The Marshall-Olkin-Kumaraswamy-G family of distributions

Transcription

1 Th Mrshll-Ol-Kumrswmy- mly o dsruos L Hdu d Sur Chrory* Drm o Sscs Drurh Uvrsy Drurh-7864 Id *Corrsod uhor: Eml: sur_ss@dru.c. Auus 6 Asrc A w mly o couous dsruo s roosd y us Kumrswmy- Cordro d d Csro dsruo s h s l dsruo h Mrshl-Ol Mrshll d Ol 997 cosruco. A umr o ow dsruos r drvd s rculr css. Vrous rors o h roosd mly l ormulo o h d s dr mxur o xod sl dsruos ordr sscs moms mom r uco Ry roy ul uco d rdom sml ro hv vsd. Asymos shs d sochsc ordr r lso vsd. Th rmr smo y mhods o mxmum llhood hr lr sml sdrd rrors d codc rvls d mhod o mom r lso rsd. Two mmrs o h roosd mly r comrd wh corrsod mmrs o Kumrswmy- Mrshl-Ol- mly Alzdh l. 5 y o wo rl l d ss. Ky words: Kumrswmy- dsruo Mrshll-Ol mly Exod mly AIC BIC Powr whd mom. Prr

2 . Iroduco O o h rrrd r o rsrch h ld o h roly dsruo s h o r w dsruos sr wh s l dsruo y dd o or mor ddol rmrs ol mo hm r Azzl s swd mly Azzl 985 Mrshll-Ol xdd MOE mly Mrshll d Ol 997 xod mly E o dsruos u l. 998 or y comos mhods o com wo or mor ow com dsruo hrouh rsormos l -rd mly Eu l. Jos 4 mm-rd mls Zoros Blrsh 9 Rsc d Blrsh Kumrswmy - Kw mly Cordro d d Csro Ndrh l. Huss 3 McDold- Mc Alxdr l. xdd- mly Cordro l. Kumrswmy - rlzd mly Pscm l. xod rsormd rsormr mly ET Alzhl l. 3 xod rlzd Ex mly Cordro l. 3 omrc xol-posso mly Ndrh l. 3 rucdxol sw symmrc mly Ndrh l. 3 losc-rd Lo mly Tor d Mozr 4 d Kumrswmy Mrshl - Ol mly Alzdh l. 5. Whl h ddol rmrs r mor lxly h sm m hy lso comlcs h mhmcl orm o h rsul dsruo o cosdrly ouh o rdr o ml o urhr lycl d umrcl mulos. Bu wh h dv o sohscd owrul mhmcl d sscl sowrs ul s ow dys mor d mor comlx dsruos r ccd s vl modls o d lyss. Thr l. 5 rovdd dl ccou o how w mls o uvr couous dsruos r rd hrouh roduco o ddol rmrs.. ormuls d oos Hr w ls som ormuls o usd h susu scos o hs rcl. I T s couous rdom vrl wh d d cd P T h s Survvl uco s: P T Hzrd r uco hr: h / Rvrs hzrd r uco rhr: r / Cumulv hzrd r uco chr: H lo. h r Powr Whd Mom PWM: r d r

3 Eroy: Th Réy roy o T s v y I R lo d.. Mrshll-Ol Exdd MOE mly o dsruos Sr wh v sl dsruo wh roly dsy uco d cumulv dsruo uco cd Mrshll d Ol 997 roosd w lxl sm rmrc mly o dsruos d dd w survvl uco s MO y roduc ddol rmr. Th s MO o h MOE mly o dsruos s dd y MO whr d. Th rmr ' ' s ow s h l rmr s sc h hr o h w mly s shd low ov or h hr o h s l dsruo MO Nd d Ds. Th s or ll h h wh d h MO h wh whr h MO d h r h hr s o h MOE d sl dsruos rscvly. Now MO MO or MO d MO d d d d or 3 Whr d. I h w hv MO. Ohr rlly msurs l h hr rhr d chr ssocd wh r h MO MO MO h r MO MO MO r or r d H MO lo or lo rscvly. Whr h d r s h hr d rhr o h sl dsruo. 3

