A note on Kumaraswamy Fréchet distribution

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AENSI Jourls Austrl Jourl of Bsc d Appld Sccs ISSN:99-878 Jourl hom pg: wwwswcom A ot o Kumrswmy Frécht dstruto Md M E d 2 Ad-Eltw A R Dprtmt of Sttstcs Fculty of Commrc Zgzg Uvrsty Egypt 2 School of

llhood stmto A B S T R A C T Th modlg d lyss of lftms s mportt spct of sttstcl wor wd vrty of sctfc d tchologcl flds For th frst tm th clld Kumrswmy Frécht dstruto troducd d studd Th dstruto hs umr

1 AENSI Jourls Austrl Jourl of Bsc d Appld Sccs ISSN: Jourl hom pg: wwwswcom A ot o Kumrswmy Frécht dstruto Md M E d 2 Ad-Eltw A R Dprtmt of Sttstcs Fculty of Commrc Zgzg Uvrsty Egypt 2 School of Busss Sccs Zgzg Egypt A R T I C L E I N F O Artcl hstory: Rcvd 25 Ju 204 Rcvd rvsd form 8 July 204 Accptd 25 August 204 Avll ol 4 Octor 204 Kywords: Kumrswmy Frécht dstruto Epottd Frécht dstruto Mmum llhood stmto A B S T R A C T Th modlg d lyss of lftms s mportt spct of sttstcl wor wd vrty of sctfc d tchologcl flds For th frst tm th clld Kumrswmy Frécht dstruto troducd d studd Th dstruto hs umr of w d wll-ow lftm spcl su-modls For ths modl som of ts sttstcl proprts r studd Th mthod of mmum llhood s usd for stmtg th modl prmtrs d th osrvd formto mtr s drvd Two pplctos to rl dt sts r gv to llustrt th pottlty of ths dstruto 204 AENSI Pulshr All rghts rsrvd ToCt ThsArtcl:Md ME d Ad-Eltw AR A ot o Kumrswmy Frécht Dstruto Aust J Bsc & Appl Sc 8(5): INTRODUCTION Th Frécht dstruto hs pplctos rgg from cclrtd lf tstg through to rthqus floods hors rcg rfll quus suprmrts s currts wd spds d trc rc rcords Th oo of Kotz d Ndrh (2000) dmostrts th pplclty of th Frécht dstruto svrl flds Cordro d d Cstro (200) troducd th Kumrswmy-grtd fmly of dstrutos Th cumultv dstru- to fucto (cdf) of th Kumrswmy grlzd dstrutos (KG) s gv y F G () Th prolty dsty fucto (pdf) corrspodg to () ts th form f g G G (2) Svrl grlzd dstrutos from (2) hv dfd d vstgtd th ltrtur cludg th Kumrswmy Wull dstruto y Cordro t l(200) th Kumrswmy grlzd gmm dstruto y d Cstro t l (20) d th Kumrswmy grlzd hlf-orml dstruto y Cordro t l (202) I ths ppr w propos th Kumrswmy Frécht (KF) dstruto whch tds th Frécht (F) d th pottd Frécht (EF) dstrutos th sm st-up Cordro d d Cstro (200) wth th hop tht t wll ttrct wdr pplcto coomcs ddtolly s svrl rs of study 2 Th Kumrswmy Frécht dstruto: Th cdf d pdf of th Frécht dstruto r rspctvly - G( ; ) 0 0 (3) d - g( ; ) 0 0 (4) Th cdf of th KF dstruto c dfd y susttutg G ( ; ) to quto () Hc t wll ( ) ( ; ) F (5) d th corrspodg pdf s dfd s Corrspodg Author: Md ME Dprtmt of Sttstcs Fculty of Commrc Zgzg Uvrsty Egypt

