The Beta Generalized Marshall-Olkin-G Family of Distributions. (21 August 21, 2016)

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1 Th a azd Masha-O- ay o Dsuos Laa Hadu ad Suaa Chaaoy* Da o Sascs Duah Uvsy Duah-7864 da *Cosod Auho. Ea suaa_sas@du.ac. Auus 6 Asac hs a w oos a w ay o dsuo cosd azd Masha-O dsuo as h as dsuo h a- ay o Cosuco. Th w ay cuds a- Eu a. ad Jos 4 ad E Jayaua ad Mahw 8 as as acua cass. Poay dsy uco d ad h cuuav dsuo uco cd a xssd as xu o h Masha-O Masha ad O 997 dsuo. Ss xasos o d o h od sascs a aso oad. Mos o a uco Réy os ua ow ss ado sa ao ad asyos a aso vsad. Paa sao y hod o axu hood ad hod o o a aso sd. ay oosd od s coad o h azd Masha-O Kuaasway xdd ay Hadu ad Chaaoy 5 y cosd h daa xas wh a daa ss. Ky wods a ad ay azd Masha-O ay Exoad ay AC C ad Pow whd os.. oduco H w y oduc h a- Eu a. ad Jos 4 ad azd Masha-O ay Jayaua ad Mahw 8 o dsuos.. So ouas ad oaos H s w s so ouas o usd h susu scos o hs ac. T s a couous ado vaa wh d ad cd P T h s Suvva uco s P T Hazad a uco h h / Rvs hazad a uco h /

2 Cuuav hazad a uco ch H o h Pow Whd Mo PWM d Réy oy R o. a- ay o dsuos d. Th cd o a- Eu a. ad Jos 4 ay o dsuo s Wh v v d v o x x dx dos h co a uco ao. Th d cosod o s Wh d d. s h h h ch H o So o h w ow a ad as a h a-ad a- ay Eu a. Jos 4 a xdd ay Codo a. Kuaasway a azd ay Psc a. a azd wu dsuo Sa a. a azd Rayh dsuo Codo a. 3 a xdd ha oa dsuo Codo a. 4 a o-osc dsuo Lo 4 a azd vs Wu dsuo ahah a. 4 a Masha-O ay o dsuo Azadh a. 5 ad a ad Kuaasway- ay o dsuo Hadu ad Chaaoy 6a ao ohs..3 azd Masha-O Exdd E ay o dsuos Jayaua ad Mahw 8 oosd a azao o h Masha ad O 997 ay o dsuos y us h Lha scod aav Lha 953 o oa h s h E ay o dsuos y xoao h s o E ay o dsuos as o

3 3 3 wh ad s a addoa sha aa. Wh ad o. Th cd ad d o h E dsuo a scvy 4 ad 5 Ray asus h h h ad ch assocad wh a h h { H o o Wh ad h a scvy h d cd s ad h o h as dsuo. W do h ay o dsuo wh d 3 as E a whch o ducs o E a. So o h oa dsuos dvd Masha-O Exdd xoa dsuo Masha ad O 997 Masha-O Exdd uo dsuo Ksha Jos ad Ksha Masha-O Exdd ow o oa dsuo u 3a Masha- O Exdd o osc dsuo u 3 Masha-O Exdd Essch asod Laac dsuo o ad o 3 Masha-O Kuaasway- auy o dsuo Hadu ad Chaaoy 5a azd Masha-O Kuaasway- ay o dsuo Hadu ad Chaaoy 5 ad Kuaasway azd Masha- O ay o dsuo Hadu ad Chaaoy 6.

4 4 hs ac w oos a ay o a ad dsuo y cosd h azd Masha-O ay Jayaua ad Mahw 8 as h as dsuo h a- ay Eu a. ad Jos 4. Ths w ay d o as h a azd Masha-O ay o dsuo s vsad o so s a os. Th s o hs ac s oazd sv scos. sco h w ay s oducd ao wh s hysca ass. oa sca cass o h ay ao wh h sha ad a ay chaacscs a sd h x sco. sco 4 w dscuss so a sus o h oosd ay. D hods o sao o aas ao wh h coaav daa od acaos a sd sco 5. Th ac ds wh a cocuso sco 6 oowd y a adx o dv asyoc codc ouds.. Nw azao a azd Masha-O- ay o dsuos H w oos a w a xdd ay y cosd h cd ad d o Jayaua ad Mahw 8 dsuo 4 ad 5 as h ad scvy h a ouao ad ca dsuo. Th d o s v y 6 a Say susu o uao 4 w h cd o scvy as 7 Th s h h ad ch o dsuo a scvy v y s h 8 h h

