Generalizedextended Weibull Power Series Family of Distributions

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1 Arca Rvw o Mathatcs ad Statstcs Dcbr 205 Vol. 3 No. 2 pp SSN: (Prt (Ol Copyrght Th Author(s. All Rghts Rsrvd. Publshd by Arca Rsarch sttut or Polcy Dvlopt DO: /ars.v32a8 URL: Gralzdxtdd Wbull Powr Srs Faly o Dstrbutos Sad H. Alkar Abstract ths study w troduc a w alyo odls or lt data calld gralzd xtdd Wbullpowr srs aly o dstrbutos by copoudggralzdxtdd Wbull dstrbutos ad powr srs dstrbutos. Th copoudg procdur ollows th sa stup carrd out by Adads (998. Th proposd aly cotas all typs o cobatos btw trucatd dscrt wth gralzd ad ogralzd Wbull dstrbutos. So xstg powr srs ad subclasss o xd lt dstrbutos bco spcal cass o th proposd aly such as th copoud class o xtdd Wbull powr srsdstrbutos proposd by Slva t al. (203 ad th gralzd xpotal powr srs dstrbutostroducd by Mahoud ad Jaar (202.So athatcal proprts o th w class ar studd cludgth cuulatv dstrbuto ucto dsty ucto survval ucto ad hazard rat ucto. Th thod o axu lklhood s usd or obtag a gral stup or statg th paratrs o ay dstrbuto ths class. A xpctato-axzato algorth s troducd or statg axu lklhood stats.spcal subclasss ad applcatos or so odls aral datastar troducd to dostrat th lxblty ad th bt o ths w aly. Kyword: Gralzd xtdd Wbull powr srs dstrbutos Wbull powr srs dstrbuto gralzd powr srs dstrbutos. troducto Th odlg ad aalyss o lts s a portat aspct o statstcal work a wd varty o sctc ad tchologcal lds such as publc halth actuaral scc bodcal studs dography ad dustral rlablty. rsk odlg th lt assocatd wth a partcular rsk s ot obsrvabl asoly th axu or th u lt valu aog all th rsks ca b obsrvd. rlablty w obsrv oly th axu copot lt o a paralll syst ad th caus o alur. Lt data odlg s troducd by copoudg ay cotuous dstrbuto ad powr srs dstrbutos. Th Wbull dstrbuto s xhaustvly usd or dscrbg hazard rats du to ts gatvly ad postvly skwd dsty shaps. Chahkad ad Gajal (2009proposd th xpotal powr srs aly o dstrbutosthatgralzto a two-paratr xpotal powr srscalld th Wbull powr srs (WPS class o dstrbutos by Moras ad Barrto-Souza (20. Th WPSdstrbutos ca hav a crasg dcrasg ad upsd-dow bathtub alur rat ucto. th sa ar th xpotatd Wbull powr srs dstrbuto ad tsapplcatos wr prstd by Mahoud ad Shra (202. Rctly th xpotatdwbull Posso dstrbuto ad ts applcatos wrtroducd by Mahoud ad Spahdar (203. Th gralzd xpotal powr srs (GEPS dstrbutos wr proposd by Mahoud ad Jaar (202ollowg th sa approach dvlopd by Moras ad Barrto-Souza (20 by copoudg th gralzd xpotal ad th powr srs dstrbutos. Lahu t al. (203 proposd th powr srs dstrbutos th lt as th axu or u o th sapl wth a powr srs dstrbutd sz. Dpartt o Quattatv Aalyss Kg Saud Uvrsty Ryadh Saud Araba salkar@ksu.du.sa