2 295 Md ME d Ad-Eltw AR f ( ; ) 0 0 (6) whr s scl prmtr d th othr prmtrs d r shp prmtrs Plots of th KF for slctd prmtr vlus 08 & 5 d dffrt vlus of d r gv Fgur =8 =22 =2=22 =5 =2 =2 =05 =08= Fg : Som possl shps of th KF dsty fucto Th survvl fucto S() hzrd rt fucto h() rvrsd hzrd rt fucto r() d th cumultv hzrd rt fucto H() of KF r gv y S( ) F( ) ( ) f( ) - h( ) ( ) S( ) - - f( ) r ( ) F ( ) ( ) d H( ) S( ) ( ) Plots of th HRF for slctd prmtr vlus & 6 d dffrt vlus of r gv Fgur =05 =008 =5 =009 =20 =25 =2 =3 =67= Fg 2: Som possl shps of th HRF 3 Som Spcl Css Not tht th KF dstruto hs svrl wll ow modls s spcl css whch m t of dstgushl sctfc mportc from othr dstrutos - If w t quto (6) rducs to th pottd Frécht (EF) dstruto prstd y Ndrh d Kotz (2003) 2- Sttg w ot th Frécht dstruto 3- If 2 th dsty (6) corrspods to th Kumrswmy vrs Rylgh (KIR) dstruto 4- If 2 d quto (6) coms th pottd vrs Rylgh (EIR) dstruto 5- For th cs 2 d th dsty (6) gvs th vrs Rylgh (IR) dstruto 6- Sttg quto (6) ylds th Kumrswmy vrs potl (KIE) dstruto

3 296 Md ME d Ad-Eltw AR If w gt th pottd vrs potl (EIE) dstruto 8- Wh th pdf (6) rducs to th vrs potl (IE) dstruto 9- Sttg p th dsty (6) coms th Kumrswmy Guml typ-2 (KGuII) dstruto (Shhz t l 202) 0- Gusmáo t l (20) troducd thr prmtrs lftm dstruto co-clld th grlzd vrs Wull dstruto to td som wll-ow dstrutos th lftm ltrtur Thy cocludd tht thr w dstruto s much mor fll th th vrs Wull dstruto d could hv crsg dcrsg d umodl hzrd rts whch r qut rllty d ologcl studs For qc w gt th Kumrswmy grlzd vrs Wull (KGIW) dstruto whch s th tso of grlzd vrs Wull dstruto dfd y Gusmáo t l (20) wth pdf - c q - c q f ( ; c ) c q 0 c 0 (7) For th cs quto (7) ylds th pottd grlzd vrs Wull (EGIW) dstruto ddto to t rducs to grlzd vrs Wull dstruto Tl provds som spcl su-modls of th KF dstruto 4 Epsos for th cumultv d dsty fuctos d sttstcl proprts: Hr w provd smpl psos for th cdf of th KF dstruto dpdg o whthr th prmtr (or ) s rl o-tgr or tgr W cosdr th srs pso ( ) ( ) z z (8) 0! ( ) vld for z < d 0 rl d o-tgr Applcto of (8) to (5) f s rl o-tgr gvs ( ) ( ) F ( ; ) (9) 0! ( ) Ag usg quto (6) w ot ( ) F( ; ) d! ( ) y ( ) dy! ( ) ( ) 0! ( ) J For tgr th sum (9) smply stops t Ag usg (8) w c rwrt (6) s ( ) f ( ; ) 0! ( )( ) whr 0 g ; 0 ( )! ( )( ) d g ; dots th Frécht dsty fucto wth shp prmtr d scl prmtr s tgr th sum (0) s ft d stops t ( ) If 0 Now w dscuss som sttstcl proprts of th KF dstruto (0) -Th mod: Th mod m m m m for KF dstruto s gv y th soluto of ()

4 297 Md ME d Ad-Eltw AR 204 -Th qurtl: p Th qurtl of ordr p for th KF dstruto s gv y th soluto of p ( ) ( p) (2) spcl qurtls my otd y (2) for mpl f p 2 th md s gv -Th momt: As wth y othr dstruto my of th trstg chrctrstcs d fturs of th KF dstruto th c studd through th momts W ot mmdtly th r momt of th KF dstruto Usg th quto (0) th r momt out zro c otd s r r E( ) g ; d 0 0 Bsd o th trsformto y ov prsso d smplfyg th rth momt of KF f r r s r r r E( ) r r (3) 0 Th m vrc Swss d Kurtoss c otd from (3) If 0 s tgr d r th sum stops t If d r quto (3) gvs th rth momt of th EF dstruto wth prmtrs d dstruto -Momt grtg fucto: Now w c drv th momt grtg fucto of th KF dstruto Rcll tht (y Tylor's srs t pso of out zro) t t 0! so th momt-grtg fucto (MGF) of th KF dstruto s gv y t t ( t) E( ) f ( ) d 0 t ( ) d - 0! 0 0! ( ) t ( ) d!! ( ) Bsd o th trsformto y w gt t t ( ) ( ) ( t) E( ) 0 0!! ( ) ( ) 5 Estmto d Fshr formto mtr: Th mmum llhood stmto (MLE) s o of th most wdly usd stmto mthod for fdg th uow prmtrs Lt 2 dpdt rdom smpl from KF Th totl logllhood s gv y ( ) ( ) ( ) ( ) ( ) ( ) ( Z ) ( ) ( W ) whr Z ( ) Z ( ) D d W D Th scor vctor hs compots (4)

5 298 Md ME d Ad-Eltw AR 204 Z Z D W ( ) ( ) W ( ) ( ) ( ) Z ( Z ) ( ) D Z W ( Z ) d ( ) Z Z D W Th mmum llhood stmts (MLEs) of th uow four prmtrs c otd y solvg th systm of olr qutos 0 trtvly For trvl stmto d hypothss tsts o th modl prmtrs w rqur th osrvd formto mtr J J J J J J J J ( ) J J J whos lmts r 2 2 ( ) Z 2 DW Z DW ( ) ( ) ( ) Z DW ( Z ) Z ( DW ) Z ( Z ) Z DW Z DW Z 2 Z DW ( Z ) Z DW 2 2 ( ) ( ) ( ) ( ) 2 Z Z D Z W Z Z D W ( W ) ( D Z W Z Z Z D W W Z Z ( ) Z ( ) 2 2 D W Z D W ( W ) Z 2 ( ) ( ) ( ) ( ) 6 Applcto mpls: To llustrt th w rsults prstd ths ppr w ft th KF dstruto to two mpls of rl dt Th frst mpl s ucsord dt st from Nchols d Pdgtt (2006) cosstg of 00 osrvtos o rg strss of cro frs ( G) Th dt r s follows : Th scod dt st s otd from Smth d Nylor (987) Th dt r th strgths of 5 cm glss frs msurd t th Ntol Physcl Lortory Egld Ufortutly th uts of msurmt r ot gv th ppr Th dt r s follows:

6 299 Md ME d Ad-Eltw AR Ths dt wr prvously studd y Souz t l (20) for t Frécht (BF) pottd Frécht (EF) d Frécht dstrutos I th followg w shll compr th proposd KF dstruto (d thr sumodls KIR KIE EF d F dstrutos ) wth svrl othr thr- d four-prmtr lftm dstrutos mly: th Zogrfos-Blrsh log-logstc (ZBLL) (Zogrfos d Blrsh 2009) th t Frécht (BF) (Ndrh d Gupt 2004 d Souz t l 20) d rctly th Kumrswmy Prto (KP) ( Bourgugo t l 203) modls wth corrspodg dsts: 2 ZBLL : fzbll ( ; ) ( ) ( ) 0 ( ) 0 B( ) ( ) ( ) ( ) BF : fbf ( ; ) ( ) KP : fkp( ; ) ( ) ( ) whr 0 Tl 2 lsts th MLEs of th modl prmtrs for KF KIR KIE BF KP ZBLL EF d F dstrutos th corrspodg stdrd rrors (gv prthss) d th sttstcs ( ˆ ) (whr ( ˆ ) dots th logllhood fucto vlutd t th mmum llhood stmts) A formto crtro (AIC) th Bys formto crtro (BIC) d H-Qu formto crtro (HQIC) Sc th KF dstruto hs th lowst ( ˆ ) AIC BIC d HQIC vlus mog ll th othr modls d so t could chos s th st modl Addtolly t s vdt tht th KIR dstruto prsts th worst ft to th frst dt Smlrly th rsults gv Tl 3 llustrt tht th KF d F dstrutos r th st d th worst modls rspctvly ccordg to th scod dt Th rqurd umrcl vlutos r mplmtd usg th MATHCAD PROGRAM Tl : Som spcl su-modls of th KF dstruto dstruto 2 Kumrswmy vrs Rylgh Kumrswmy vrs potl Epottd Frécht (Ndrh d Kotz2003) 2 Epottd vrs Rylgh Epottd vrs potl Frécht 2 Ivrs Rylgh Ivrs potl p Kumrswmy Guml typ-2(kguii) dstruto (Shhz t l 202) qc Grlzd vrs Wull (Flp t l20) qc Kumrswmy grlzd vrs Wull qc Epottd grlzd vrs Wull Tl 2: MLEs (stdrd rrors prthss) d th sttstcs ( ˆ ) AIC BIC d HQIC; frst dt st Modl KF( ) (2393) KIR( ) (3733) KIE( ) (2663) BF( ) (0236) KP( ) (0502) ZBLL( ) (004) EF( ) (3954) F( ) 8956 (02) (6863) 3974 (064) (2077) (3552) (49552) (00093) (4666) (04) Estmts ( ˆ ) Sttstcs AIC BIC HQIC (2259) (0028) (902) (3006) (252) (09) (0045) (0288) (00897)

