The Kumaraswamy Generalized Marshall-Olkin-G family of distributions
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- Theresa Nelson
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1 Th Kurswy rzd Mrsh-O- y o dsruos L Hdqu d Sur Chrory* Dr o Sscs Drurh Uvrsy Drurh-7864 Id *Corrsod Auhor. E sur_ss@dru.c. Auus 6 Asrc Aohr w y o couous roy dsruo s roosd y us rzd Mrsh-O dsruo s h s dsruo h Kurswy- dsruo. Ths y cuds Kw Cordro d d Csro d MO Jyur d Mhw 8 s sc cs sds udr o ohr dsruos. Th roy dsy uco d d h survv uco s r xrssd s srs o osrv s xur o h rzd Mrsh-O dsruo. Srs xsos d o ordr sscs r so od. Mos o r uco Réy ros qu uco rdo s ro d syos r so vsd. Prr so y hod o xu hood d hod o o r so rsd. y h roosd od s cord o h rzd Mrsh-O Kurswy xdd y Hdqu d Chrory 5 y cosdr our xs o r d od. Ky words Kurswy- dsruo rzd Mrsh-O y Exod y AIC BIC d Powr Whd Mos.. Iroduco Rcy so ors hv d o d w s o couous dsruos o xd w-ow dsruos d h s rovd r xy od d rcc. So svr csss y dd o or or rrs o r w dsruos hv roosd h ssc rur. Th sc ovo o hs wors s o r or xy h od dr y o d rd ro r suo. Nwy hr s rwd cvy hs r o roos d vs w s o dsruos. A rc rvw r y Thr. 5 rovds d ccou o vrous
2 our chqus o r w s o uvr couous dsruos hrouh roduco o ddo rrs. Vry rc coruos hs cud h Kurswy Mrsh - O y roosd y Azdh. 5 Mrsh - O Kurswy- y roducd y Hdqu d Chrory 5 rzd Mrsh - O Kurswy- y roducd y Hdqu d Chrory 5 d B rd Kurswy- y roducd y Hdqu d Chrory 6 o ohrs. I hs rc w roos ohr y o couous roy dsruo y r h rzd Mrsh O y Jyur d Mhw 8 s h s dsruo h Kurswy- y Cordro d d Csro. Ths w y rrrd o s h Kurswy rzd Mrsh-O- KwMO y o dsruo s vsd. W hv dscussd s hc rors d d o r d or usro o h xy o h od. Th rs o hs rc s orzd sv scos. I sco w ry roduc so or chrcrscs o roy dsruos h Kurswy- Cordro d d Csro d h rzd Mrsh-O y Jyur d Mhw 8 o dsruos. Th roosd w y s dd wh s hysc ss sco 3. Ior sc css o h y wh hr sh d ry chrcrscs r rsd h x sco. I sco 5 w dscuss so r rsus o h roosd y wh dr hods o so o rrs wh our corv d od cos r rsd sco 6. Th rc ds wh cocuso sco 7 oowd y dx o drv syoc codc ouds.. So orus d oos Hr rs w s so orus o usd h susqu scos o hs rc. I T s couous rdo vr wh d d cd P T h s Survv uco s P T Hzrd r uco hr h / Rvrs hzrd r uco rhr r / Cuuv hzrd r uco chr H h q r q Powr Whd Mo PWM q r d r
3 Réy roy I R d.. Kurswy- Kw y o dsruo or s cd wh d Cordro d d Csro dd Kw dsruo wh cd d d Kw d Kw d Whr d r sh rrs ddo o hos h d s dsruo. Th s hr rhr d chr o hs dsruo r rscvy v y h r Kw Kw Kw Kw Kw Kw Kw Kw Kw d H.. rzd Mrsh-O Exdd MOE y o dsruo Jyur d Mhw 8 roosd rzo o h Mrsh d O 997 y o dsruos y us h Lh scod rv Lh 953 o o h s MO y o dsruos s MO o h MOE y o dsruos y xoo h s o MOE whr d s ddo sh rr. Wh MO MO d or MO. Th cd d d o h MOE dsruo r rscvy 3 d MO MO 4 5 3
4 4 Ry surs h hr rhr d chr ssocd wh r h MO MO MO h r MO MO MO H MO Whr d h r rscvy h d cd s d hr o h s dsruo. W do h y o dsruo wh d 5 s MOE whch or rducs o MOE. 3. Kurswy rzd Mrsh-O- KwMO y o dsruo W ow roos w xso o h Kw Cordro d d Csro y y cosdr h cd d d o MO Jyur d Mhw 8 dsruo 4 d 5 s h d rscvy h Kw oruo d c KwMO dsruo. Th rsu xrsso or h d o KwMO s v y KwMO 6 Th cd s hr rhr d chr o KwMO dsruo r rscvy v y cd KwMO 7 s KwMO
5 5 hr 8 KwMO h rhr KwMO r 9 chr KwMO H Rr KwMO rducs o so ow s o dsruos s or KwMO KwMO Azdh. 5 or Kw KwMO or MO KwMO Jyur d Mhw 8 d or Kw KwMO Cordro d d Csro. 3. Sh o h dsy d hzrd uco Hr w hv od h d d hr o h KwMO or so chocs o h rrs o sudy h vry o shs ssud y h y.
