The Generalized Marshall-Olkin-Kumaraswamy-G family of distributions
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1 rzd Mrs-O-Kumrswmy- my o dsruos L Hdqu d Sur Crory* Drm o Sscs Drur Uvrsy Drur-7864 Assm Id *Corrsod Auor. Em: sur_ss@dru.c. Auus 6 Asrc A w my o dsruo s roosd y us Kumrswmy- Cordro d d Csro dsruo s s dsruo rzd Mrs-O Jyumr d Mw 8 Cosruco. A umr o sc css r rsd. By xd roy dsy uco d survv uco s srs roosd my s s s mxurs o Kumrswmy- Cordro d d Csro dsruo. Dsy uco d s srs xsos or ordr sscs r so od. Ordr sscs moms mom r uco Réy roy qu uco rdom sm ro symos ss d socsc ordrs r so vsd. mods o rmr smo y mod o mxmum ood d mod o mom r rsd. Lr sm sdrd rror d codc rvs or ms r so dscussd. O r co o comrv d w som o mor su mods o my d som or mods s cosdrd. Ky words: Mrs - O Kumrswmy- dsruo rzd Mrs-O my Exod my AIC BIC d Powr Wd Moms. Prr
2 . Iroduco r w dsruos sr w s dsruo y dd o or mor ddo rmrs rou vrous mcsms s r o rsrc d o roy dsruo wc v s o o wor o. sc movo o s wors s o r mor xy mod dr y o d rd rom r suo. Rcy r s rwd cvy s r o roos d vs w ms o dsruos. A rc rvw r y r. 5 rovds d ccou o vrous our cqus o r w ms o uvr couous dsruos rou roduco o ddo rmrs. Vry rc coruos s cud Kumrswmy Mrs-O my roosd y Azd. 5 d Mrs - O Kumrswmy- my roducd y Hdqu d Crory 5 mo ors. I s rc w roos or my o couous roy dsruo y r Kumrswmy- my Cordro d d Csro s s dsruo rzd Mrs O Exdd my Jyumr d Mw 8. s w my rrrd o s rzd Mrs-O Kumrswmy- MOKw my o dsruo s vsd or som s r rors rmr smo d r cos. rs o s rc s orzd sv scos. I sco w ry roduc som mor crcrscs o roy dsruos rzd Mrs-O Jyumr d Mw 8 d Kumrswmy- Cordro d d Csro my o dsruos. roosd w my s dd o w s ysc ss sco 3. Imor sc css o my o w r s d m ry crcrscs r rsd x sco. I sco 5 w dscuss som r rsus o roosd my w dr mods o smo o rmrs o comrv d mod xm r rsd sco 6. rc ds w cocuso sco 7 oowd y dx o drv symoc codc ouds.. Som ormus d oos Hr rs w s som ormus o usd susqu scos o s rc. I s couous rdom vr w d d cd P s Survv uco s: P Hzrd r uco r: / Rvrs zrd r uco rr: r / Cumuv zrd r uco cr: H
3 3 r q Powr Wd Mom PWM: d r q r q Réy roy: d I R.. rzd Mrs-O Exdd MOE my o dsruos Jyumr d Mw 8 roosd rzo o Mrs d O 997 my o dsruos y us Lm scod rv Lm 953 o o s MO o MOE my o dsruos y xoo s o MOE my o dsruos s MO wr d s ddo s rmr. W MO MO d or MO. cd d d o MOE dsruo r rscvy MO d MO 3 Ry msurs r rr d cr ssocd w r MO MO MO r MO MO MO H MO Wr d r rscvy d cd s d r o s dsruo.