7 300 Md ME d Ad-Eltw AR 204 Tl 3: MLEs (stdrd rrors prthss) d th sttstcs ( ) AIC BIC d HQIC; scod dt st Modl KF( ) (7982) KIR( ) (905) KIE( ) 2699 (33675) BF( ) (85) KP( ) (0524) ZBLL( ) 6444 (032) EF( ) (663) F( ) (0059) Estmts (53948) (632) (3473) (8238) (92579) (00039) (2945) (0234) ˆ ( ˆ ) Sttstcs AIC BIC HQIC 2623 (4555) (007) (2955) (37523) (085) (08) (0083) (0632) (036) Cocluso: I fct th KF dstruto rprsts grlzto of som dstrutos prvously cosdrd th ltrtur such s th KGIW KIR KIE EF (Ndrh d Kotz 2003) d Frécht dstrutos Som of ts mthmtcl d sttstcl proprts r studd Prmtr stmto s pprochd y mmum llhood d th osrvd formto mtr s drvd Two umrcl mpls llustrt tht th KF dstruto provds ttr fts th thr su-modls (KIR KIE EF d F dstrutos) d th th othr modls slctd from th ltrtur W hop tht th proposd tdd modl my ttrct wdr pplctos survvl lyss REFERENCES Bourgugo M RB Slv LM Z d GM Cordro 203 Th Kumrswmy Prto dstruto J of Stt Thory d Applctos 2: Cordro GM d M d Cstro 200 A w fmly of grlzd dstrutos Jourl of Sttstcl Computto d Smulto 8: Cordro GM EMMOrtg d S Ndrh 200 Th Kumrswmy Wull dstruto wth pplcto to flur dt J Frl Ist 347: Cordro GM R Pscm d EMM Ortg 202 Th Kumrswmy grlzd hlf- orml dstruto for swd postv dt J Dt Sc 0: Cstro MAR EMM Ortg d GM Cordro 20 Th Kumrswmy grlzd gmm dstruto wth pplcto survvl lyss Stt Mthodol 8: Gusmáo FR EM Ortg d GM Cordro 20 Th grlzd vrs Wull dstruto Sttstcl Pprs Kotz S d S Ndrh 2000 Etrm vlu dstrutos: Thory d pplctos Imprl Collg Prss Ndrh S d AK Gupt 2004 Th t Frécht dstruto Fr Est Jourl of Thortcl Sttstcs 4: 5-24 Ndrh S d S Kotz 2003 Th pottd Frécht dstruto Avll t Itrstt sttourls t Nchols MD d W J Pdgtt 2006 A ootstrp cotrol chrt for Wull Prctls Qulty d Rllty Egrg Itrtol 22: 4-5 Smth RL d JC Nylor 987 A comprso of mmum llhood d Bys stmtors for th thr-prmtr Wull dstruto Appld Sttstcs 36: Shhz MQ S Shhz d NS Butt 202 Th Kumrswmy vrs Wull dstruto Pst Jourl of Sttstcs d Oprto Rsrch 8: Souz WM GM Cordro d AB Sms 20 Som rsults for t Frécht dstruto Commu Sttst Thory-Mth 40: Zogrfos K d N Blrsh 2009 O fmls of t- d grlzd gmm-grtd dstrutos d ssoctd frc Stt Mthod 6:

Statistical properties and applications of a Weibull- Kumaraswamy distribution

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