2 54 Arca Rvw o Mathatcs ad Statstcs Vol. 3(2 Dcbr 205 Th copltary xpotal powr srs dstrbuto wth crasg alur rat was troducd by Jos t al. (203 as a copltto th xpotal powr srs odl proposd by Chahkady ad Gajal (2009. Slva t al. (203troducd thpowr srs dstrbutos o th copoud class o th xtdd Wbull dstrbuto. Rctly Bourgugo t al. (204 proposd a w class o atgu l dstrbuto as th Brbau Saudrs powr srs class o dstrbutos. W cob th GEPS dstrbutos troducd bymahoud ad Jaar (202ad th copoud class o th xtdd Wbull powr srs dstrbutos (EWPS proposd by Slva t al. (203 to a or gral aly calld th gralzd xtdd Wbullpowr srs (GEWPS. Cosdr a syst wth N copots whr N (th ubr o copots s a dscrt rado varabl wth support { 2...}. Th lt o th th ( 2... N copot s th ogatv cotuous rado varabl say thdstrbuto o whch blogs to o o th lt dstrbutos such as xpotal gaa Wbull ad Parto aog othrs. Th dscrt rado varabl N ca hav svraldstrbutos such as zrotrucatd Possogotrc boal logarthc ad th powr srs dstrbutos gral. Th o-gatv rado varabl dotg th lt o such a syst s dd by { } { } or basd o whthr th copots ar srs or paralll.by takg a syst wth paralll copots whch th rado varabl N has th powr srs dstrbutos ad th rado varabl ollows th gralzd Wbulldstrbuto w troduc th GEWPS class o dstrbutos that cota th GEPSad th EWPS dstrbutos as spcal cass. Ths study as to gralz th EWPS dstrbutos to obta a w ad or lxbl aly to dscrb rlablty data. Th proposd aly ca b appld to othr lds cludg busss vrot actuaral scc bodcal studs dography ad dustral rlabltyad ay othr lds. Ths aly cotas svral subclasss ad lt odls as spcal cass. Ths papr s orgazd as ollows. Scto 2 w d th class o Wbull ad gralzd Wbull dstrbutos ad dostrat th ay xstg odls that ca b dducd as spcal cass o th proposd ud odl. Scto 3 w d th GEWPS class o dstrbutos trs o dstrbuto uctos ad spcal cass o so xstg classs. Scto 4 w provd th gral proprts o th GEWPS class cludg th dsts ad thsurvval ad hazard rat uctos. Quatls ots ad ordr statstcs o GEWPS ar dscussd Scto 5. Th stato o th GEWPS paratrs s vstgatd Scto 6 usg th axu lklhood thod wthxpctato-axzato(em algorth ad a larg sapl rc. Scto 7 spcal subclasss ad so spcal dstrbutos ar troducd alog wth th lxbl athatcal ors o thrproprts. Scto 8 two odls ar prstd ad appld to llustrat how to us th proposd aly.fally so cocludg rarks ar addrssd Scto Th class o Wbull ad Gralzd Wbull dstrbuto Wbull dstrbuto s o o th ost wdly usd lt dstrbutos trs o rlablty. A larg ubr o odcatos hav b suggstd or th Wbull dstrbutoto prov th shap o th hazard rat ucto. Pgad Ya (204 prstd ay rrcs o ths attr. Th class o xtdd Wbull dstrbutos (EW was proposd by Gurvch t al. (997. Ths class s llustratd by th ollowg dto. Dto : Arado varabl dstrbuto ucto (cd s gv by G x G x x - H ( x ; W ( ; W ( - ; 0 s a br o th Wbull class o dstrbuto ts cuulatv whr H( x ; H(x s a o-gatv ootocally crasg ucto that dpds o th paratr vctor 0. Th corrspodg probablty dsty ucto (pd bcos g x g x h x x (2 H ( x W ( ; W ( ( ; 0