4 W do my o dsruo w d s MOE Jyumr d Mw 8 wc or rducs o MOE Mrs d O Kumrswmy- Kw my o dsruos or s cd w d Cordro d d Csro dd Kw dsruo w cd d d 4 d 5 d Wr d r s rmrs ddo o os d s dsruo. s r rr d cr o s dsruo r rscvy v y r d H. 3. rzd Mrs-O Kumrswmy- MOKw my o dsruos W ow roos w xso o MO my y cosdr cd d d o Kw dsruo 4 d 5 s d rscvy MO ormuo 3 d c MOKw dsruo. rsu xrsso or d o MOKw s v y MO 6 cd s r rr d cr o MOKw dsruo r rscvy v y MO d MO 7 r: MO 8 4
5 rr : r MO cr: H MO or w c corrsod xrssos or MOKw dsruo o Hdqu d MO MO Crory 5 Jyumr d Mw 8 d MO or. 3. ss o dsruo or... wr s r... squc o..d. rdom vrs w survv uco d N. N s omrc dsruo w rmr dd o ' s mm... N s dsrud s MOKw - or. N s omrc dsruo w rmr / dd o ' s mmx... Proo. N s dsrud s MOKw -. L N squc o..d. rdom vrs w survv uco d... suos N s omrc dsruo w rmr dd o s. c sy sow W m... N d V mx... N r dsrud s MOKw - d MOKw - rscvy.. or N s omrc dsruo w rmr Pm W W... W P W P W... P W P W s s s o MOKw. MO. or N s omrc dsruo w rmr Pm V V... V 5
6 P V P V... P V P V MO.. S o dsy d zrd ucos Hr w v od d d r o MOKw or som cocs o rmrs o sudy vry o ss ssumd y my. MOKw-E MOKw-W Dsy =5.5=.99==.5=.9 =3==.6=.6=.6 =5=.5=.5=.5=.5 =.45=.45==.5=.5 =.35=.6==.5=.5 =4=3.5=.5=.5=.5 Dsy =.99=.9==.5=.9=.5 ==3.5=.5=.6=.5=.5 =6=3.5=.5=.5=.5=.5 =.5=.45==.5=.5=.5 =.5=.6==.5=.5=.4 =4=9.5=.5=.5=.5= MOKw-L MOKw-r Dsy =4=.5==.5=.5=.5 =3.5=.5==.5=.=.5 ==3.5=.5=.5=.5=.5 =.5=.45==.5=.5=.5 =.5=.6==.5=.5=.4 =33=3.5=.5=.5=.5=.5 Dsy =.3===.5=.9=.5 =.=.==.4=.9=.4 =5=5.5=.9=.=.5=.8 =3.5=5.=.9=.=.5=.7 =.=.4==.5=.9=.5 =.3=.5=.5=.6=.9= c d 6
7 : Dsy os o MOKw r dsruos. MOKw E MOKw W c MOKw L d d MOKw-E MOKw-W =5.5=.4==.5=.5 =45=.6=.9=.9=.4 =.5=.36=.5=.9=.5 =.5=.5=.5==.3 =5=.55=.9==.4 x =5=.4==.5=.6 x =.3=.39=.8=3.=.4=.9 =.6=.49= =3.5=.3=.5 =.8=.69= =.9=.4=.7 =.4=.59=.9=3.5=.3=.5 =.5=.69=.9=3.7=.3=.5 =.6=.9=.8=3.5=.4= MOKw-L MOKw-r =5.3=45.5==.5=.9=.3 =55=.5=.5=.5=.4=.55 =.3=4.5==.5=.9=.5 =.3=.89=3.5=.9=.3=.5 =6=3.5=.5=.5=.5=.5 =.5=.99=.5=.9=.3=.5 =.5=.9=.=4.5=.5=.6 =.6=.59=.9=.39=.4=.5 =.5=.6==.5=.5=.4 =.5=.69=.9=.9=.3=.5 Dsy =45=3.5=.5=.5=.5=.5 x =3=3.5=.5=.5=.4= c d : Hzrd r uco os o MOKw r dsruos. MOKw E MOKw W c MOKw L d d rom os ur d c s my s vry x d c or my dr ys o ss. I ors IR DR v u sd zrd r. 4. Som sc MOKw Som sc css o dsruo MOKw my o dsruos r rsd s sco. 7
8 4. MOKw xo MOKw E dsruo L s dsruo xo w rmr : d : or MOKw E mod w d: MOKwE cd: MOKwE s: r : MOKwE MOKwE rr: cr: r MOKwE H MOKwE 4. MOKw Lomx MOKw L dsruo Cosdr Lomx dsruo y. 7 w rscv d d cd : d : MOKw L dsruo s od w MOKwL d: cd: MOKwL s: MOKwL r: MOKwL rr : r MOKwL 8