4 I s ovous h my w mls c drvd rom Mrshll-Ol s u y cosdr dr s l dsruo h uo. Ths w mls r usully rmd s Mrshll- Ol xdd dsruo. or dls s Thr l. 5. Jyumr d Mhw 8 rlzd h Mrshll-Ol s u y xo h Mrshll-Ol survvl uco MO. Ths mhod s clld mhod o Lhm lrv du o Lhm 953 Thr l. 5 roos ohr rlzo hrouh Lhm lrv du o Lhm 953 y xo h Mrshll-Ol cd s MO. or mor o Mrshll - Ol mly chc Br l. 9 Jos Krsh Brro-Souz l. 3 Cordro l. 4 Alzdh l. 5.3 Kumrswmy- Kw mly o dsruos or sl cd wh d Cordro d d Csro dd Kw dsruo wh cd d d Kw 4 d Kw 5 d Whr d r sh rmrs ddo o hos h sl d dsruo whch rly ovr swss d vro l whs. or lm rdom vrl '' h s hr rhr d chr or dsruo 4 r v rscvly y h r Kw Kw Kw Kw Kw Kw Kw Kw Kw d H lo. 4

5 Rcly Alzdh l. 5 roosd h Kumrswmy Mrshl-Ol mly o dsruos y us h Mrshl-Ol Mrshll d Ol 997 cd h Kumrswmy- Cordro d d Csro mly d sudd s my rors. Th m movo hd h rs rcl s o roos ohr mly o couous roly dsruo h rlzs h Kumrswmy- Cordro d d Csro mly d h Mrshll-Ol Exdd mly Mrshll d Ol 997 y r h ormr s h s l dsruo h lr. W cll hs w mly h Mrshll-Ol Kumrswmy- MOKw mly o dsruo whch comsss my ow mls o dsruos d sudy som s rl rors rmr smo d rl l lco h rs rcl. Th rs o hs rcl s orzd sv scos. I sco h w mly s dd lo wh s hyscl ss. Nx sco s dvod o rs som mor scl css o h mly lo wh hr sh d m rlly chrcrscs. I sco 4 w dscuss som mor rl rsuls o h roosd mly whl dr mhods o smo o rmrs r rsd sco 5. I sco 6 w rs wo xmls o comrv d. Th rcl ds wh coclud dscusso d rmr h l sco ollowd y dx o drv symoc codc ouds.. Nw Mrshll-Ol Kumrswmy- MOKw mly o dsruos W ow roos w xso o h MO Mrshll d Ol 997 mly y cosdr h cd d d o Kw Cordro d d Csro dsruo 4 d 5 s h d rscvly h MO ormulo 3 d cll MOKw s v y MOKw { MOKw dsruo. Th d o 6 Smlrly us uo 4 w h cd d hc h s o MOKw d MOKw MOKw MOKw rscvly s 7 8 hr: h MOKw MOKw MOKw 5

6 9 rhr : r MOKw MOKw MOKw MOKw MOKw chr: H lo lo or h d 6 rducs o h o Kw Cordro d d Csro h d 6 rducs o h o MO Mrshll d Ol ss o h dsruo L N suc o.. d. rdom vrls wh survvl uco d T l N osv r rdom vrl dd o h T s v y h o omrc dsruo wh rmr sy x x x. Th h vrs o x coms x x.w c vry h uo 8 coms rom MOKw { or d MOKw { or. or oh css uo 8 rrss h survvl uco o h mmum d mxmum o N whr N hs roly r uco x. wh rmr d rscvly.. Sh o h dsy d hzrd ucos Hr w hv lod h d d hr o h MOKw or som chocs o h dsruo d rmr vlus o sudy h vry o shs ssumd y h mly. T 6

7 MOKw-E MOKw-W Dsy =7.5=.5=.5=.8 =.4=.7=.5=.9 =6.5=.5=.5=5.5 =.5=.5=.7=.6 =.=.4=.=.5 =.5=.5=.5=.5 Dsy ==.9==.5=.9 =.=.5==.5=. =4=3.5=.5=.5=.5 =.5=.45==.5=.5 =.5=.6==.5=.5 =33=3.5=.=.5= MOKw-L MOKw-r Dsy =.55=.5=5=.5=.99 =5.5=.5==.5=. =.5=.45==.5=.5 =46=.5=.5=.5=.5 =.5=.6==.5=.5 =3.55=.5=5=.5=.99 Dsy =.=4.7=.3=9.3=65.5 ==5.9=.5==5.4 =.55=6.9=.=4=5.4 =.7=8.9=.=4=.4 ==75=.39==.5 ==9=.399== c d : Dsy los MOKw E MOKw W c MOKw L d d MOKw r dsruos. 7