3 Sad H. Alkar 55 Whr h(x h( x ; s th rst drvatv o H( x.th dstrbutos o ost Wbull typsca b rwrtt or dpdg o th choc o th ucto H( x. Slva t al. (203 ad Gurvch t al. (997lstd so xapls or ths class. By usg th da o Gupta ad Cudu (999 th gralzd xpotal o ths class ca b odd as ollows: Dto 2: Arado varabl gv by - H (x G ( x ; G ( x (- ; x 0 (3 blogs to th gralzd xtdd Wbull dstrbuto class ts cd s whr H( x s a o-gatv ootocally crasg ucto thatdpds o a paratrvctor. Th corrspodg pd bcos H ( x - H ( x g ( x ; g ( x h ( x (- ; x 0 (4 whr h( x s th rst drvatv o H( x G ( x ( G (.O ca s that W x ad thus g ( x ( GW ( x gw ( x.th dstrbutos o ost Wbull ad xpotatd Wbull typsca b wrtt or (3 dpdg o th choc o th ucto H( x ad. Tabl dsplays usul H ( x ad corrspodg paratr vctorsor so xstg dstrbutos. Tabl : Spcal dstrbutos ad th corrspodg H( x ; ad vctor. Dstrbuto H ( x Rrcs Expotal x - Johso t al.(994 ( x Expotal powr [ ] Sth ad Ba (975 Burr ( x 0 log( c x c Rodrguz (977 Wbull ( x 0 x Johso t al.(994 x Modd Wbull x ( x / [ ] La t al. (2003Wbull xtso [ ] [ ] t al. (2002 Expotal powr xp[( x ] [ ] Sth ad Ba (975Parto ( x k log( x / k kjohso t al.(994 Gralzd xpotal x - Gupta ad Kudu (2000 ExpotatdWbull x Nassar ad Essa (2003 ( x / Exp. Mod. Wbull xt. [ ] [ ] Sarha ad Apaloo (203 Expotatd Raylgh x 2 - Surls ad Padgtt ( Th GEWPS aly ths scto w drv th aly o GEWPS dstrbutos by copoudg th gralzd xtdd Wbull class ad powr srs dstrbutos. Lt N b a zro trucatd dscrt rado varabl havg a powr srs dstrbuto wth th ollowg probablty ass ucto:

4 56 Arca Rvw o Mathatcs ad Statstcs Vol. 3(2 Dcbr 205 a p p( N 2... c( (5 0 whr a c ( a dpds oly o ad (0 s s chos such a way that c( s t.gv N lt N b dpdt ad dtcally dstrbutd (d rado varabls ollowg (3. Lt ( ax{ } N ( N.Thth cd o s gv by G - H ( x (x [- ] x 0. ( N ( N That s has agralzd xtddwbull class o dstrbuto wth paratrs ad basd o th sa ucto H( x. Th gralzdpowr srs dstrbutodotd by GEWPSwth acrasg ( alur rat s dd by th argal dstrbuto (cd o : a c F x ( c( c( - H ( x - H ( x ( (- ( (- x 0 whch ca b wrtt as c( G ( x F ( x p ( ( x 0. ( G x c( Not that H( x x th odl s rducd to th GEPS troducd by Mahoud ad Jaar (202. { } N Rarks.Lt th F Th cdo s c( G ( x c( ( G ( x ( x. c( c( Not that th th cd o F s: - H ( x c( ( GW ( x c( ( x c( c( whch s calld th EWPS proposd byslva t al. (203. ad H ( x F x x c( ( x c( th th cd o (6 (7 bcos Whch s th xpotal powr srs dstrbutos dvlopd by Chahkadad Gajal (2009 thatclud th lt dstrbutos class proposd by Adads ad Lukas (998 Kus (2007Tahasb ad Rza (2008.

5 Sad H. Alkar 57 (2 Lt Y G ( G ( whrg s th vrs ucto o G th Y has a GEWPS dstrbuto as F y P Y y P G G y P G G y Y ( ( ( ( ( ( ( ( c( G ( y c( F ( G ( G ( y. a ( (x ad Basd o th choc o c H wth or (6 ths class covrs th tr copoud trucatd dscrt dstrbutoswth all o thcotuous ltdstrbutos th ltratur. 4. Dsty survval ad hazard uctos Th probablty dsty uctos assocatd wth (6 ad (7 rspctvly ar gv by ad ( c ( x g(x ( G(x c( c ( (- c ( ' - H ( x H ( x - H ( x = h( x (- (8 c ( x g(x ' ( ( G ( x c ( c ( ( (- c ( ' - H ( x H ( x - H ( x = h( x (-. (9 Th survval uctos ar gv by s ( Ad s - H ( x c( G ( x c ( (- ( x (0 c( c( - H ( x c( ( G ( x c( ( (- ( x. ( c( c( Th corrspodg hazard rat uctos ar (x c ( G ( x s ( x c ( c ( G ( x ( ( x g ( x ( ( ad - H ( x H ( x - H ( x c ( (- = h ( x (- (2 - H ( x ' (x ( c ( ( G ( x ( x g(x s ( x c ( ( G ( x c ( c ( (- - H ( x H ( x - H ( x c ( ( ( - = h ( x ( -. (3 - H ( x c ( ( ( -

