaos: Oba h smaos by mmzg 5 Y G Wh spc o h ukow paams Tho cas o dsbuo h las squas smaos o ad say LSE LSE ad LSE spcvly ca b obad by mmzg Wh spc o ad Mhod Wghd Las Squas Esmaos: Th wghd las squas smaos ca b obad by mmzg 5 G Y w Wh spc o h ukow paams wh G Y V w Tho cas o dsbuo h wghd las squas smaos o ad say WLSE WLSE ad WLSE spcvly ca b obad by mmzg w wh spc o h ukow paams oly

10 I J Pu Appl Sc Tchol Esmao ad Ic I hs sco w dm h mamum lklhood smas MLEs o h paams o h dsbuo om compl sampls oly L X X X b a adom sampl o sz om Th lklhood uco o h vco o paams Φ ca b w as 6 L Φ Π Φ Π Π Takg h log-lklhood uco o h vco o paams Φ w g 6 log log log log log L Th log-lklhood ca b mamzd h dcly o by solvg h ola lklhood quaos obad by dag 6 Th compos o h sco vco a gv by { } 6 log L { } 64 log L Ad { } 65 log L W ca d h smas o h ukow paams by mamum lklhood mhod by sg

11 I J Pu Appl Sc Tchol hs abov o-la quaos 6 ad 65 o zo ad solv hm smulaously Tho w hav o us mahmacal packag o g h MLE o h ukow paams Also all h scod od dvavs s Thus w hav h vs dspso ma s gv by Wh V V V N V V V V V V V V V V V V V E V V V 66 V V L V L V L V L L By solvg hs vs dspso ma hs soluos wll yld asympoc vaac ad covaac's o hs ML smaos o ad Usg 66 w appoma γ % codc vals o ad a dmd spcvly as ± z V ± z V ad ± z V γ γ γ wh z γ s h upp γ h pcl o h sadad omal dsbuo Rcs [] GR Ayall ad CP Tsokos Tasmud Wbull dsbuo: A galzao o h Wbull pobably dsbuo Euopa Joual o Pu ad Appld Mahmacs [] GR Ayall ad CP Tsokos O h asmud m valu dsbuo wh applcaos Nola Aalyss: Thoy Mhods ad Applcaos [] GR Ayall Tasmud log-logsc dsbuo J Sa Appl Po - [4] HS Bakouch BM Al-Zaha AA Al-Shoma VAA Mach ad F Louzada A dd Ldly dsbuo J Koa Sa Soc [5] EG Dz ad EC Oda Th dsc Ldly dsbuo-pops ad applcaos J Sa Compu Smul [6] ME Ghay B Ah ad S Nadaaah Ldly dsbuo ad s applcaos Mah Compu Smul [7] ME Ghay ad DK Al-Mua Esmao mhods o h dsc Posso-Ldly dsbuo J Sa Compu Smul 79 8a -9 [8] ME Ghay ad DK Al-Mua Sz-basd Posso-Ldly dsbuo ad s applcaos Mo- I J Sa LXVI 8b 99- [9] ME Ghay DK Al-Mua ad S Nadaaah Zo-ucad Posso-Ldly dsbuo

12 I J Pu Appl Sc Tchol ad s applcaos Mah Compu Smul 79 8c [] ME Ghay F Al-qalla DK Al-Mua ad HA Hussa A wo paam wghd Ldly dsbuo ad s applcaos o suvval daa Mah Compu Smul [] DV Ldly Fducal dsbuos ad Bays' hom J Royal Sa Soc Ss B [] J Mazuchl ad JA Achca Th Ldly dsbuo appld o compg sks lm daa Compu Mhods Pogams Bomd [] MS Kha ad R Kg Tasmud modd Wbull dsbuo: A galzao o h modd Wbull pobably dsbuo Euopa Joual o Pu ad Appld Mahmacs [4] M Sakaa Th dsc Posso-Ldly dsbuo Bomcs [5] W Shaw ad I Buckly Th alchmy o pobably dsbuos: byod Gam- Chal pasos ad a skw- kuoc- omal dsbuo om a ak asmuao map axv pp axv [6] H Zakzadah ad A Dola Galzd Ldly dsbuo J Mah E - 5