9 cr: H MOKwL 4.3 MOKw Wu MOKw W dsruo Cosdr Wu dsruo y. 5 Z d 7 w rmrs d v d d cd MOKw W dsruo w d rscvy w MOKwW d: cd: MOKwW s: MOKwW r: rr : MOKwW r MOKwW cr: H MOKwW 4.4 MOKw rc MOKw r dsruo Suos s dsruo s rc dsruo Krs. 3 w d d cd v y d corrsod d o MOKw r dsruo s od w rscvy d MOKwr d: cd: MOKwr 9
10 s: MOKwr r: MOKwr rr: r MOKwr : cr: H MOKwr 4.5 MOKw omrz o MOKw dsruo Cosdr omrz dsruo sr. 998 w d d cd d rscvy w o MOKw dsruo w d: MOo cd: MOo s: MOo r: MOo rr: r MOo cr: H MOo
11 4.6 MOKw Exdd Wu MOKw EW dsruo xdd Wu EW dsruos o urvc. 997 s dd y cd : x E : D R wr E : s o-v mooocy crs uco wc dds o rmr vcor. corrsod d s : x E : : wr : s o cours drvv o E :. My mor dsruos c od y coos dr xrssos or E :. d: MOKw EW s drvd y cosdr EW s s dsruo w x E : : x E : x E : MOKwEW cd: MOKwEW s: MOKwEW x E : x E : x E : x E : x E : r: MOKwEW x E : : x E : x E : x E : rr: r MOKwEW x E : : x E : x E : x E : cr: H MOKwEW x E : x E : x E :. x E : 4.7 MOKw Exdd Modd Wu MOKw EMW dsruo modd Wu MW dsruo o Sr d Zd 3 s cd d d s v y x d x rscvy. corrsod MOKw EMW s od w
12 d: MOKwEMW x x x x cd: MOKwEMW x x s: MOKwEMW x x r: MOKwEMW x x x x rr : r MOKwEMW x x x x x x cr: H MOKwEMW x x 4.8 MOKw Exdd Exod Pro EEP MOKw dsruo d d cd o xod Pro dsruo o Ndr 5 r v rscvy y d x d. us EEP MOKw dsruo s dd w d: MOKwEEP cd: MOKwEEP s: MOKwEEP r: MOKwEEP
13 rr: r MOKwEEP cr: H MOKwEEP 4.9 MOKw Exdd Powr MOKw EP dsruo cd d d o xdd owr dsruo r rscvy v y d d. corrsod d: MOKw EP dsruo s v y MOKwEP cd: MOKwEP s: MOKwEP r: rr: MOKwEP r MOKwEP cr: H MOKwEP 5. r rsus or rzd Mrs-O Kumrswmy- MOKw my o dsruos I s sco w drv som r rsus or roosd MOKw my smr mr s do Brro-Souz. 3 Cordro. 4 d Azd. 5 mo ors. 5. Exsos W ow z z z d 9 wr. s mm uco. I us 9 6 w o 3
14 4 MO Α A A d d A d d A d d A A Wr rom s ormuo - MOKw c cd o cr Kw Cordro d d Csro dsruo. Smry xso or survv uco o MOKw or c drvs s MO
15 5 A A 3 Arvy w c xd d s MO 4 Wr A Aor xso o dsy uco 6 c od y xrss d s MO Hc or us 9 w C C 5 Wr A C C or survv uco o MOKw c xrssd s
16 6 MO / O us 9 w C Wr C C. 5. Ordr Sscs Suos... s rdom sm rom y MOKw dsruo. L : do ordr sscs. d o : c xrssd s MO MO MO : MO MO MO MO MO Now us r xso o MOKw dsruo d d s w d o ordr sscs or o MOKw or s : A A wr d r dd rr o A A
17 7 M 6 / d d M d d M d d M M 7 Wr A A M d / M M I rcur or 6 rducs o d o ordr sscs or o - MOKw v y : c c c H Wr c c H d c c A us r xso o MOKw dsruo d d s w d o ordr sscs or o MOKw or s C C : C C Wr C C d dd sco 5.