6 KwMO-E KwMO-W Dsy =5.5=.5=.4=.5=.9 =.5=.5==.4=.5 =4.5=.4==.4=.5 =.5=.45==.5=.5 =.5=.6==.5=.5 =.9===.5=.9 Dsy =5.5=.5=.5=.5=.5=.9 =6.5=.5=.=.5=.5=.9 =.7=.5=.5=.9=.5=.5 =.5=.4=.5=.9=.5=.6 =.5=.6==.5=.5=.5 ==.4=.5=.9=.4= X X KwMO-L KwMO-r Dsy =.5=3=5=.5=.99=4.5 =.99=.=8=.5==9.5 =3.8=3.=5.=.5=.99=4.5 =.49=.45==.5=.5=.5 =6.5=55=.5=.5=.5=.6 =33=3.5=8.=.5==9.5 Dsy =.=.8=.=.4=.5=.6 =.9=.7=.8=.8=.89=.6 =.3=.3=.8=.6=.9=.5 =.4=.3=.6=.7=.9=.5 =.9=.33=.8=.8=.8=.6 =.9=.33=.8=.7=.8= X c d Dsy os KwMO E KwMO W c KwMO L d d KwMO r dsruos. X 6
7 MOKw-E KwMO-W hx =.5=.9=8.5=.6=.5 =4.5=.6=4.5=.8=.8 ==3.8=.4=.4=.5 =.4=.45=.5=.5=.5 =.9=.55=.4=.4=.4 =.5=.39=.7=.5=.8 hx =.=.5==4=.5=.7 =.=.35=. =3=.5=.9 =.=.==3.8=.5=.6 =.=.45=. =.79=.5=.9 =.3=.5=. =3.4=.5=.6 =.=.9==3.8=.5= X X KwMO-L KwMO-r hx =.5=.45==.5=.5=.5 =9=.5=4.7=6=.5=.7 =.=.48==.5=.5=.5 =3.5=.5=8=.5==9.5 =4=.9=4.7=6=.5=.7 =.9=.5=8=.5==9.5 hx =.5=.9=.=.5=.5=.5 =.3==.5=.5=.59=.5 =5=3=.9=.9=.5=.5 =.3=.3=.8=.6=.9=.5 =.9=.33=.8=.8=.8=.6 =.=.5=.5=.5=.7= X c d Hzrd os KwMO E KwMO W c KwMO L d d KwMO r dsruos. ro h os ur d c s h h y s vry x d c or y dr ys o shs o dsy d hzrd r uco cud h h u shd r hzrd. 4. So sc KwMO dsruos I hs sco w rovd so sc css o h s hr dsruo chrcrscs. 4. Th KwMO xo KwMO E dsruo X KwMO y o dsruos d 7
8 8 L h s dsruo xo wh rr d h or h E KwMO od w h d d cd rscvy s KwMOE cd KwMOE s KwMOE hr h KwMOE rhr KwMOE r chr KwMOE H 4. Th KwMO Lox L KwMO dsruo Cosdr h Lox dsruo hy. 7 wh d d cd v y d d h d d cd o h L KwMO dsruo r v y KwMOL
9 9 cd KwMOL s KwMOL hr KwMOL h rhr r KwMOL chr KwMOL r 4.3 Th KwMO Wu W KwMO dsruo Cosdr h Wu dsruo hy. 5 h d X 7 wh rrs d hv d d cd d rscvy w h d d cd o W KwMO dsruo s KwMOW
10 cd KwMOW s KwMOW hr KwMOW h rhr KwMOW r chr KwMOW H 4.4 Th KwMO rch r KwMO dsruo Suos h s dsruo s h rch dsruo Krsh. 3 wh d d cd v y d rscvy d h h corrsod d d cd o r KwMO dsruo cos KwMOr
11 cd KwMOr s KwMOr hr KwMOr h rhr r KwMOr chr KwMOr H 4.5 Th KwMO orz o KwMO dsruo Nx y h orz dsruo sr. 998 wh d d cd d rscvy w h d d cd o o KwMO dsruo s KwMOo
12 cd KwMOo s KwMOo hr KwMOo h rhr KwMOo r chr KwMOo H 4.6 Th KwMO Exdd Wu EW KwMO dsruo Th d d h cd o h xdd Wu EW dsruos o urvch. 997 s v y x z d x
13 3 R D whr s o-v ooocy crs uco whch dds o h rr vcor. d z s h drvv o. By cosdr EW s h s dsruo w drv d d cd o h EW KwMO s x x x z KwMOEW x x x x cd KwMOEW x x s KwMOEW x x hr x x x z h KwMOEW x x x x rhr x x x z r KwMOEW x x x x x x chr KwMOEW o H x x Ior ods c s s rcur css wh dr chocs o = xo dsruo.