6 58 Arca Rvw o Mathatcs ad Statstcs Vol. 3(2 Dcbr 205 Th ltg dstrbuto o th GPS wh c( G ( x l F ( x l l c ( ( c c - H ( x c ( ( (- s a ( G ( x c c ac ( G ( x a ( G ( x c l 0 c a a G x c a whr c { a 0} Th dsts o GEWPS ca b xprssd as a t ubr o lar cobatos o dsts o th c ( a ordr statstcs. Gv that thror ' c ( G (x ( x g(x (N g ( ; ( p Y x ( c( g Y ( x ; Y ( whr s th dsty ucto o ( ax( Y... Y gv by g Y ( x ; g ( x ( G ( x h( x ; (- ( Morovr H ( x - H ( x ' c ( G (x ( x g(x p(n g ( ; Y x c( g Y ( x ; Y whr s th dsty ucto o ( Y... Y. gv by g ( x ; g ( x ( G ( x h( x (- 5. Quatls ots ad ordr statstcs ( H ( x - H ( x Lt Q b a rado varabl wth cd as (6. Th quatl ucto.. F ( (p pp (0 Q ( (. Thror c ((p c( Q ( p H og (. ( p s dd by

7 Sad H. Alkar 59 wth th cd as (7 c (( p c( Q ( p H og whr 0 p (0. Th ot gratg ucto s obtad as ollows: tx tx ( ( ( 0 0 M ( t ( x dx P ( N g ( x dx tx k P N g ( x dx P N E ( 0 ( ( ( (Y whch ca b obtad th ucto H( x. Slarly tx tx k 0 0 M ( t ( x dx P ( N g ( x dx P ( N E (Y. Ordr statstcs saog th ost udatal tools o-paratrc statstcs ad rc.t trs statoprobls ad hypothss tsts ay ways.th probablty dstrbuto ucto o th th ordr statstcs ro a rado sapl... wthdsty ucto (8 s gv by c( G ( x c( G ( x ( : ( x ( x x ( ( c( c( Usg th boal xpaso th abov orula ca b wrtt as ollows: ( : j! j c( G ( x j 0 j ( x ( x ( x 0. ( (!(! c( For th dsty ucto (9 c( G ( x c( G ( x : ( x ( x x ( c( c( Ths xprsso ca b wrtt as : 6. Estato ad rc j! j c( G ( x ( x ( x ( x 0. ( (!(! j 0 j c( Lt... x b a rado sapl wth th obsrvd valus... x obtad ro th GEWPS wth paratrs ad. Lt ( b th p paratr vctor. Th log lklhood ucto s gv by

8 60 Arca Rvw o Mathatcs ad Statstcs Vol. 3(2 Dcbr 205 H ( x l [log log log log(c( ] H ( x ( log( H ( x log(c ( ( ( ; Cosdr p H x. Th th scor ucto s gv by U ( ( l/ l/ l/ l/ T. l p c( p c( c( p c( l ( ( ( ( p H ( x ( p p c( p H x h x p c( p l p log( p ( c( p log( l k p c( p logh( x H ( x p c( p [ ( ( p p ]. p c( p k k Th axu lklhood stato (MLE o say s obtad by solvg th olar syst U (x; 0. Ths olar syst o quatos dos ot hav a closd or. For th trval stato ad hypothss tsts o th odl paratrs w rqur th ollowg obsrvd orato atrx: T T T ( ar th scod partal drvatvs o U (. Udr th stadard rgular codtos or th larg sapl approxato (Cox ad Hkly 974 ullld or th proposd odl th dstrbuto o N ( ( s approxatly p J J ( E[ ( ]. whr Whvr th paratrs ar th whr th lts o tror o th paratr spac but ot o th boudary th asyptotc dstrbutoo ( s N p (0 J ( J ( l ( whr s th ut orato atrx ad p s th ubr o paratrs N p ( ( o th dstrbuto. Th asyptotc ultvarat oral dstrbuto o ca b usd to approxat th codc trval or th paratrs th hazard rat ad th survval uctos. A 00( asyptotc codc trval or paratr s gv by ( Z Z 2 2

9 Sad H. Alkar 6 Whr s th ( dagoal lt o stadard oral dstrbuto. 6. EM algorth ( or... Z p ad 2 s th quatl / 2 o th Basd o th udrl dstrbuto th MLE o th paratrs ca b oud aalytcally usg them algorth. Th Nwto Raphso algorth s o o th stadard thods to dtr th MLEs o th paratrs. To us th algorth th scod drvatvs o th log-lklhood ar rqurd or all tratos. Th EM algorth s a vry powrul tool orhadlg th coplt data probl (Dpstr t al.977; McLachla ad Krsha 983. t s a tratv thod thatrpatdly rplacs ssg data wth statd valus ad updatsth paratr stats. t s spcally usul th coplt datast s asy to aalyz. As potd out by Lttl ad Rub (983 th EM algorth covrg rlably but rathr slowly copard wthth Nwto Raphso thod wh th aout o orato th ssg data s rlatvly larg. Rctly EM algorth has b usd by such rsarchrs asadads ad Loukas (998 Adads (999 Ng t al. (2002 Karls (2003 ad Adads t al. (2005. statg th EM algorth s a rcurrt thod whch ach stp cossts o a stat o th xpctd valu o a hypothtcal rado varabl ad latr axzs th log-lklhood o th coplt data. Lt th coplt data b wth th obsrvd valus x x ad th hypothtcal rado varabl N N. Th jot probablty ucto s such that th argal dsty o s th lklhood o trst. Th w T ( d a hypothtcal copltdata dstrbuto or ach N wth a jot probablty ucto th or o z a ( - ( N xz; z H x H x z z h( x (- c( whr x R ad z N. Thror t s straghtorward to vry that th E-stp o a EM cycl ( r Z ; ( r ( r ( r ( r ( r rqurs th coputato o th codtoal xpctato o whr ( s th currt stat ( th rth trato o Θ. Th th EM cycl s copltd wth th M-stp whch s coplt data axu lklhood ovr E Z ; wth th ssg Z s rplacd by thr codtoal xpctatos (Adads ad Loukas998 whr p Z z N x z ; a z z ( - ' - H ( x ( x c ( ( - z c( c( z az ad sc z E Z z - H ( x z 2 a z (- ts xpctd valu s z 2 - H ( x z z ' - H ( x ' - H ( x z c ( (- c ( (- z ' - H ( x - H ( x - H ( x c ( (- (- c (- ' - H ( x c ( (- ( - c (-. ' - H ( x c ( (- - H ( x - H ( x z a [ (- ] 2 - H ( x z z

10 62 Arca Rvw o Mathatcs ad Statstcs Vol. 3(2 Dcbr Spcal subclasss ths scto w prst our spcal subclasss o th GEWPS aly o dstrbutos. W provd th ors o th cuulatv dsty survval ad hazard rat uctos or ad (. 6. A copoud class o th Posso ad lt dstrbutos Th copoud class o th Posso dstrbuto (CP (Alkar ad Oraby 202 s a subclass o th GEWPS ( a c ( 0 aly o dstrbuto wth. W assu that... N ar dtcally dpdt rado varabls wth a dstrbuto ucto as (3 ad wth N ollowga trucatd Posso dstrbuto at zro. Tabl 2 shows th cssary uctos or ths class. Tabl 2: Cd pd survval ad hazard rat uctos or th CL class (... N ( G(x F ( x g ( x ( x G(x s ( x g ( x ( x G(x 7.2 A copoud class o logarthc ad lts dstrbutos G(x G(x ax(... N F ( x g ( x ( x ( G(x s( x g ( x ( x (G(x (G(x Th copoud class o logarthc dstrbuto (CL (Alkar 202 s a subclass o th GEWPS aly o a ( log( (0 dstrbuto wth c. W assu that... N ar dtcally dpdt rado varabls wth a dstrbuto ucto as (3 ad wth N ollowga trucatd logarthc dstrbuto at zro. Tabl 3 shows th cssary uctos or ths class. ( G(x ( G(x Tabl 3: Cd pd survval ad hazard rat uctos or th CL class (... N ( log( ( G ( x F ( x log( g ( x ( x ( ( G(x log( log( ( G ( x s ( x log( g ( x ( x log( ( G(x[ ( G 7.3 A copoud class o gotrc ad lt dstrbutos ax(... N log( G ( x F ( x log( g ( x ( x ( G(x log( log( G ( x s ( x log( g ( x ( x ( G(x[log( G(x log Th copoud class o gotrc dstrbuto (CG (Alkar 203 s a subclass o th GEWPS aly o a ( ( dstrbuto wth c (0.