5 5 9 ch o H Ra Th ducs o Azadh a. 5 o Jayaua ad Mahw 8 Masha ad O 997 wh ad v Eu a. Jos 4 o.. ss o h dsuo ad a oh s h h oay dsuo as h sa o as h od sascs o h ado vaa T. Poo L... T T T a suc o..d. ado vaas wh cd a. Th h d o h h od sascs T s v y { {!!! { { { {. Sha o h dsy uco H w hav od h d o h o so chocs o h dsuo ad aa vaus o sudy h vay o shas assud y h ay.

6 -E -W Dsy =.6===.5=.9 =.3=.9=.6=.6=. =3.5==.5=.6=.5 =9.5=.9=.9=.6=.6 =9.5=.9=.9=.6=.5 =.9=.=.5=.5=.5 Dsy ===.5=.=.9=.6 =3=3.5=.5=.6=.5=.5 =5=8.5=.=.89=.5=.5 =.5=.45=.4=.5=.5=.5 =.=.4=3.5=.5=.5=.4 =4=9.5=.5=.5=.5= X X a -L - Dsy =8.=.9=5.55=3.5=.48=.8 =7.=.9=5.45=3.=.4=.7 =5=.56=.5=.5=.5=.5 =5.=.56=4.5=.5=.5=.5 =8.=.9=5.55=3.=.45=.6 ==3.5=.5=.5=.5=.5 Dsy =.=.5=.4=3.5=.5=.5 =.3=.=.3=4.5=.5=.5 =.3=4.45=.3=4.5=.5=.5 =.6=3.35=.4=4.5=.3=.5 =.5=.5=.=4.5=.5=.5 =.=.5=.3=4.5=.5= X X c d Dsy os a Dsuos E W c L ad d 6

7 -E -W hx =.5=.9=.6=.9=.7 =3.5=.99=.58=.6=.5 =.4=.99=.58=.6=.5 =9.5=.84=.9=.8=.9 =5.5=.9=.8=.6=.5 =.89=.78=.48=.5=.5 hx =.=.65= =.9=.9=.8 =.8=.3=. =.83=.8=.7 =.4=.5=.7==.8=.8 =.5=.45=.5=.8=.8=.8 =.8=.5= =.=.8=.8 =.=.65=.9=.9=.9= X X a -L - hx ==.35 =.5=5.5=.9=.9 =7.=.9=5.45=3.=.4=.7 =5=.56 =.5=.5=.5=.5 =4=.56 =4.5=.5=.5=.5 =4.=.99=6.55=9.=.45=.6 =.5=3.8=.9=3.5=.6=.9 hx =.6=.9=.5=5.5=.8=.9 =.99=.5=3=3.5=.5=.5 =5.3=4.65=.3=6.7=.5=.5 =.9=.5=3=3.5=.63=.7 =.3=.8=.=3.5=.5=.8 =7.3=4.95=.3=6.7=.4= c Hazad os os a Dsuos X E W d c L ad d o h os u ad ca s ha h ay s vy x ad ca o ay d ys o shas o dsy ad hazad a uco cud h ah u shad hazad. 3. So sca dsuos So sca cass o h ay o dsuos a sd hs sco. 3. Th xoa E dsuo L h as dsuo xoa wh aa ad h o h E od w h d ad cd scvy as X 7

8 8 E ad E s E h E h h E ch o H E 3. Th Loax L dsuo Cosd h Loax dsuo hay a. 7 wh d ad cd v y ad ad h d ad cd o h L dsuo a v y L ad L s L

9 9 h h L { h L { ch o H L 3.3 Th Wu W dsuo Cosd h Wu dsuo hay a. 5 ha ad X 7 wh aas ad hav d ad cd ad scvy w h d ad cd o W dsuo as W ad W s W h h W

10 h W ch o H W 3.4 Th ch dsuo Suos h as dsuo s h ch dsuo Ksha a. 3 wh d ad cd v y ad scvy ad h h cosod d ad cd o dsuo cos ad s h h h

11 ch o H 3.5 Th oz o dsuo Nx y a h oz dsuo s a. 998 wh d ad cd ad scvy w h d ad cd o o dsuo as o ad o s o h h o h o

12 ch o H o 3.6 Th Exdd Wu EW dsuo Th d ad h cd o h xdd Wu EW dsuos o uvch a. 997 s v y x z ad x R D wh s a o-av ooocay cas uco whch dds o h aa vco. ad z s h dvav o. y cosd EW as h as dsuo w dv d ad cd o h EW as x x x x x{ x{ x EW z ad EW x x oa ods ca s as acua cass wh d chocs o = xoa dsuo. = Rayh u y-x dsuo. = o Pao dsuo v = x oz dsuo. s EW x x

13 3 h h EW x x x x x x x{ x{ x z h EW x x x x x x x{ x{ x z ch o x x H EW 3.7 Th Exdd Modd Wu EMW dsuo Th odd Wu MW dsuo Saha ad ad 3 wh cd ad d s v y x ad x scvy. Th cosod d ad cd o EMW a v y x x x x x{ x{ x EMW ad EMW x x s EMW x x h h EMW

14 4 x x x x x x x{ x{ x h EMW x x x x x x x{ x{ x ch o x x H EMW 3.8 Th Exdd Exoad Pao EEP dsuo Th d ad cd o h xoad Pao dsuo o Nadaaah 5 a v scvy y ad x ad. Thus h d ad h cd o EEP dsuo a v y { { { { { { EEP ad EEP { { s EEP { { h h EEP { { { { { { { {

15 5 h EEP { { { { { { { { ch o { { H EEP 4. a sus o h a azd Masha-O ay o dsuos hs sco w dv so a sus o h oosd ay. 4. Exasos y us oa xaso 6 w oa { { d d

16 6 d d d d Wh ad Aavy w ca xad h d as 3 d d d d Wh ad W ca xad h cd as s co a uco o Mah Wod A Woa W Rsouc. h//ahwod. Woa.co/ co a uco. h 4!!! syo. Pochha a s x Wh! a a a z z a z z a z z a z a a z Us 4 7 w hav

17 7 { { { y xcha h dcs ad h su syo w hav ad h 5 Wh Say a xaso o h cd o ca dvs as { {

18 8 6 Wh 4. Od sascs Suos T T T... s a ado sa o ay dsuo. L T do h h od sascs. Th d o T ca xssd as {!!!!!! Now us h a xaso o h dsuo d ad cd w h d o h h od sascs o o h s v y!!! Wh ad dd aov Now d Wh c c c d c d Nadaaah. a 5 Tho h dsy uco o h h od sascs o dsuo ca xssd as!!! d d!!! 7

19 9 Wh d!!! ad d ad dd aov. 4.3 Poay whd os Th oay whd os PWMs s oosd y wood a. 979 a xcaos o ca ucos o a ado vaa whos a xss. Th h PWM o T s dd y d o uaos ad 3 h h s o o T ca w h as d T E s s d s d s s o d T E s s d s s Wh d s h PWM o M O dsuo. Tho h os o h - ca xsss s o h PWMs o M O Masha ad O 997. Th PWM hod ca ay usd o sa aas uas o azd dsuos. Ths os hav ow vaac ad o sv ass ad hy coa avouay wh saos oad y axu hood.

20 Pocd as aov w ca dv h s o o h h od sasc T a ado sa o sz o o us uao 7 as s T E s Wh ad dd aov 4.5 Mo a uco Th o a uco o ay ca asy xssd s o hos o h xoad Masha ad O 997 dsuo us h sus o sco 4.. o xa us uao ca s ha d d d d E s M s s st T s M d d d X s Wh s M X s h o a Masha ad O 997 dsuo. 4.6 Ry Eoy Th oy o a ado vaa s a asu o ucay vaao ad has usd vaous suaos scc ad. Th Réy oy s dd y d R o wh ad o uhs das s So. Us oa xaso 6 w ca w { {

21 Thus h Réy oy o T ca oad as R o d Wh 4.7 Qua ow ss ad ado sa ao Th ua uco T Q u u ca oad y v 7. L z Q h a ua uco. Th { Q u Q u Q { Q u s oss o oa a xaso o Q u h Woa ws as s oss o oa a xaso o Q u as z Q u u u s Pow ss o MahWod--A Woa W Rsouc. h//ahwod.woa.co/powss.h Wh d ad d d d d d Th owy swss Ky ad K 96 asus ad Moos uoss Moos 988 asu a ous ad ss ssv o ous ad xs v o dsuos whou os. o ay hs asus a v y Q3 4 Q 4 Q ad M Q3 4 Q 4 Q3 8 Q 8 Q7 8 Q5 8 Q6 8 Q 8 o xa h xoa dsuo wh aa hav d ad cd as ad scvy. Th h h ua s oad as / o. Tho h h ua o E s v y

22 4.8 Asyos { Q o { Q H w vsa h asyoc shas o h oosd ay oow h hod oowd Azadh a. 5. Pooso. Th asyos o uaos 6 7 ad 8 as a v y ~ { ~ as { as h ~ { as Pooso. Th asyos o uaos 6 7 ad 8 as a v y ~ as ~ as h ~ as 5. Esao 5. Maxu hood hod Th od aas o h dsuo ca sad y axu hood. L T... a ado sa o sz o wh aa T T T vco wh... cosods o h aa vco o h as dsuo. Th h o-hood uco o s v y o o o o o { o o o { Ths o-hood uco ca o sovd aaycay caus o s cox o u ca axzd ucay y oy oa ozao hods avaa wh sowa s R SAS Mahaca o y sov h oa hood uaos oad y da 8. 8

23 3 y a h aa dvavs o h o-hood uco wh sc o ad coos o h sco vco T T U U U U U U ca oad as oows U { o U { o U o o o { { o { { { o { U { { U { { Wh. s h daa uco. 5. Asyoc sadad o ad codc va o h s Th asyoc vaac-covaac ax o h MLEs o aas ca oad y v h sh oao ax whch ca dvd us h scod aa dvavs o h ohood uco wh sc o ach aa. Th h s o a v y E 3 Th xac vauao o h aov xcaos ay cuso. acc o ca sa y h osvd sh s oao ax ˆ Î s dd as

24 Î ˆ 3 Us h a hoy o MLEs ud so uay codos o h aas as h asyoc dsuo o ˆ s N V V wh v. Th asyoc V ˆ havou as vad V Î ˆ s acd y. Ths su ca usd o ovd a sa sadad os ad aso cosuc codc vas o h od aas. Thus a aoxa sadad o ad / % codc va o h o h aa a scvy v y ˆ ˆ wh / s h / o o sadad oa vˆ ad / v dsuo. As a usao o h MLE hod s a sa sadad os codc va h cas o E s dscussd a adx. 5.3 Ra acaos hs susco w cosd o h a daa ss o show ha h oosd dsuo ca a od ha Kw Hadu ad Chaaoy 5 y a as Wu dsuo as. W hav sad h aas y uca axzao o ohood uco ad ovdd h sadad os ad 95% codc vas us a sa aoach s adx. od o coa h dsuos w hav cosdd ow ca AC Aa oao Co C aysa oao Co CAC Coss Aa oao Co ad HQC Haa-Qu oao Co. ay od ha C o CAC AC ad HQC oo AC wh s h u o aas h sasca od h sa sz ad s h axzd vau o h o-hood uco ud h cosdd od. hs acaos hod o axu hood w usd o oa h sa o aas. Exa Th oow daa s vs h o au 3 h o uocha o o y o v Xu a. 3. {

25 Ta MLEs sadad os ad 95% codc vas ahss ad h AC C CAC ad HQC vaus o h daa s. Paas â ˆ ˆ ˆ ˆ ˆ Kw W W o-hood ax AC C CAC HQC Esad d's Esad cd's Dsy W Kw-W cd W Kw-W X X a 3 Pos o h osvd a hsoa ad sad d s ad sad cd s o h ad Kw o xa. 5

26 Exa Th oow daa s aou 346 co asus ad o sva ads o cas 998. Th daa hav cocd y h da Tad Cosso whch s a dd acy o h US ov whos a sso s h ooo o cosu oco. h // os/oacco o h // w.co.co/ davs/ so. h. {

27 Ta MLEs sadad os ad 95% codc vas ahss ad h AC C CAC ad HQC vaus o h co asus daa. Paas â ˆ ˆ ˆ ˆ ˆ Kw W W o-hood ax AC C CAC HQC Esad d's Esad cd's Dsy W Kw-W cd W Kw-W X X a 4 Pos o h osvd a hsoa ad sad d s ad sad cd s o h ad Kw o xa. 7

28 Exa Ths daa s cosss o osvaos o a sss o cao s a v y Nchos ad Pad 6. { Ta 3 MLEs sadad os ad 95% codc vas ahss ad h AC C CAC ad HQC vaus o h a sss o cao s daa. Paas â ˆ ˆ ˆ ˆ ˆ Kw W W o-hood ax AC C CAC HQC

29 Esad d's Esad cd's Dsy W Kw-W cd W Kw-W X a 5 Pos o h osvd a hsoa ad sad d s ad sad cd s o h ad Kw o xa. Tas ad 3 h MLEs s s ahss ad 95% codc vas ahss o h aas o h d dsuos ao wh AC C CAC ad HQC vaus a sd o xa ad scvy. h xas cosdd h asd o h ows vaus o h AC C CAC ad HQC h dsuo ha X W dsuo us ou o a Kw W dsuo. A vsua coaso o h cosss o h d dss wh h osvd hsoa o h daa o xa ad a sd h us 3 4 ad 5 scvy. Ths os dca ha h oosd dsuos ovd a cos o hs daa. 6. Cocuso a azd Masha-O ay o dsuos s oducd ad so o s oa os a sudd. Th axu hood ad o hod o sa h aas a aso dscussd. Acao o h a daa shows ood su avou o h oosd ay wh coad o azd Masha-O Kuaasway xdd ay o dsuos. s ho xcd ha hs ay o dsuo w a oa addo o h xs au o dsuo. Adx Maxu hood sao o E Th d o h E dsuo s v y E 9

30 3 o a ado sa o sz o hs dsuo h o-hood uco o h aa vco T s v y o o o o o { o { o o Th coos o h sco vco T a U { o U { o U o o o { { o { { { o { U { { U { { Th asyoc vaac covaac ax o s o h uow aas o E dsuo s sad y

31 3 ˆ Î va ˆ covˆˆ covˆˆ ˆ covˆ ˆ covˆ covˆˆ va ˆ covˆˆ ˆ covˆ ˆ covˆ covˆˆ covˆˆ va ˆ ˆ covˆ ˆ covˆ cov ˆˆ cov ˆˆ cov ˆˆ ˆ va ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ va Wh h s o h oao ax ˆ ˆ ˆ ca dvd us h oow scod aa dvavs { { o { { { o { 3 3 { - { - { { - { { { { - { { - { { { { - { { { { -

32 3 { { - { { { { 3 { - - { { { - { - { { { - { { { { 3 { - { { { o { - { { { { - { { { { { o { { -

33 33 { { - { { - - { { { - { - { { o { { - { - - { { o { { - { - { o { { - - { { { - { - { { o { { - { - - { { o { { - { - { { - { { -

34 { { { { 3 3 { { { { { { { { { { { { { { { { { - { Wh. s h dvav o h daa uco. Rcs. Azadh M. Tah M.H. Codo.M. ua M. ad Hada.. 5. Th Kuaasway Masha-O ay o dsuos. Joua o h Eya Mahaca Socy ss.. oa oa ad Wss Ec W. Pow ss o MahWod-A Woa W Rsouc. h//ahwod.woa.co/powss.h. 3. ao-souza W. Lo A.J. ad Codo.M. 3. a sus o Masha ad O's ay o dsuos. A Acad as Cc ahah L.A. Mousa S.A. Aaah M.A. ad Eya S.H. 4. Th a azd vs Wu dsuo. sh J Mah Cou Sc Codo.M. Sva.O. ad Oa E.M.M. a. Th a xdd Wu dsuo. J Poa Sa Sc Codo.M. Cso C.T. Hashoo E.M. ad Oa E.M.M. 3. Th a azd Rayh dsuo wh acaos o daa. Sa Pa Codo.M. Sva.O. Psc R.R. ad Oa E.M.M. 4c. a os o h a xdd ha-oa dsuo. J Sa Cou Su Eu N. L C. ad aoy.. a-oa dsuo ad s acaos. Cou Sas Tho Mh

35 9. o D. ad o S. 3. Masha-O Essch asod Laac dsuo ad ocsss. az J Poa Sas hay M.E. A-Awadh.A. ad Ahaa L.A. 7. Masha-O xdd Loax dsuo ad s acao o csod daa. Cou Sa Thoy Mhod hay M.E. A-Hussa E.K. ad A-Jaaah R.A. 5. Masha-O xdd Wu dsuo ad s acao o csod daa. J A Sa s P.W. Cha M.N. Rao P.V. Shus J.J. ad Pu J Mod cu as us h oz od wh covaa oao. Sa Md wood J.A. Ladwh J.M. Maaas N.C. ad Was J.R Poay whd os do ad ao o aas o sva dsuos xssa vs o. Wa Rsou Rs u W. 3a. A Masha-O ow o-oa dsuo ad s acaos o suvva daa. J Sas Poa u W. 3. Masha-O xdd o-osc dsuo ad s acao cao ocsss. A Mah Sc uvch M. Ddo A. ad Raad S A w sasca dsuo o chaacz h ado sh o aas. J. Ma. Sc Hadu L. ad Chaaoy S. 5a. Th Masha-O-Kuaasway- ay o dsuos. axv59.88 ah.st. 8. Hadu L. ad Chaaoy S. 5. Th azd Masha-O-Kuaasway- ay o dsuos. axv5.84 ah.st. 9. Hadu L. ad Chaaoy S. 6. a ad Kuaasway- ad oh w as o dsuos. axv ah.st Ud Rvw.. Hadu L. ad Chaaoy S. 6. Th Kuaasway azd Masha-O ay o dsuos.. Jos M.C. 4. as o dsuos as o h dsuos o od sascs. Ts Ky J.. K ad E.S. 96. Mahacs o Sascs Pa. Va Nosad Nw Jsy 3d do. 3. Ksha E. Jos K.K. ad Rs c M. 3. Acaos o Masha-O éch dsuo. Co. Sa. Sua. Cou Lha E.L Th ow o a ss. A Mah Sas Lo A.J. 4. Th a o-osc dsuo. az J Poa Sas

36 6. Masha A. ad O A w hod o add a aa o a ay o dsuos wh acaos o h xoa ad Wu as. oa Mad M.E. Ay A.. Hada.. ad hosh. 6. Th a Exoa ch Dsuo wh Acaos o aa Ausa Joua o Sascs. 8. Moos J.J.A A ua aav o uoss. Th Sasca Nadaaah S. 5. Exoad Pao dsuos. Sascs Nadaaah S. Codo.M. ad Oa E.M.M. 5. Th oaos-aasha ay o dsuos ahaca os ad acaos Cou. Sa. Thoy Mhods Nchos M.D. ad Pad W.J.A. 6. A oosa coo cha o Wu cs. Quay ad Ray E aoa v Psc R.R. Codo.M. Do C... Oa E.M.M. ad Nadaaah S.. Th w cass o Ku a azd dsuos. SORT Saha A.M. ad Aaoo J. 3. Exoad odd Wu xso dsuo. Ra E Sys Say Sa N. Ja K. ad Shaa S.K.. Th a azd Wu dsuo Pos ad acaos. Ra E Sys Say So K.S.. Réy oao ohood ad a sc dsuo asu. EM J Sa Pa Wss Ec W. co a uco o MahWod-A Woa W Rsouc. h//ahwod.woa.co/coauco.h. 37. Xu K. X M. Ta L.C. ad Ho S.L. 3. Acao o ua wos ocas syss ay. Ad So Cou ha T. ad X M. 7. au daa aayss wh xdd Wu dsuo. Coucao Sascs Suao ad Couao

The Log-Gamma-Pareto Distribution

The Log-Gamma-Pareto Distribution aoa Joa of Scc: Bac ad Appd Rach JSBAR SSN 37-453 P & O hp:odphp?oajoaofbacadappd ---------------------------------------------------------------------------------------------------------------------------