18 8 Now d C Ndr. 5 Wr d C C d ror dsy uco o ordr sscs o MOKw dsruo c xrssd s : C d d C 8 Wr d C Rmr. Equos -8 rv dsy ucos o MOKw dsruo d o s ordr sscs c xrssd s dr Kw Cordro d d Csro dsy mud y owr srs o cd o Kw Cordro d d Csro d so s v om mxur o Kw Cordro d d Csro dsruos udr Lm rvs. s rsus r xcd o y mor ros o o xc xrssos or moms d mom r uco m o MOKw dsruo d os o s ordr sscs r s u d or sc mods us corrsod rsus o Kw Cordro d d Csro dsruos. 5.3 Proy wd moms
19 9 roy wd moms PWMs rs roosd y rwood. 979 r xcos o cr ucos o rdom vr wos m xss. r q PWM o s dd y d r q r q rom quos 4 d 5 s mom o or d c wr r s s s d A E d A s s A or d E s s d B s d B s s B or s s d C E d C s d C s s C Wr d r q r q
20 s PWM o Kw - dsruo. ror moms o MOKw - c xrsss rms o PWMs o Kw - Cordro d d Csro. PWM mod c ry usd or sm rmrs qus o rzd dsruos. s moms v ow vrc d o svr ss d y comr voury w smors od y mxmum ood. Procd s ov w c drv s mom o ordr ssc : rdom sm o sz rom MOKw or d o us quos 6 d 8 s E s M s d E s s Wr A B C d M dd sco 5. d 5. rscvy. 5.4 Mom r uco mom r uco o o xod MOKw my c sy xrssd rms o os Kw Cordro d d Csro dsruo us rsus o sco 5.. or xm us quo c s M s E A s s d d s d s A d d d A M s Wr M s s m o Kw 5.5 Réy Eroy Cordro d d Csro dsruo. roy o rdom vr s msur o ucry vro d s usd vrous suos scc d r. Réy roy s dd y I R d wr d or urrs ds s So. or us xso 9 6 w c wr MO d
21 us or Réy roy o - MOKw c od s d L I R Wr L L A dsy uco 6 c xrssd s MO or us xso 9 w us or Réy roy o MOKw so c drvd s d R I R Wr R R 5.6 Qu uco d rdom sm ro W s ow rs ormu or r MOKw rdom vr y us vrso mod y vr cd or survv uco. MO MO MO MO MO MO MO MO MO
22 MO MO MO MO MO MO o r rdom vr rom MOKw rs r ~ U u us u u 9 Qu or MOKw c sy od rom 9 s or xm s dsruo xo w rmr v d d cd s : d : rscvy. ror Qu o E MOKw s v y 5.7 Asymos d ss Hr w vs symoc ss o roosd my oow mods oowd Azd. 5. Prooso. symos o quos 6 7 d 8 s r v y ~ s ~ s ~ s Prooso 3. symos o quos 6 7 d 8 s r v y ~ s
23 3 ~ s ~ s ss o dsy d zrd r ucos c dscrd ycy. crc os o MOKw dsy uco r roos o quo: d d r my mor o roo o. I s roo o corrsods o oc mxmum oc mmum or o o xo dd o wr or wr d d MO MO MO MO crc os o r roos o quo d d r my mor o roo o. I s roo o corrsods o oc mxmum oc mmum or o o xo dd o wr
24 4 or wr d d MO MO MO MO 5.8 Socsc ordrs I s sco w sudy ry rors d socsc ordr o MOKw dsruos Socsc ordr rors v cos dvrs ds suc s coomcs ry survv yss surc c cur d mm sccs Sd d Sumr 7. L d Y wo rdom vrs w cds d rscvy survv ucos d d corrsod d s. s sd o smr Y ood ro ordr Y r s dcrs socsc ordr Y s or zrd r ordr Y r s dcrs rvrsd zrd r ordr Y rr s dcrs. s our socsc ordrs r rd o c or s Y Y Y Y s r r rr orm: L Y ~MOKw - d ~ MOKw -. I Y r Y Y Y s rr r.
25 5 Proo: Sc d d Wc s wys ss. Hc / s dcrs. s Y r. rm sms oow rom mcos. 6. Esmo 6. Mxmum ood mod mod rmrs o MOKw dsruo c smd y mxmum ood. L... rdom sm o sz rom MOKw w rmr vcor θ wr q... corrsods o rmr vcor o s dsruo. -ood uco or θ s v y θ 3
26 6 s -ood uco c o sovd ycy cus o s comx orm u c mxmzd umrcy y moy o omzo mods v w sowr s R SAS Mmc or y sov or ood quos od y dr 3. By r drvvs o -ood uco w rsc o d w o comos o scor vcor U U U U U U θ U U U U U S s quos o zro U U U U U U θ d sov m smuousy yds mxmum ood sm MLE ˆ ˆ ˆ ˆ ˆ ˆ θ o θ. 6. Asymoc sdrd rror d codc rv or ms: symoc vrc-covrc mrx o MLEs o rmrs c od y vr sr ormo mrx I θ wc c drvd us scod r drvvs o -ood uco w rsc o c rmr. ms o I θ r v y q E 3 I θ xc vuo o ov xcos my cumrsom. I rcc o c sm I θ y osrvd sr s ormo mrx ˆ Î θ s dd s:
27 Î θ θθˆ 3 q Us r ory o MLEs udr som rury codos o rmrs s symoc dsruo o ˆ θ θ s N V wr V v I θ. symoc vour rms vd V s rcd y ˆ V Î ˆ θ. s rsu c usd o rovd r sm sdrd rrors d so cosruc codc rvs or mod rmrs. us roxm sdrd rror d / % codc rv or m o rmr r rscvy v y o o sdrd orm dsruo. ˆ ˆ wr Z / s / vˆ d Z / v As usro o MLE mod s r sm sdrd rrors codc rv cs o MOKw E s dscussd dx. 6.3 Esmo y mod o moms Hr rv mod o smo o rmrs s dscussd. Sc moms r o cosd orm smo y usu mod o moms my o rc. ror oow Brro-Souz. 3 w E d d L u du d L / u u u u du u v dv / / u v v du du u dv u / du v dv 7
28 8 / v / / / or w corrsod rsu o - MOKw Hdqu d Crory 5. / E d d L u d du du u u du u u B or Wr du u u m B x m x s com uco. or w corrsod rsu o - MOKw v v. Hdqu d Crory 5. ror w v
29 E or rdom sm / v B... or / v or v rom ouo w survv uco 7 mod rmrs c smd us 4 y sov quos 6.4 R cos / or v v B or v 3... W cosdr o r d o usr suy o MOKw W dsruo rsd Sco 4.. d s s sus o d rord Br. wc corrsods o survv ms yrs o rou o s v cmory rm o. d coss o survv ms yrs or 46 s r: Hr w v smd rmrs y umrc mxmzo o ood uco d rovdd r sdrd rrors d 95% codc rvs us r sm roc s dx. W v comrd MOKw W dsruo w som o s su mods my Mrs-O Kumrswmy Wu MOKw W Hdqu d Crory 5 Kumrswmy Wu d urr w BKw W Hdqu d Crory 6 d B W Kw d Mrs-O Wu MO W mods s w. s mod s cos s o v ows AIC A Iormo Crro. I my od AIC wr s umr o rmrs ssc mod d s mxmzd vu o -ood uco udr cosdrd mod. mod w ows AIC vu s cosdrd s s. Lood Ro s or sd mods: 9
30 MOKw - W dsruo rducs o MOKw - W w o Kw - W o MO - W or. Hr w v moyd ood ro crro o s oow u yoss: H : s sm s rom MOKw - W H : s sm s MOKw - W. H : s sm s rom Kw - W H : s sm s MOKw - W. H : s sm s rom MO - W H : s sm s MOKw - W. * Lˆ x Wr ood ro s ssc s v y LR = wr Lˆ x * ˆ s rsrcd ML sms udr u yoss H d ˆ s ursrcd ML sms udr rv yoss H. Udr u yoss H LR crro oows C-squr dsruo w drs o rdom d d d u. u yoss s rcd or -vu ss.5. I MLEs d sdrd rrors d 95% codc rvs rss o rmrs or d mod o w AIC vus d Lood ro ssc r rsd. MOKw W mod w ows vus o AIC s s s s mod. Morovr LR s rcs r su mods vour o MOKw W. os o d dss d d cd s o w osrvd os r dsyd urs 3 d 4 dc MOKw W rovds s o d cosdrd r. 3
31 : MLEs sdrd rrors d 95% codc rvs rss o w Dsruo â ˆ ˆ ˆ ˆ ˆ ood d AIC vus or survv ms o ccr s d. MO W Kw W MOKw W MOKw W B W BKw W mx AIC LR vu Esmd d's Dsy MO-W Kw-W B-W MOKw-W MOKw-W BKw-W 3 4 : 3 Pos o osrvd sorm d smd d s or MO W Kw W MOKw W d MOKw W mods d B W BKw W mods. 3
32 Esmd cd's cd MO-W Kw-W B-W MOKw-W MOKw-W BKw-W 3 4 :4 Pos o osrvd ov d smd cd s or MO W Kw W MOKw W d MOKw W mods d B W BKw W mods. 7. Cocuso rzd Mrs-O xdd Kumrswmy rzd my o dsruos s roducd d som o s mor rors d rmr smo r sudd. Acos w o r d s s sow mods d so som or comors w rsc o AIC vus. MOKw W dsruo s r s su Adx: Mxmum ood smo or MOKw E d o MOKw E dsruo s v y MOKwE or or rdom sm o sz rom s dsruo -ood uco or rmr vcor θ θ s v y 3
33 33 comos o scor vcor θ r θ θ θ θ θ symoc vrc covrc mrx or ms o uow rmrs θ o E MOKw dsruo s smd y ˆ Î θ ˆ vr ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ vr ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ cov ˆ ˆ cov ˆ vr ˆ ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ vr ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ vr Wr ms o ormo mrx θ θ θ θ I ˆ ˆ ˆ c drvd us oow scod r drvvs:
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35 35
36 36 Wr. s drvv o dmm uco. Rrcs. Azd M. r M.H. Cordro.M. Zur M. d Hmd.. 5. Kumrswmy Mrs-O my o dsruos. Jour o Ey Mmc Socy I rss.. Brro-Souz W. Lmo A.J. d Cordro.M. 3. r rsus or Mrs d O's my o dsruos. A Acd Brs Cc 85: Br A. Roux J. d Mosr P.. A rzo o comoud Ry dsruo: us Bys mods o ccr survv ms Commu. S. ory Mods 9: Cordro.M. d D Csro M.. A w my o rzd dsruos. J S Comu Smu 8: Cordro.M. Lmo A.J. d Or E.E.M. 4. Mrs-O my o dsruos: Mmc rors d w mods. J S or Prc 8: y M.E. A-Awd.A. d A L.A. 7. Mrs-O xdd Lomx dsruo d s co o csord d. Commu S ory Mod 36: y M.E. A-Huss E.K. d A-Jr R.A. 5. Mrs-O xdd Wu dsruo d s co o csord d. J A S 3: sr P.W. C M.N. Ro P.V. Susr J.J. d Pu J Mod cur rs us omrz mod w covr ormo. S Md 78: urvc M. DBdo A. d Rd S A w ssc dsruo or crcrz rdom sr o r mrs. J. Mr. Sc. 3:
37 . rwood J.A. Ldwr J.M. Ms N.C. d Ws J.R Proy wd moms: do d ro o rmrs o svr dsruos xrss vrs orm. Wr Rsour Rs 5: Hdqu L. d Crory S. 5. Mrs-O-Kumrswmy- my o dsruos. rv:59.88 m.s Udr Rvw.. Hdqu L. d Crory S. 6. B rd Kumrswmy- d or w ms o dsruos. rv: m.s Udr Rvw. 3. Jyumr K. d Mw. 8. O rzo o Mrs-O scm d s co o Burr y II dsruo. S P 49: Krs E. Jos K.K. Rs c M. 3. Acos o Mrs-O réc dsruo. Comm. S. Smu. Comu. 4: Lm E.L owr o r ss. A M Ss 4: Mrs A. d O I A w mod or dd rmr o my o dsruos w cos o xo d Wu ms. Bomr 84: Ndr S. 5. Exod Pro dsruos. Sscs 39: Sr A.M. d Aoo J. 3. Exod modd Wu xso dsruo. R E Sys Sy : Ndr S. Cordro.M. d Or E.M.M. 5. Zoros-Brs my o dsruos: mmc rors d cos Commu. S. ory Mods 44: Nym J. d Prso E.S. 98. O us d rro o cr s crr or uros o ssc rc Bomr A 75-4 d Sd M. d Sumr J.. 7. Socsc Ordrs Srr Nw Yor.. So K.S.. Réy ormo ood d rsc dsruo msur. EM J S P Ir 93: r M.H. d Ndr S. 5. Prmr duco couous uvr dsruos: W-ssd ms. As d Acdm Brsr d Cêcs As o Brz Acdmy o Sccs 87: Z. d M. 7. ur d yss w xdd Wu dsruo. Commuco Sscs-Smuo d Comuo 36:
The Kumaraswamy Generalized Marshall-Olkin-G family of distributions
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More informationThe Marshall-Olkin-Kumaraswamy-G family of distributions
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