8 MOKw-E MOKw-W hx =.5=.45=.9=.5 =.5=.5=.9=.5 =.5=.6=.5=.5 =.5=.5=.5=.5 =.5=.5==.5 =.5=.5==.5 hx =5=9.5=.55=.5=5.5 =45==.5=.5=5.5 =.6=.5=.5=4.5=.4 =.5=.45=.5=4.7=.9 =.=.5=.5=4.5=.7 =.=.9=.5=6.= MOKw-L MOKw-r hx =53=9.55=.65=.5=5 =5.6=.5 =.=.5=. =.5=.6=.=.9=. =6.5=.6 ==.5=. =5=9.6 =.65=.5=55 =.6=.4=.3=.9=.5 hx ==5.9=.5==5.4 ==4.9=.55=5=4.4 =.8=.9=.=39.5=3.4 =.7=5.9=.=4=5.5 ==4.9=.49==9.5 =.7=5.9=.=4= c d : Hzrd los MOKw E MOKw W c MOKw L d d MOKw r dsruos. rom h los ur d c s h h mly s vry lxl d c or my dr ys o shs o dsy d hzrd r uco clud h h u shd or hzrd. 3. Scl modls I hs sco w rovd som scl css o h MOKw mly o dsruos d ls hr m dsruol chrcrscs. 8

9 3. Th MOKw xol MOKw E dsruo L h s l dsruo xol wh rmr d h or h MOKw E modl w d: MOKwE cd : MOKwE s : MOKwE hr : h MOKwE rhr: : r MOKwE MOKwE MOKwE chr : H lo lo 3. Th MOKw Lomx MOKw L dsruo { Cosdr h Lomx dsruo hy l. 7 wh d d cd v y d d h d o h MOKw L dsruo s v y MOKwL d: cd: MOKwL s: MOKwL hr: h MOKwL { { { { { { rhr: r MOKwL { { { MOKwL MOKwE { chr: H lo lo { 3.3 Th MOKw Wull MOKw W dsruo 9

10 Cosdr h Wull dsruo hy l. 5 Zh d 7 d Cro wh rmrs d hv d d cd d rscvly w h d o W MOKw dsruo s MOKwW = cd: MOKwW /{ { s: MOKwW /{ { hr: h MOKwW /{ { rhr: r MOKwW chr: lo H MOKwW MOKwW lo 3.4 Th MOKw rch r MOKw dsruo Suos h s l dsruo s h rch dsruo Krsh l. 3 wh d d cd v y d rscvly d h h corrsod d o r MOKw dsruo coms d: MOKwr cd: MOKwr s: MOKwr hr: h MOKwr rhr: r MOKwr chr: lo H MOKwr MOKwr lo 3.5 Th MOKw omrz o MOKw dsruo Nx y h omrz dsruo sr l. 998 wh d d cd d rscvly w h d o o MOKw dsruo s

11 d: MOKwo { { cd: MOKwo { { s: MOKwo { { hr: h MOKwo rhr: r MOKwo { { { MOKwo MOKwo chr: H lo { lo { 3.6 Th MOKw EW dsruo { Th xdd Wull EW dsruos o urvch l. 997 hs h cd x E : D R whr E : s o-v moooclly crs uco whch dds o h rmr vcor. Th corrsod d s x E : : whr : s h drvv o E :. Som mor rculr css o EW c s s ollows: E : = : xol dsruo. E : = : Rylh Burr y- dsruo. E : = lo : Pro dsruo v E : = x : omrz dsruo. Hr w drv d: MOKwEW MOKw EW y cosdr EW s h s l dsruo s x E : : x E : { x E : { x E : cd: MOKwEW { x E : { x E :

12 s: MOKwEW hr: h MOKwEW rhr: r MOKwEW { x E : { x E : x E : : x{ E : { x E : { x E : x E : : x{ E : { x E : { x E : chr: H MOKwEW { x E : lo { x E : { x E : 3.7 Th MOKw Exdd modd Wull MOKw EMW dsruo Th cd d d o h modd Wull MW dsruo Srh d Zd 3 s v rscvly y x d x. Th corrsod d o MOKwEMW MOKwEMW s: MOKwEMW hr: h MOKwEMW MOKw EMW s v y x x x { x { x { x rhr: r MOKwEMW { x { x x x x { x x x x { x { x cd:

13 3 chr: lo H MOKwEMW MOKwEMW x { x { lo 3.8 Th MOKw Powr Lo-orml PL MOKw dsruo Powr Lo-orml dsruo roducd y Nlso d Dosoy 99 y xd h Loorml dsruo d s dsy d cd r rscvly v y l l d l. or rducs o h usul Lo-orml dsruo. Wh hs s h s l dsruo w h d o PL MOKw dsruo s MOKwPL l l l l l cd: MOKwPL l l s: MOKwPL l l hr: h MOKwPL l l l l l

14 4 rhr: r MOKwPL l l l l l l chr: lo H MOKwPL MOKwPL l l lo or rducs o h MOKw -Lo-orml dsruo. 3.9 Th MOKw Exdd Exod Pro EEP MOKw dsruo Th d d cd o h xod Pro dsruo o Ndrh 5 r v rscvly y d x. Thus h d o EEP MOKw dsruo s v y MOKwEEP { { { { cd: MOKwEEP { { s: MOKwEEP { { hr: h MOKwEEP rhr: r MOKwEEP { chr: H MOKwEEP = { { lo 3. Th MOKw Exdd Powr EP MOKw dsruo

15 Th cd d d o h xdd owr dsruo r d. Th corrsod d o MOKw EP d: ` { MOKwEP d dsruo s h v y cd: MOKwEP s : MOKwEP rh: h MOKwEP rhr: r MOKwEP chr: H MOKwEP ` lo ` { 4. rl rsuls or h Mrshll-Ol s Kumrswmy- MOKw mly o dsruos I hs sco w drv cr rl rsuls or h roosd MOKw mly ollow h mhods dscrd Brro-Souz l. 3 Cordro l. 4 d Alzdh l Srs xsos Cosdr h srs rrso z z.! Ths s vld or z d whr. s h mm uco. I us 6 w o MOKw { Kw Kw / d d Kw 5

16 6 Whr / / d Kw Kw s h s o Kw dsruo. A { MOKw { { Kw Kw Kw Kw Kw d d / Kw d d 3 Whr / / d Kw s h cd o Kw Cordro d d Csro dsruo uo 4. Smlrly xso or h survvl uco o MOKw or c drvs s MOKw { {! Kw Kw { Kw Kw

17 7 Th dsy uco 6 c lso xrssd s MOKw Hc or us w Kw Kw 4 / Kw d d 5 whr d / / or h survvl uco o MOKw c xrssd s MOKw / O us w! Kw Kw Kw Kw C whr C. 4. Ordr sscs

18 8 Suos T T T... s rdom sml rom y MOKw dsruo. L T : do h h ordr sscs. Th d o T : c xrssd s MOKw MOKw MOKw!!! : MOKw MOKw MOKw!!! MOKw MOKw!!! Now us h rl xso o h MOKw dsruo d d s w h d o h h ordr sscs or o h MOKw or s : Kw Kw!!! Kw whr d r dd rlr.!!! Kw Kw o Kw Kw 6 / Kw d d Kw d d Kw d d Kw 7 Whr d /. A us h rl xso o h d d s o MOKw dsruo w h d o h h ordr sscs or o h MOKw or s

19 9 Kw Kw Kw Kw C!!! :!!! Kw Kw Kw Kw C Whr C d dd sco 4. Now Kw Kw d C Ndrh. l 5 Whr h h h d C h C d Th dsy uco o h h ordr sscs o MOKw dsruo c xrssd s :!!! Kw Kw Kw Kw d Kw Kw Kw d!!! Kw Kw Kw 8 Whr d Rmr I: Euos -8 rvl h h dsy ucos o h MOKw dsruo d h o s ordr sscs c xrssd s roduc o h sl dsy x wh owr srs o. d lso s mxur o xod- dsruos udr Lhm lrvs. Ths rsuls ly mor rol d my usd o o xlc xrssos or h moms d

20 mom r uco m o h MOKw dsruo d o s ordr sscs rl rmwor d lso or scl modls us h corrsod rsuls o xod - dsruos. 4.3 Proly whd moms Th roly whd moms PWMs rs roosd y rwood l. 979 r xcos o cr ucos o rdom vrl whos m xss. Th hv s l cd dd y h r PWM o T s r r d rom uos d 4 h s h s mom o T or d c wr hr s E T s s d d E or s s s T d s d or s s s s E T s d d Whr r s h PWM o s { { r MOKw h or h sl dsruo. d