11 Sad H. Alkar 63 W assu that... N ar dtcally dpdt rado varabls wth adstrbuto ucto as (3 ad wth N ollowga trucatd gotrc dstrbuto at zro. Tabl 4 showsth cssary uctos or ths class. Tabl 4: Cd pd survval ad hazard rat uctos or th CG class N (... ( ( ( G(x F x ( G ( x ( g ( x ( x 2 ( ( G ( x ( ( G(x s ( x ( G ( x g ( x ( x ( ( G ( x ( G( 7.4 A copoud class o boal ad lt dstrbutos ( N ax(... ( G(x F ( x G ( x ( g ( x ( x 2 ( G ( x ( G(x s ( x G ( x ( g ( x ( x ( G ( x ( G(x Th copoud class o boal dstrbuto (CB (Alkar 203 s a subclass o th GEWPS aly o dstrbutos wth ( ( c. W assu that... N ar dtcally dpdt rado varabls wth a dstrbutoucto as (3 ad wth N ollowg a trucatd boal dstrbuto at zro. Tabl 5 shows th cssary uctos or ths class. Tabl 5: Cd pd survval ad hazard rat uctos or th CB class (... N ( ( ( G ( x F ( x ( g(x( ( G ( x ( x ( ( ( G ( x s ( x ( g(x( ( G ( x ( x ( ( G ( x Tabl 6 llustrats xapls o so xstg dstrbutos wth b obtad drctly ro th prvous tabls. ax(... N ( G ( x F ( x ( g(x( G ( x ( x ( ( G ( x s ( x ( g(x( G ( x ( x ( ( G ( x H ( x ad c(. Th othr uctos ca

12 64 Arca Rvw o Mathatcs ad Statstcs Vol. 3(2 Dcbr 205 Tabl 6: Spcal dstrbutos wth cd ad th corrspodg H ( x ad c( Dstrbuto H ( x a c( F ( x ( Rrcs x ( ( x Gralzd gotrc xpotal x (( Mahoud ad Jaar (202ExpotatdWbull- ( x ( ( gotrc ( x ( x (( Mahoud ad Shra(202 x ( Gralzd Posso xpotal x Mahoud ad Jaar (202 ExpotatdWbull Posso ( x Gralzd boal xpotal Gralzd logarthc xpotal ( x ( Mahoud ad Spahdar (203 x ( ( x ( ( Mahoud ad Jaar (202 x log( ( x log( log( Mahoud ad Jaar (202 ( x log( ( x log( ( ExpotatdWbull-logarthc log( Mahoud ad Spahdar (204 ( k / x Parto Posso log( x / k ; x k Slva t al. (203 Posso-Loax ( x ( x Al-Zahra ad Sagor ( Subodls ad applcatos ths scto two odls ar dscussd wth ral data as xapls o th GEWPS aly. Gotrc xpotal dstrbuto (GE ad gralzd xpotalgotrc (GEG dstrbuto ar ttd or ral data.by ( ax{ } N substtutgdrctly th ors oud Tabl 4 ro Scto 6 or w obta th ollowg pds ad hazard uctos: GE GE GEG x ( ( x ; x 2 [ ( ] ( ( x ; x ( x x ( ( ( x ; x 2 [ ( ] x x ( ( ( x ;. ( ( ( ( GEG x x

13 Sad H. Alkar 65 Fgs. ad 2 show th dsts ad hazard uctos o th GEad GEGdstrbutos or th slctd paratr valus. Fg.. Plots o th dsty ad hazard rat ucto o th GE or drt valus o ad. Fg. 2. Plots o th dsty ad hazard rat ucto o th GEG or ad drt valus ad.