14 4 = Ryh Burr y-x dsruo. 3 = Pro dsruo 4 = x orz dsruo. 4.7 Th KwMO Exdd Modd Wu EMW KwMO dsruo Th odd Wu MW dsruo Srh d d 3 wh cd d d s v y x d x rscvy. Th corrsod d d cd o EMW KwMO r v y x x x x x x x KwMOEMW Th cd s hr rhr d chr o EMW KwMO dsruo r rscvy v y cd KwMOEMW x x s KwMOEMW x x hr x x x x x x x KwMOEMW h rhr r KwMOEMW x x x
15 5 x x x x x x chr KwMOEMW H x x 4.8 Th KwMO Exdd Exod Pro EEP KwMO dsruo Th d d cd o h xod Pro dsruo o Ndrh 5 r v rscvy y d d. Thus h d d h cd o EEP KwMO dsruo r v y KwMOEEP cd KwMOEEP s KwMOEEP hr h KwMOEEP rhr r KwMOEEP
16 6 chr KwMOEEP H 5. r rsus or h Kurswy rzd Mrsh-O- KwMO y o dsruos I hs sco w drv so r rsus or h roosd KwMO y. 5. Exsos o d d s By us o xso 6 w o KwMO MO MO MO MO A MO d d A MO d d A Whr A d A A Arvy w c xd h d s
17 7 MO MO A MO MO A MO MO B MO d d B MO d d B 3 d d B MO B MO 4 Whr o A B d B B Sry xso or h survv uco o KwMO c drvs s KwMO MO MO C 5 Whr C A KwMO MO C
18 8 6 C MO MO MO Whr C 5. Ordr sscs Suos T T T... s rdo s ro y KwMO dsruo. L T do h h ordr sscs. Th d o T c xrssd s KwMO KwMO KwMO!!! KwMO KwMO KwMO!!! KwMO KwMO!!! Us h r xso o h KwMO dsruo d d s w h d o h h ordr sscs or o h KwMO s MO MO MO B!!! Whr B d dd sco 5.!!! MO MO MO B MO MO B!!! MO MO 7
19 9 MO d d MO d d d d MO MO 8 Whr B d 5.3 Mos Th roy whd os PWMs rs roosd y rwood. 979 r xcos o cr ucos o rdo vr whos xss. Th h r q PWM o T s dd y d r q r q ro quos d h h s o o T c wr hr s MO MO A MO MO B d A T E MO MO s s d A s s A or d B T E MO MO s s d B s s B
20 Whr q r q r d s h PWM o MO dsruo. Thror h os o h KwMO - c xrsss rs o h PWMs o MO Jyur d Mhw 8. Th PWM hod c ry usd or s rrs qus o rzd dsruos. Ths os hv ow vrc d o svr ss d hy cor voury wh sors od y xu hood. Procd s ov w c drv rdo s o sz ro E T s h s o o h KwMO o us quos 7 s s h ordr ssc T Whr A B d dd sco 5. d 5. rscvy. 5.4 Mo r uco Th o r uco o KwMO y c sy xrssd rs o hos o h xod MO Jyur d Mhw 8 dsruo us h rsus o sco 5.. or x us quo 3 c s h M T s E B st s d d s MO d s B d d d B M MO X Whr M X s s h o MO Jyur d Mhw 8 dsruo. 5.5 Réy Eroy Th roy o rdo vr s sur o ucry vro d hs usd vrous suos scc d r. Th Réy roy s dd y I R whr d or urhrs ds s So. Us o xso 6 w c wr KwMO d s d
21 MO MO MO MO MO MO MO MO MO MO MO MO MO MO MO Thus h Réy roy o T c od s d d I MO MO MO MO R Whr 5.6 Qu uco d rdo s ro W sh ow rs oru or r KwMO rdo vr y us vrso hod y vr h cd or h survv uco. KwMO KwMO KwMO KwMO KwMO KwMO KwMO KwMO KwMO
22 KwMO KwMO KwMO KwMO o r rdo vr ro KwMO rs r ~ U u h us u u 9 Th h Qu or KwMO c sy od ro 9 s or x h s dsruo xo wh rr hv d d cd s d rscvy. Thror h h Qu o E KwMO s v y 5.7 Asyos Hr w vs h syoc shs o h roosd y oow h hods oowd Azdh. 5. Prooso. Th syos o quos 6 7 d 8 s r v y ~ s ~ s ~ h s Prooso 3. Th syos o quos 6 7 d 8 s r v y ~ s ~ s ~ h s
23 3 6. Eso 6. Mxu hood hod Th od rrs o h KwMO dsruo c sd y xu hood. L T... rdo s o sz ro KwMO wh rr vcor T T θ whr T q... corrsods o h rr vcor o h s dsruo. Th h -hood uco or θ s v y θ Ths -hood uco c o sovd ycy cus o s cox or u c xzd urcy y oy o ozo hods v wh sowr s R SAS Mhc or y sov h or hood quos od y dr. By h r drvvs o h -hood uco wh rsc o d w o h coos o h scor vcor T T U U U U U U θ U
24 4 U U U U r r 6. Asyoc sdrd rror d codc rv or h s Th syoc vrc-covrc rx o h MLEs o rrs c od y vr h shr oro rx I θ whch c drvd us h scod r drvvs o h -hood uco wh rsc o ch rr. Th h s o I θ r v y q E 3 I θ Th xc vuo o h ov xcos y curso. I rcc o c s I θ y h osrvd shr s oro rx ˆ Î θ s dd s q 3 Î ˆ θ θ θ
25 Us h r hory o MLEs udr so rury codos o h rrs s h syoc dsruo o ˆ θ θ s N V whr V I θ. Th syoc hvour rs vd v V s rcd y ˆ V Î ˆ θ. Ths Ths rsu c usd o rovd r s sdrd rrors d so cosruc codc rvs or h od rrs. Thus rox sdrd rror d / % codc rv or h o h rr r rscvy v y s h / o o sdrd or dsruo. / ˆ ˆ whr vˆ d / v As usro o h MLE hod s r s sdrd rrors codc rv h cs o KwMO E s dscussd dx. 6.3 R cos I hs susco w cosdr o our r d ss o cor h roosd KwMO dsruo wh ohr rcy roducd od MOKw Hdqu d Chrory 5 y s Wu s h s dsruo. Th s o h rrs r y od y urc xzo o hood uco. Sdrd rrors s d 95% codc rvs r so drvd us r s roch s dx. I ordr o cor h dsruos w hv cosdrd ow crr AIC A Ioro Crro BIC Bys Ioro Crro CAIC Coss A Ioro Crro d HQIC H-Qu Ioro Crro. I y od h AIC BIC CAIC AIC / d HQIC. Whr h ur o rrs s h s sz d s h xzd vu o h -hood uco udr h cosdrd od. I hs cos hod o xu hood w usd o o h s o rrs. Ex I Hr w cosdr h oow d s o 346 co surs d ro svr rds o crs 998. Th d hv cocd y h dr Trd Cosso whch s dd cy o h US ovr whos sso s h rooo o cosur roco. h // rors/occo or h // w.co.co/ rdvs/ so. h