21 Th PWM c rlly usd or sm rmrs uls o rlzd dsruos. Ths moms hv low vrc d do ossss svr ss d hy comr vourly wh smors od y mxmum llhood Alzdh l. 5. Procd s ov w c drv h s mom o h h ordr ssc T: rdom sml o sz rom MOKw or d o us uos 6 d 8 s E T s s d s E T s rscvly. Whr h us d r dd sco 4. d Mom r uco Th mom r uco o xod MOKw mly c sly xrssd rms o hos o h Kw Cordro d d Csro dsruo us h rsuls o sco 4.. or xml us uo 5 c s h s s s / M s E d T / Whr ollows xod 4.5 Ry Eroy s d d d d Kw d Kw / d M s Kw Cordro d d Csro dsruo. Th roy o rdom vrl s msur o ucry. Th Réy roy s dd s I R lo d whr d or urhrs dls s So. or us xso 6 w c wr { Thus or h Réy roy o T c od s I R lo {! { d Whr!

22 Th dsy uco 6 c xrssd s MOKw Thror { or us xso w! { Thus or h Réy roy o T lso c od s d r I R lo Whr! r r 4.6 Qul uco d rdom sml ro W shll ow rs ormul or r MOKw rdom vrl y us vrso mhod y vr h cd or h survvl uco. MO MO MO MO MO MO MO 9 Now or h Kw w hv. Thror lo lo lo lo { { Now com 9 d w or MOKw u u Whr ~ U u

23 Th h Qul or MOKw c sly od rom s or xml l h xol dsruo wh rmr hv d d cd s : d : rscvly. Th h h ul s od s / lo. Thror h h ul o MOKw E s v y lo 4.7 Asymos d shs Hr w vs h symoc shs o h roosd mly ollow h mhod ollowd Alzdh l. 5. Prooso. Th symos o uos 6 7 d 9 s r v y ~ s ~ s h ~ s Prooso. Th symos o uos 6 7 d 9 s r v y ~ s ~ s h ~ s Th shs o h dsy d hzrd r ucos c dscrd lyclly. Th crcl os o h MOKw dsy uco r h roos o h uo: d lo d 3

24 4 Th umr o roos o my mor h o. I rculr s roo o h corrsods o locl mxmum locl mmum or o o lxo dd o whhr or whr lo d d h MOKw h MOKw { h MOKw { h MOKw Th crcl os o h r h roos o h uo d h d lo 3 Th umr o roos o 3 my mor h o. I rculr s roo o 3 h corrsods o locl mxmum locl mmum or o o lxo dd o whhr or whr lo h d d

25 5 h MOKw { h h MOKw MOKw { h MOKw 4.8 Sochsc ordrs I hs sco w sudy h rlly rors d sochsc ordr o h MOKw dsruos Sochsc ordr rors hv lcos dvrs lds such s coomcs rlly survvl lyss surc c curl d mm sccs Shd d Shhumr 7. L d Y wo rdom vrls wh cds d rscvly survvl ucos d d corrsod d s. Th s sd o smllr h Y h llhood ro ordr Y lr s dcrs sochsc ordr Y s or ll hzrd r ordr Y hr s dcrs rvrsd hzrd r ordr Y rhr s dcrs. Ths our sochsc ordrs r rld o ch ohr s Y Y Y Y s hr lr rhr 4 Thorm: L ~ ~ MOKw Y d MOKw. I h Y lr Y Y Y s rhr hr. Proo: Sc d d { { { 3 { { { {

26 6 3 { { 3 { {. Whch s lwys lss h. Hc / s dcrs. Th s Y lr.th rm sms ollow rom h mlcos Esmo 5. Mxmum llhood smo or MOKw Th modl rmrs o h MOKw dsruo c smd y mxmum llhood. L T... rdom sml o sz rom MOKw wh rmr vcor T T θ whr T... corrsods o h rmr vcor o h sl dsruo. Th h lo-llhood uco or θ s v y lo lo lo θ lo lo lo lo lo lo 5 Ths lo-llhood uco c o solvd lyclly cus o s comlx orm u c mxmzd umrclly y mloy lol omzo mhods vll wh sowrs l R Mhmc. By h rl drvvs o h lo-llhood uco wh rsc o d comos o h scor vcor T T U U U U U θ c od s ollows: U U lo lo lo U lo lo