14 66 Arca Rvw o Mathatcs ad Statstcs Vol. 3(2 Dcbr 205 Both odls ar ttd or th data troducd by Brbau ad Saudrs (969 o th atgu l o 606- T6 aluu coupos cut paralll wth th drcto o rollg ad oscllatd at 8 cycls pr scod. Th data arlstd Tabl 3 whch cossts o 0 obsrvatos. Tabl 7: Fatgu l o 606-T6 aluu coupos Th EMalgorth s usd to stat th odl paratrs. Th MLEs o th paratrs th axzd log lklhood th Kologorov Srov statstcs wth ts rspctv p-valu th Akak orato Crtro (AC ad Baysa orato Crtro (BC or th GE ad GEG odls ar gv Tabl 8. Th ttd dsts ad th prcal dstrbuto vrsus th ttd cds o th GE ad GEG odls oths data ar show Fg. 4. Thy dcat that th GEG dstrbuto ts th data bttr tha th GE dstrbuto. Th KS tst statstc taks th sallst valu wth th largst valu o ts corrspodg p-valu or th GEG dstrbuto. Morovr ths cocluso s cord ro th AC ad BC valus or th ttd odls gv Tabl 8.Thr dsts ad cuulatv dstrbutos ar plottd Fg.4. Tabl 8: Paratr stats KS statstc P-valu AC ad BC o th Brbau ad Saudrs data. Dst. MLE(std. K-S p-valu -log(l AC BC ˆ GE ˆ ˆ ˆ GEG ˆ Fg. 3: Plots o ttd GEG ad GE o th Brbau ad Saudrs data.

15 Sad H. Alkar Cocludg rarks W d a w aly o lt dstrbutos calld th GEWPS aly o dstrbutos whch gralzs th xtdd Wbull powr srs class ad th gralzd powr srs xpotal dstrbutos troducd by Slva t al. (203 admahoud ad Jaar (202 rspctvly. Th GEWPS class cotas ay lt subclasss ad dstrbutos. Varous stadard athatcal proprts wr drvd such as dsty adsurvval ad hazard uctos wr troducd lxbl ad usulors. Paratr stato usg themalgorth was coductd usg th axu lklhood thod. Fally w ttd so o th GEWPS odls to a ral datast to show th lxblty ad th bts o th proposd class. Ackowldgts Th author s gratul to th Dashp o Sctc Rsarch at Kg Saud Uvrstyas rprstd by th Rsarch Ctr o thcollg o Busss Adstratoor acally supportg ths study. Rrcs Alkar S. ad Oraby A. "A copoud class o Posso ad lt dstrbutos" J. Stat. Appl. Pro. vol. pp Alkar S. "Nw aly o logarthc lt dstrbutos" J. athatcs ad statstcs Vol. 8(4 pp Alkar S. "A copoud class o gotrc ad lts dstrbutos" Th op stat. ad prob. joural Vol. 5 pp Alkar S. "A class o trucatd boal lt dstrbutos" Op joural o stat. Vol. 3 pp Adads K. "A EM algorth or statg gatv boal paratrs" Austral Nw Zalad Statst. vol. 4 (2 pp Adads K. Dtrakopoulou T. ad Loukas S. "O a xtso o th xpotal gotrc Dstrbuto" Statst. Probab. Ltt. vol.73 pp Adads K. ad Loukas S. "A lt dstrbuto wth dcrasg alur rat" Statstcs ad Probablty Lttrs vol.39 pp Al-Zahra B. ad Sagor H. "Th Posso-loax dstrbuto" Rvsta Colobaa d Estadstca vol.37 pp Brbau Z. ad Saudrs S. " Estato or a aly o l dstrbutos wth applcatos to atgu" j. o appld prob. Vol. 6 pp Bourgugo M. Slva R. ad Cordro G. "A w class o atgu l dstrbutos" Joural o statstcalcoputato &sulato vol.84 pp Chahkad M. Gajal M O so lt dstrbutos wth dcrasg alur rat. Coputatoal Statstcs ad Data Aalyss Cox D. ad Hkly D. "Thortcal Statstcs " Chapa ad Hall Lodo 974. Statstcs ad Data Aalyss vol. 53 pp Dpstr A. Lard N. ad Rub D. "Maxu lklhood ro coplt data va th EM Algorth " J. Roy. Statst. Soc. Sr. B vol. 39 pp Flors Borgs J. Cacho P. ad Louzada G. "Th copltary xpotal powr srs dstrbuto" Brazla Statstcal Assocato Gupta R. ad Kudu D. "Gralzd xpotal dstrbuto: drt thod o statos" J. Statst. Coput. Sul. Vol. 00 pp Gurvch M. Dbdtto A. Raad S. " A w statstcal dstrbuto or charactrzg th rado strgth o brttl atrals J. Matr. Sc. Vol.32 pp Karls D. "A EM algorth or ultvarat Posso dstrbuto ad rlatd odls" J. Appl. Statst. vol. 30 pp Kus C A w lt dstrbuto. Coputatoal Statstcs ad Data Aalyss

16 68 Arca Rvw o Mathatcs ad Statstcs Vol. 3(2 Dcbr 205 Flors J. Borgs P. Cacho V. ad Louzada F. "Th copltary xpotal powr srs dstrbuto" Brazla Joural o probablty ad statstcs vol.27 pp Lahu A. Mutau B. ad Cataracuc S. "O th lt as th axu or u o th sapl wth powr srs dstrbutd sz" ROMA joral vol. 9 pp Lttl R. ad Rub D. "coplt data. : Kotz S. Johso N.L. (Eds. Ecyclopda o Statstcal Sccs" vol. 4 Wly NwYork983. Mahoud E. ad Jaar A. "Gralzd xpotal- powr srs dstrbutos" Coputatoal Statstcs ad Data Aalyss vol. 56 pp Mahoud E. ad Spahdar A. " Expotatd Wbull-Posso dstrbuto: odl proprts ad applcatos" Mathatcs ad coputrs sulato vol. 92 pp Mahoud E. Spahdar A. ad Lot A. "Expotatd Wbull-logarthc dstrbuto: odl proprts ad applcatos" prt arv: /204. Mahoud E. ad Shra M. "Expotatd Wbull powr srs dstrbutos ad ts applcatos" prt arv: /202. Mahoud E. ad Shra M. "Expotatd Wbull-gotrc dstrbuto ad ts applcatos"prt arv: /202. McLachla G. ad Krsha T. "Th EM Algorth ad Extso" Wly Nw York 997. Moras A. ad Barrto-Souza W. "A copoud class o Wbull ad powr srs dstrbutos" Coputatoal Statstcs ad Data Aalyss vol. 55 pp Nassar M. ad Essa F. "O th xpotatd Wbull dstrbuto"cou. Stat. Thory Mth Vol.32: pp Ng M. Cha P. ad Balakrsha N. "Estato o paratrs ro progrssvly csord data usg EM algorth" Coput. Statst. Data Aal. vol. 39 pp Pg. ad Ya Z. "Estato ad applcato or a w xtdd Wbull dstrbuto" Rlablty grg ad syst saty vol.2 pp Slva B. Bourgugo M. Das C. ad Cordro G. Th copoud class o xtdd Wbull powr srs dstrbutos" Coputatoal Statstcs ad Data Aalyss vol. 58 pp Surls J. ad Padgtt W. "rc or rlablty ad strss-strgth or a scald Burr Typ dstrbuto" Lt Data Aalyss vol. 7 pp Tahasb R. Rza S A two-paratr lt dstrbuto wth dcrasg alur rat. Coputatoal Statstcs ad Data Aalyss