The Odd Generalized Exponential Modified. Weibull Distribution

Transcription

1 Itatoal Mathmatcal oum Vol. 6 o HIKARI td Th Odd Galzd Epotal Modd Wbull Dstbuto Yassm Y. Abdlall Dpatmt o Mathmatcal Statstcs Isttut o Statstcal Studs ad Rsach Cao Uvsty Egypt Copyght 6 Yassm Y. Abdlall. Ths atcl s dstbutd ud th Catv Commos Attbuto cs whch pmts ustctd us dstbuto ad poducto ay mdum povdd th ogal wo s poply ctd. Abstact I ths pap w popos a w dstbuto calld th odd galzd potal modd Wbull dstbuto. Som mathmatcal popts o th w dstbuto a studd. Th mthod o mamum llhood s usd o stmatg th modl paamts ad th obsvd sh's omato mat s dvd. W llustat th usulss o th poposd modl by applcato to al data. Kywods: Modd Wbull dstbuto momts mamum llhood stmato od statstcs. Itoducto Th modd Wbull MW dstbuto s o o th most mpotat dstbutos ltm modlg ad som wll-ow dstbutos such as th potal Raylgh la alu at ad Wbull dstbutos a spcal cass o t. Ths dstbuto was toducd by a ad Muthy 3 to whch w th ad o a dtald dscusso as wll as applcatos o th MW dstbuto patcula th us o th al data st pstg alu tms to llustat th modlg ad stmato pocdu. Also Saha ad Zad 9 toducd th modd Wbull dstbuto. It ca b usd to dscb sval lablty modls. It has th paamts two scal ad o shap paamts. Rctly Caasco t al. 8 tdd th MW dstbuto by addg aoth shap paamt ad toducg a ou paamt galzd MW GMW ad log-gmw GMW.

2 944 Yassm Y. Abdlall Rctly Tah t al. 5 poposd a w class o uvaat dstbutos calld th odd galzd potal OGE amly ad studd ach o th OGE- Wbull OGE-W dstbuto th OGE-écht OGE- dstbuto ad th OGE-Nomal OGE-N dstbuto. Ths mthod s lbl bcaus th hazad at shaps could b casg dcasg bathtub ad upsd dow bathtub. I ths atcl w pst a w dstbuto om th odd galzd potal dstbuto ad modd Wbull dstbuto calld th Odd Galzd Epotal-Modd Wbull OGE-MW dstbuto usg w amly o uvaat dstbutos poposd by Tah t al. 5. A adom vaabl s sad to hav galzd potal GE dstbuto wth paamts th cumulatv dstbuto ucto cd s gv by. Th Odd Galzd Epotal amly by Tah t al. 5 s dd as ollows. t G s th cd o ay dstbuto dpds o paamt ad thus th suvval ucto s G G th th cd o OGE-amly s G dd by placg CD o GE Equato by to gt G G G. Wh a two addtoal paamts. Ths pap s outld as ollows. I Scto w d th cumulatv dstbuto ucto dsty uctolablty ucto ad hazad ucto o th Odd Galzd Epotal-Modd Wbull OGE-MW dstbuto. I Scto 3 w toduc th statstcal popts clud th quatl ucto th mda adth momts. Scto 4 dscusss th dstbuto o th od statstcs o OGE-MW dstbuto. Moov mamum llhood stmato o th paamts s dtmd Scto 5. ally a applcato o OGE-MW usg a al data st s pstd Scto 6.. Th OGE-MW Dstbuto. OGE-MW spccatos I ths scto w d w v paamts dstbuto calld Odd Galzd Epotal-Modd Wbull dstbuto wth paamts ad wtt as OGE-MWΘ wh th vcto Θ s dd by Θ =.

3 Odd galzd potal modd Wbull dstbuto 945 A adom vaabl s sad to hav OGE-MW wth paamts ad ts cumulatv dstbuto ucto cd gv as ollows. wh a scal paamts ad a shap paamts. Hc th cospodg pobablty dsty ucto pd s. 3. Suvval ad hazad uctos I a adom vaabl has cd th th cospodg suvval ucto s gv by. S Th hazad ucto o OGE-MW s dd as ollow. S h gu ad 3 llustats som o th possbl shaps o th pd cd ad hazad ucto o OGE-MW dstbuto o som valus o th paamts ad

4 946 Yassm Y. Abdlall gu. Th pd s o vaous OGE-MW dstbutos gu.th cd o vaous OGE-MW dstbutos. 4 h t h t h t h t h t t gu 3. Th hazad ucto o vaous OGE-MW dstbutos.

5 Odd galzd potal modd Wbull dstbuto 947 Not that thoge-mw dstbuto s vy lbl modl that appoachs to dt dstbutos wh ts paamts a chagd. Th OGE-MW dstbuto cotas as spcal-modls wth th ollowg wll ow dstbutos. I patcula o w hav th odd galzd potal- Wbull OGE-W dstbuto as dscussd Tah t al. 5. Th odd galzd potal-potaloge-e dstbuto s claly a spcal cas o ad as dscussd Mat ad Pama 5.o / ad w hav th odd galzd potal-la alu at OGE- R dstbuto as dscussd El-Damcs t al. 5. Wh ad th th sultg dstbuto s th odd galzd potal-raylgh OGE-R dstbuto. 3. Statstcal Popts Ths scto s dvotd o studyg somstatstcal popts o th odd galzd potal-modd Wbull OGE-MW spccally quatl mda ad th momts. 3. Quatl ad Mda o OGE-MW Th quatl ucto q o OGE-MWΘ dstbuto s gv by usg q q 4 Substtutg om to 4 q s th al soluto o th ollowg quato q q l q l q Th abov quato has o closd om soluto q so w hav to us a umcal tchqu such as a Nwto- Raphso mthod to gt th quatl. By puttg q. 5 Equato 5 w ca gt th mda o odd galzd potal modd Wbull dstbuto. 3. Momts Momts a cssay ad mpotat ay statstcal aalyss spcally applcatos. It ca b usd tostudy th most mpotat atus ad chaactstcs o a dstbuto.g. tdcy dspso swss ad utoss. I ths subscto w wll dv th th momts o th OGE-MWΘ dstbuto as t ss paso. 5

6 948 Yassm Y. Abdlall Thom. Th th momt o a adom vaabl ~ OGE-MWΘ wh Θ = s gv ' Poo: Th th momt o a adom vaabl wth pd s dd by ' d 6 Substtutg om 3 to 6 w obta. ' d 7 Sc o w obta. 8 Substtutg om 8 to 7 w gt. ' d Usg ss paso o w obta. ' d Usg bomal paso o w obta

7 Odd galzd potal modd Wbull dstbuto 949 '. d Usg ss paso o w obta '. d d By usg th dto o gamma ucto th om s Zwllg 4. z dt t z z t z ally w obta th th momt o OGE-MW as ollows. ' Ths complts th poo. 4. Od Statstcs t... b a smpl adom sampl o sz om OGE-MWΘwth cumulatv dstbuto ucto ad pobablty dsty ucto gv by ad 3 spctvly. t : : :... dot th od statstcs obtad om ths sampl. Th pobablty dsty ucto o : s gv by : B 9

8 95 Yassm Y. Abdlall wh ad a th pd ad cd o OGE-MWΘ dstbuto gv by ad 3 spctvly ad B.. s th bta ucto also w d st od statstcs... m : ad th last od statstcs as... ma :. Sc o w ca us th bomal paso o gv as ollows. Substtutg om to 9 w obta. : B Substtutg om ad 3 to w obta. : Rlato shows that : s th wghtd avag o th odd galzd potal-modd Wbull wth dt shap paamts. 5. Estmato ad Ic Now w dscuss th stmato o th OGE-MW paamts by usg th mthod o mamum llhood basd o a complt sampl. 5. Mamum llhood stmatos t... b a adom sampl o sz om OGE-MWΘ wh Θ = th th llhood ucto l o ths sampl s dd as. l 3 Substtutg om 3 to 3 w gt. l Th log-llhood ucto bcoms:

9 Odd galzd potal modd Wbull dstbuto 95. l l l l 4 Th mamum llhood stmats o th paamts a obtad by Dtatg th log-llhood ucto wth spct to th paamts ad sttg th sult to zo. l l l l l 8 ad l 9 Wh th ola uctos ad a gv by.

10 95 Yassm Y. Abdlall om quato 9 w obta th mamum llhood stmat o a closd om as ollow. l Substtutg om to ad 8 w gt th MEs o by solvg th ollowg systm o o-la quatos l l l l l wh ad. Ths quatos caot b solvd aalytcally ad statstcal sotwa ca b usd to solv th quatos umcally. W ca us tatv tchqus such as Nwto Raphso typ algothm to obta th stmat.

11 Odd galzd potal modd Wbull dstbuto Asymptotc codc bouds I ths subscto w dv th asymptotc codc tvals o th uow paamts ad. As th sampl sz th appoachs a multvaat omal vcto wth zo mas ad aac mat I wh I s th vs o th obsvdomato mat whch dd as ollows I Va Va Va Va Va Th scod patal dvatvs cludd I a gv as ollows

12 954 Yassm Y. Abdlall l

13 Odd galzd potal modd Wbull dstbuto 955 l l l l l l l

14 956 Yassm Y. Abdlall l l l l l l l l l wh ad. Th asymptotc % codc tvals o ad a Va z Va z Va z Va z ad

15 Odd galzd potal modd Wbull dstbuto 957 z Va spctvly wh stadad omal dstbuto. 6. Data Aalyss z s th upp th pctl o th I ths scto w pom a applcato to al data to llustat that th OGE- MW ca b a good ltm modl compag wth may ow dstbutos such as th Epotal E Galzd Epotal GE a alu Rat R ad Wbull W. Cosd th data hav b obtad om Aast [] ad wdly potd may ltatus. It psts th ltms o 5 dvcs ad also possss a bathtub-shapd alu at popty Tabl. Tabl : Th data om Aast [] Th mamum llhood stmats MEs o th uow paamts o th v modls s gv Tabl. Tabl.MEs o th paamts Th Modl E GE R W OGEMW ME o th paamts Th valus o log-llhood uctos - Aa Iomato Cta AIC Baysa Iomato Cta BIC ad th Cosstt Aa Iomato Cta CAIC a gv Tabl 3 o th v modls od to vy whch dstbuto ts btt to ths data.

16 958 Yassm Y. Abdlall Tabl 3.Th - AIC BIC ad CAIC o dvcs data. Th Modl Masus AIC BIC CAIC E GE R W OGE-MW Basd o Tabl ad 3 t s show that OGE-MW modl povd btt t to th data ath tha oth dstbutos whch w compad wth bcaus t has th smallst valu o AIC BIC CAIC. Substtutg th MEs o th uow paamts to w gt stmato o th vaac aac mat as th ollowg: I Th appomat 95% two sdd codc tvals o th uow paamts ad a [.74.78] [.43] [.] [ ] ad [ ] spctvly. 7. Coclusos I ths pap w hav toducd a w v-paamt modl calld odd galzd potal modd Wbull OGE-MW dstbuto ad studd ts dt popts. Som statstcal popts o ths dstbuto hav b dvd ad dscussd. W povd th pd th cd ad th hazad at ucto o th w modl also w povd a plct psso o th momts. Th dstbutos o th od statstcs a dscussd. Both pot ad asymptotc codc tval stmats o th paamts a dvd usg mamum llhood mthod ad w obtad th obsvd sh omato mat. W us applcato o st o al data to compa th OGE-MW wth oth ow dstbutos such as Epotal E Galzd Epotal GE a alu Rat R ad Wbull W. Applcatos o st o al data showd that th OGE-MW s th bst dstbutoo ttg ths data sts compad wth oth dstbutos cosdd ths atcl.

17 Odd galzd potal modd Wbull dstbuto 959 Rcs [] M. V. Aast How to dty a bathtub hazad at IEEE Tasactos o Rlablty o [] J. M. Caasco E. M. Otga ad G. M. Codo A galzato modd wbull dstbuto o ltm modlg Computatoal Statstcs ad Data Aalyss 53 8 o [3] M. A. El-Damcs A. Mustaa B. S. El-Dsouy ad M. E. Mustaa Th odd galzd potal la alu at dstbuto Joual o Statstcs Applcatos & Pobablty [4] C. D. a M. ad D.N.P. Muthy A modd Wbull dstbuto IEEE Tasactos o Rlablty [5] S. S Mat ad S. Pama Odds galzd potal-potal dstbuto Joual o Data Scc [6] A. M. Saha ad M. Zad Modd Wbull dstbuto Appld Sccs [7] M. H. Tah G. M. Codo M. Alzadh M. Masoo M. Zuba ad G. G. Hamda Th odd galzd potal amly o dstbutos wth applcatos Joual o Statstcal Dstbutos ad Applcatos 5 o [8] D. Zwllg Tabl o Itgals Ss ad Poducts Elsv Ic. 4. Rcvd: July 6 Publshd: Octob 3 6