27 7 U Solv h uos T T U U U U U smulously vs h mxmum llhood sm MLE T T ˆ ˆ ˆ ˆ ˆ o T T. 5. Asymoc sdrd rror d codc rvl or h mls Th symoc vrc-covrc mrx o h MLEs o rmrs c od y vr h shr ormo mrx I θ whch c drvd us h scod rl drvvs o h lollhood uco wh rsc o ch rmr. Th h lms o I θ r v y l E 3 I θ Th xc vluo o h ov xcos my cumrsom. I rcc o c sm I θ y h osrvd shr s ormo mrx ˆ Î θ s dd s: l 3 Î ˆ θ θ θ Us h rl hory o MLEs udr som rulry codos o h rmrs s h symoc dsruo o ˆ θ θ s V N whr I θ v V. Th symoc hvour rms vld V s rlcd y ˆ Î ˆ θ V. Ths rsul c usd o rovd lr sml sdrd rrors d lso cosruc codc rvls or h modl rmrs. Thus roxm sdrd rror d % / codc rvl or h ml o h rmr r rscvly v y vˆ d v Z ˆ ˆ / whr / Z s h / o o sdrd orml dsruo. As llusro o h MLE mhod s lr sml sdrd rrors codc rvl h cs o E MOKw s dscussd dx. 5.3 Esmo y mhod o moms Hr lrv mhod o smo o h rmrs s dscussd. Sc h moms r o closd orm h smo y h usul mhod o moms my o rcl. Thror ollow Brro-Souz l. 3 w

28 E { d d L u h du d du u lou lo lo v v E { d v { u v du v u v v v lo v v E { 8 v v v 3... O c us 8 o v w mhod o smo.. or rdom sml d... rom oulo wh survvl uco 8 h modl rmrs c smd rom h uos lo v v v v v Rl l lcos I hs susco w cosdr o wo rl d ss o show h h dsruos rom h roosd MOKw dsruo c r modl h h corrsod dsruos rom KwMO Alzdh l. 5 y cosdr h rch d h Exol dsruo s our. Hr h rmrs r smd y umrcl mxmzo o lo-llhood uco d hr symoc sdrd rrors d 95% codc rvls r comud us lr sml roch. I ordr o comr h dsruos w hv cosdrd ow crr l AIC A Iormo Crro BIC Bys Iormo Crro CAIC Coss A Iormo Crro d HQIC H-Qu Iormo Crro I my od h AIC l BIC lo l CAIC AIC / d HQIC lolo l whr s h umr o rmrs h sscl modl h 8

29 sml sz d l s h mxmzd vlu o h lo-llhood uco udr h cosdrd modl. W hv usd h mxmum llhood mhod or sm h modl rmrs. Exml I: Hr w wor wh h ollow d ou 346 co msurms md rom svrl rds o crs 998. Th d hv collcd y h drl Trd Commsso whch s dd cy o h US ovrm whos m msso s h romoo o cosumr roco. h: // rors/occo or h: // w.com.com/ rdvs/ smo. hml. { I Tl h MLEs d sdrd rrors SEs d 95% codc rvls rhss o h rmrs rom ll h d dsruos lo wh AIC BIC CAIC d HQIC vlus r rsd. Accord o h lows vlus o h AIC BIC CAIC d HQIC h MOKw r MOKw E dsruo could chos s h r modl h KwMO r KwMO E dsruo rscvly. Mor ormo s rovdd or vsul comrso s rsd h orm o hsorm o h d wh h d dss urs 3. Th d cd s r lso dslyd urs 4. Ths los dc h h roosd dsruos rovd ood o hs d. 9

30 Tl : MLEs sdrd rrors d 95% codc rvls rhss d Prmrs h AIC BIC CAIC d HQIC vlus or h co msurms d. â ˆ ˆ ˆ ˆ KwMO r MOKw r KwMO E MOKw E lo-llhood AIC BIC CAIC HQIC Esmd d's Esmd d's MOKw-r KwMO-r MOKw-E KwMO-E Dsy Dsy : 3 Plos o h osrvd hsorm d smd d s or h MOKw r KwMO r d MOKw E KwMO E dsruos or xml I. 3

31 Esmd cd's Esmd cd's cd MOKw-r KwMO-r cd MOKw-E KwMO-E : 4 Plos o h osrvd ov d smd cd s or h MOKw r KwMO r d MOKw E KwMO E dsruos or xml I. Exml II: Ths d s cosss o osrvos o r srss o cro rs v y Nchols d Pd 6. { I Tl h MLEs d sdrd rrors SEs d 95% codc rvls rhss o h rmrs rom ll h d dsruos lo wh AIC BIC CAIC d HQIC vlus r rsd. Accord o h lows vlus o h AIC BIC CAIC d HQIC h MOKw r MOKw E dsruo could chos s h r modl h KwMO r KwMO E dsruo rscvly. 3