26 T MLEs sdrd rror s d 95% codc rvs d h AIC BIC CAIC d Prrs â ˆ ˆ ˆ ˆ ˆ HQIC vus or h co surs d MOKw W KwMO W hood x AIC BIC CAIC HQIC
27 Esd d's Esd cd's Dsy MOKw-W KwMO-W cd MOKw-W KwMO-W X 3 Pos o h osrvd hsor d sd d s d sd cd s or h Ex II KwMO W d MOKw W or x I. Ths d s cosss o osrvos o r srss o cro rs v y Nchos d Pd X T MLEs sdrd rrors d 95% codc rvs rhss d h AIC BIC CAIC d HQIC vus or h r srss o cro rs d. Prrs â ˆ ˆ MOKw W KwMO W
28 ˆ ˆ ˆ hood x AIC BIC CAIC HQIC Esd d's Esd cd's Dsy MOKw-W KwMO-W cd MOKw-W KwMO-W X 4 Pos o h osrvd hsor d sd d s d sd cd s or h Ex III KwMO W d MOKw W or x II. Th Cro rs d s o h 66 osrvos o h r srss o cro rs s rord Nchos d Pd 6 s cosdrd hr X 8
29 T 3 MLEs sdrd rrors d 95% codc rvs rhss d h AIC BIC CAIC d HQIC vus or h r srss o cro rs d Prrs MOKw W KwMO W â ˆ ˆ ˆ ˆ ˆ hood x AIC BIC CAIC HQIC Esd d's Esd cd's Dsy MOKw-W KwMO-W cd MOKw-W KwMO-W X 5 Pos o h osrvd hsor d sd d s d sd cd s or h KwMO W d MOKw W or x III. X 9
30 Ex IV Hr w us h oow r d ss whch vs h o ur 3 h o urochrr o o y o v Xu T 4 MLEs sdrd rrors d 95% codc rvs rhss d h AIC BIC CAIC d HQIC vus or h d s Prrs â ˆ ˆ ˆ ˆ ˆ MOKw W KwMO W hood x AIC BIC CAIC HQIC
31 Esd d's Esd cd's Dsy MOKw-W KwMO-W cd MOKw-W KwMO-W X X 6 Pos o h osrvd hsor d sd d s d sd cd s or h KwMO W d MOKw W or x IV. I Ts 3 d 4 h MLEs s s rhss d 95% codc rvs rhss o h rrs or h d dsruos wh AIC BIC CAIC d HQIC vus r rsd or x I II III d IV rscvy. I h our xs cosdrd hr sd o h ows vus o h AIC BIC CAIC d HQIC h KwMO W dsruo urs ou o r dsruo h MOKw W dsruo. A vsu corso o h cosss o h d dss wh h osrvd hsor o h d d d cds wh rc cds or x I II III d IV r rsd h urs d 6 rscvy. Ths os dc h h roosd dsruos rovd cosr o hs d. 7. Cocuso Kurswy rzd Mrsh-O xdd y o dsruos s roducd d so o s or rors r sudd. Th xu hood d o hod or s h rrs r so dscussd. our cos o r d shows ood rsu vour o h roosd y wh cord o rzd Mrsh-O Kurswy xdd y o dsruos. As such h roosd y y cosdrd s ood coruo o h xs owd. 3
32 3 Adx Mxu hood so or E KwMO Th d o h E KwMO dsruo s v y KwMOE or or rdo s o sz ro hs dsruo h -hood uco or h rr vcor T θ s v y θ Th coos o h scor vcor T θ r θ θ θ
33 33 θ θ Th syoc vrc covrc rx or s o h rrs o E KwMO dsruo s sd 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 ˆ cov ˆ cov ˆ 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 ˆ cov ˆ ˆ cov ˆ vr Whr h s o h oro rx θ θ θ θ I ˆ ˆ ˆ c drvd us h oow scod r drvvs
34
35
36 36
37 37 3 3
38 Whr. s h drvv o h d uco.
39 Rrcs. Azdh M. Thr M.H. Cordro.M. ur M. d Hd.. 5. Th Kurswy Mrsh-O y o dsruos. Jour o h Ey Mhc Socy I rss.. Brro-Souz W. Lo A.J. d Cordro.M. 3. r rsus or Mrsh d O's y o dsruos. A Acd Brs Cc Br A. Roux J. d Mosr P.. A rzo o h cooud Ryh dsruo us Bys hods o ccr survv s Cou. S. Thory Mhods Cordro.M. d D Csro M.. A w y o rzd dsruos. J S Cou Su Cordro.M. d Lo A.J. d Or E.E.M. 4. Th Mrsh-O y o dsruos Mhc rors d w ods. J S Thor Prc hy M.E. A-Awdh.A. d Ah L.A. 7. Mrsh-O xdd Lox dsruo d s co o csord d. Cou S Thory Mhod hy M.E. A-Huss E.K. d A-Jrh R.A. 5. Mrsh-O xdd Wu dsruo d s co o csord d. J A S sr P.W. Ch M.N. Ro P.V. Shusr J.J. d Pu J Mod cur rs us h orz od wh covr oro. S Md urvch M. DBdo A. d Rd S A w ssc dsruo or chrcrz h rdo srh o r rs. J. Mr. Sc rwood J.A. Ldwhr J.M. Ms N.C. d Ws J.R Proy whd os do d ro o rrs o svr dsruos xrss vrs or. Wr Rsour Rs Hdqu L. d Chrory S. 5. Th Mrsh-O-Kurswy- y o dsruos. rxv59.88 h.st.. Hdqu L. d Chrory S. 5. Th rzd Mrsh-O-Kurswy- y o dsruos. rxv5.84 h.st. 3. Jyur K. d Mhw T. 8. O rzo o Mrsh-O sch d s co o Burr y XII dsruo. S P Krsh E. Jos K.K. d Rs c M. 3. Acos o Mrsh-O réch dsruo. Co. S. Su. Cou Lh E.L Th owr o r ss. A Mh Ss
40 6. Mrsh A. d O I A w hod or dd rr o y o dsruos wh cos o h xo d Wu s. Bor Ndrh S. 5. Exod Pro dsruos. Sscs Nchos M.D. d Pd W.J.A. 6. A oosr coro chr or Wu rcs. Quy d Ry Er Iro v Srh A.M. Aoo J. 3. Exod odd Wu xso dsruo. R E Sys Sy Ndrh S. Cordro.M. d Or E.M.M. 5 Th oros-brsh y o dsruos hc rors d cos Cou. S. Thory Mhods Shd M. Shhur J.. 7. Sochsc Ordrs Srr Nw Yor.. So K.S.. Réy oro hood d rsc dsruo sur. EM J S P Ir Thr M.H. d Ndrh S. 5. Prr duco couous uvr dsruos W-sshd s. As d Acd Brsr d Cêcs As o h Brz Acdy o Sccs Xu K. X M. T L.C. d Ho S.L. 3. Aco o ur wors orcs syss ry. Ad So Cou h T. d X M. 7. ur d yss wh xdd Wu dsruo. Couco Sscs-Suo d Couo
The Generalized Marshall-Olkin-Kumaraswamy-G family of distributions
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