32 Tl : MLEs sdrd rrors d 95% codc rvls rhss d h AIC BIC CAIC d HQIC vlus or h r srss o cro rs d Prmrs KwMO r MOKw r KwMO E MOKw E â ˆ ˆ ˆ ˆ lo-llhood AIC BIC CAIC HQIC L or h ls xml hr lso w hv rsd h hsorm o h osrvd d wh h d dss urs 3 d d cd s wh mrcl cd urs 4 or vsul sco o h closss o h s. Ths los dc h h dsruos rom h roosd mly rovd closr o hs d. 3

33 Esmd d's Esmd d's Dsy MOKw-r KwMO-r Dsy MOKw-E KwMO-E : 5 Plos o h osrvd hsorm d smd d s or h MOKw r KwMO r d MOKw E KwMO E dsruos or xml II. Esmd cd's Esmd cd's cd MOKw-r KwMO-r cd MOKw-E KwMO-E : 6 Plos o h osrvd ov d smd cd s or h MOKw r KwMO r d MOKw E KwMO E dsruos or xml II. 7. Cocluso I hs r h w mly o Mrshll-Ol xdd Kumrswmy rlzd dsruos s roducd d som o s dsruol d mhmcl rors r vsd. Th mxmum llhood d mom mhod or sm h rmrs r lso dscussd. Comrv d 33

34 34 modl y wo mor mmrs o h roosd MOKw mly wh corrsod mmrs o KwMO mly rvld ormrs surory ll xmls cosdrd hr. Adx: Mxmum llhood smo or E MOKw Th d o h E MOKw dsruo s v y MOKwE or or rdom sml o sz rom hs dsruo h lo-llhood uco or h rmr vcor T θ s v y θ lo lo lo lo lo lo Th comos o h scor vcor T θ r θ θ lo lo lo θ lo lo θ. Th symoc vrc covrc mrx or mls o h rmrs o E MOKw dsruo s smd y

35 35 ˆ Î θ ˆ vr 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 vr ˆ ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ ˆ cov ˆ vr Whr h lms o h ormo mrx θ θ θ θ I ˆ ˆ ˆ l c drvd us h ollow scod rl drvvs: lo { lo { lo lo lo { lo lo { {

36 36 lo lo lo { lo lo lo { lo lo lo lo { lo lo { lo lo lo

37 { Whr. s h drvv o h dmm uco. lo lo Rrcs. Alxdr C. Cordro.M. Or E.M.M. d Sr J.M.. rlzd rd dsruos. Comu S D Al 56: Alzdh M. Thr M.H. Cordro.M. Zur M. d Hmd.. 5. Th Kumrswmy Mrshl-Ol mly o dsruos. Jourl o h Ey Mhmcl Socy I rss. 3. Alzhl A. L C. d moy. 3. Exod T- mly o dsruos wh som lcos. I J Ss Pro : Azzl A A clss o dsruos whch cluds h orml os. Scd J Ss : Br H.M. hy M.E. d AL-Huss E.K. 9. Asymoc dsruos o ordr sscs d rcord vlus udr h Mrshll-Ol rmrzo oro. Commu Ss Thor Mh 38: Brro-Souz W. Lmo A.J. d Cordro.M. 3. rl rsuls or Mrshll d Ol's mly o dsruos. A Acd Brs Cc 85: Cro C.. Ts or h Mrshll-Ol xdd orm o h Wull dsruo. S P 5: Cordro.M. Lmo A.J. d Or E.E.M. 4. Th Mrshll-Ol mly o dsruos: Mhmcl rors d w modls. J S Thor Prc 8: Cordro.M. d D Csro M.. A w mly o rlzd dsruos. J S Comu Smul 8: Cordro.M. Slv.O. d Or E.M.M.. Th xdd Wull dsruo. J Pro S Sc : Cordro.M. Or E.M.M. d DA Cuh D.C.C. 3. Th xod rlzd clss o dsruos. J D Sc : -7.. Cordro.M. Lmo A.J. d Or E.E.M. 4. Th Mrshll-Ol mly o dsruos: Mhmcl rors d w modls. J S Thor Prc 8:

38 3. Eu N. L C. d moy.. B-orml dsruo d s lcos. Commu Ss Thor Mh 3: hy M.E. Al-Awdh.A. d Alhl L.A. 7. Mrshll-Ol xdd Lomx dsruo d s lco o csord d. Commu S Thory Mhod 36: hy M.E. Al-Huss E.K. d Al-Jrllh R.A. 5. Mrshll-Ol xdd Wull dsruo d s lco o csord d. J Al S 3: sr P.W. Ch M.N. Ro P.V. Shusr J.J. d Pull J Modll cur rs us h omrz modl wh covr ormo. S Md 78: rwood J.A. Ldwhr J.M. Mls N.C. d Wlls J.R Proly whd moms: do d rlo o rmrs o svrl dsruos xrssl vrs orm. Wr Rsour Rs 5: u R.C. u P.I. d u R.D Modl lur m d y Lhm lrvs. Commu Ss Thor Mh 7: urvch M. DBdo A. d Rd S A w sscl dsruo or chrcrz h rdom srh o rl mrls. J. Mr. Sc. 3: Huss M.A. 3. Esmo o PY< or h clss o Kumrswmy dsruo. Aus J Bsc Al Sc 7: Jyumr K. d Mhw T. 8. O rlzo o Mrshll-Ol schm d s lco o Burr y II dsruo. S P 49: Jos M.C. 4. mls o dsruos rs rom h dsruos o ordr sscs. Ts 3: Jos K.K.. Mrshll-Ol mly o dsruos d hr lcos rlly hory m srs modl d srss srh lyss. Proc. ISI 58h World Ss. Cor I S Is s-6h Auus Dul Jos K.K. d Krsh E.. Mrshll-Ol xdd uorm dsruo. ProS orum 4: Krsh E. Jos K.K. d Rs c M. 3. Alcos o Mrshl-Ol réch dsruo. Comm. S. Smul. Comu. 4: Krsh E.. Mrshll-Ol rlzo o Som Dsruos d Thr Alcos. Ph. D. Thss Mhm hd Uvrsy Koym Krl Id. Uulshd. 7. Mrshll A. d Ol I A w mhod or dd rmr o mly o dsruos wh lcos o h xol d Wull mls. Bomr 84:

39 8. Mord Alzdh uss M. Cordro d Clrc rc B. Dméro 5. Th Mrshll-Ol mly o dsruos. Jourl o Sscl Dsruos d Alcos :4 do:.86/s Ndrh S. 5. Exod Pro dsruos. Sscs 39: Ndrh S. Cordro.M. d Or E.M.M.. rl rsuls or h Kumrswmy dsruo. J S Comu Smul 87: Ndrh S. Ccho V.. d Or E.M.M. 3. Th omrc xol Posso dsruo. S Mhods Al : Ndrh S. Jyumr K. d Rsc M.M. 3. A w mly o lm modls. J S Comu Smul 83: Nd A.K. d Ds S.. Sochsc ordrs o h Mrshll-Ol xdd dsruo. Ss Pro L 8: Nlso W. d Dosoy N. 99. Comur rorm owor or h owrorml d-loorml modls o l or srh d rom scms o vrous szs. NASA STI/Rco Tchcl Ror N 9: Nchols M.D. d Pd W.J.A. 6. A oosr corol chr or Wull rcls. Quly d Rlly Er Irol v. : Pscm R.R. Cordro.M. Dmro C..B. Or E.M.M. d Ndrh S.. Th w clss o Kummr rlzd dsruos. SORT 36: Rsc M.M. d Blrsh. Th mm-xod xol dsruo. J S Comu Smul 8: Srh A.M. d Aloo J. 3. Exod modd Wull xso dsruo. Rl E Sys Sy : Shd M. d Shhumr J.. 7. Sochsc Ordrs Srr Nw Yor. 4. So KS. Réy ormo lollhood d rsc dsruo msur. EM J S Pl Ir 93: Thr M.H. d Ndrh S. 5. Prmr duco couous uvr dsruos: Wll-slshd mls. As d Acdm Brslr d Cêcs Als o h Brzl Acdmy o Sccs 87: Tor H. d Mozr N.H. 4. Th losc-uorm dsruo d s lco. Commu Ss Smul Comu 43: Zh T. d M. 7. lur d lyss wh xdd Wull dsruo. Commuco Sscs Smulo d Comuo 36:

40 44. Zoros K. d Blrsh 9. O mls o - d rlzd mm rd dsruos d ssocd rc. S Mhodol 6: