Transmuted Exponentiated Gamma Distribution: A Generalization of the Exponentiated. Gamma Probability Distribution

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1 Appld Mathatcal Sccs Vol o HIKARI Ltd http//d.do.org/0.988/as Trasutd Epotatd Gaa Dstrbuto A Gralzato o th Epotatd Gaa Probablty Dstrbuto Mohad A. Hussa Dpartt o Mathatcal Statstcs Isttut o Statstcal Studs ad Rsarch Caro Uvrsty Gza Egypt Copyrght 04 Mohad A. Hussa. Ths s a op accss artcl dstrbutd udr th Cratv Coos Attrbuto Lcs whch prts urstrctd us dstrbuto ad rproducto ay du provdd th orgal work s proprly ctd. Abstract Th potatd gaa EG dstrbuto s o o th portat als o dstrbutos lt tsts. I ths papr a w gralzd vrso o ths dstrbuto whch s calld Trasutd potatd gaa TEG dstrbuto s troducd. A w dstrbuto s or lbl ad has so trstg proprts. A coprhsv athatcal tratt o th TEG dstrbuto s provdd. W drv th rth ot ad ot gratg ucto ths dstrbuto. Morovr w dscuss th au lklhood stato o ths dstrbuto. Kywords gaa dstrbuto; Hazard ucto; potatd gaa dstrbuto; Mau lklhood stato; Mots. Itroducto ad Motvato O o th portat als o dstrbutos lt tsts s th potatd gaa EG dstrbuto. Th potatd gaa EG dstrbuto has b troducd by Gupta t al. 998 thy proposd th us o th potatd gaa dstrbuto as a altratv to gaa ad wbull dstrbutos. Th cuulatv dstrbuto ucto c.d.. ad a probablty dsty ucto p.d.. o th or rspctvly;

2 98 Mohad A. Hussa 0 0 ad 0. G. whr ad ar scal ad shap paratrs rspctvly. Th corrspodg probablty dsty ucto pd s gv by. g. Shawky ad Bakoba 008 dscussd th potatd gaa dstrbuto as a portat odl o l t odls ad drvd Baysa ad o-baysa stators o th shap paratr rlablty ad alur rat uctos th cas o coplt ad typ-ii csord sapls. Also ordr statstcs ro potatd gaa dstrbuto ad assocatd rc was dscussd by Shawky ad Bakoba 009. Ghazadh t al. 0 dalt wth th stato o paratrs o th Epotatd Gaa EG dstrbuto wth prsc o k outlrs. Th au lklhood ad ot o th stators wr drvd. Ths stators ar copard prcally usg Mot Carlo sulato. Sgh t al.0 proposd Bays stators o th paratr o th Epotatd gaa dstrbuto ad assocatd rlablty ucto udr Gral Etropy loss ucto or a csord sapl. Th proposd stators wr copard wth th corrspodg Bays stators obtad udr squard rror loss ucto ad au lklhood stators through thr sulatd rsks. Kha ad Kuar 0 stablshd th plct prssos ad so rcurrc rlatos or sgl ad product ots o lowr gralzd ordr statstcs ro potatd gaa dstrbuto. Saay t l. 0 whr proposd Bays stators o th paratr o th potatd gaa dstrbuto ad assocatd rlablty ucto udr Gral Etropy loss ucto or a csord sapl. Navd ad Muhaad 0 troducd Baysa Aalyss o Epotatd Gaa Dstrbuto udr Typ II Csord Sapls. Rctly Parvz t al. 03 dscussd Classcal ad Baysa stato o paratrs o th gralzd potatd gaa dstrbuto... Trasutato Map I ths subscto w dostrat trasutd probablty dstrbuto. Lt F ad F b th cuulatv dstrbuto uctos o two dstrbutos wth a coo sapl spac. Th gral rak trasutato as gv 007 s dd as GR u F F u ad GR u F F u. Not that th vrs cuulatv dstrbuto ucto also kow as quatl ucto s dd as F y R F y or y0. Th uctos G R ad G R both ap th ut trval u u I 0 to tsl ad

3 Trasutd potatd gaa dstrbuto 99 udr sutabl assuptos ar utual vrss ad thy satsy G 0 0 ad G 0. A quadratc Rak Trasutato Map QRTM s dd as R G R u u u u.3 ro whch t ollows that th cd's satsy th rlatoshp F F F.4 whch o drtato ylds F.5 whr ad ar th corrspodg pds assocatd wth cd F ad F rspctvly. A tsv orato about th quadratc rak trasutato ap s gv Shaw t al Obsrv that at 0 w hav th dstrbuto o th bas rado varabl. Th ollowg La provd that th ucto gv.5 satss th proprty o probablty dsty ucto. May authors dalg wth th gralzato o so wll- kow dstrbutos. Aryal ad Tsokos 009 dd th trasutd gralzd tr valu dstrbuto ad thy studd so basc athatcal charactrstcs o trasutd Gubl probablty dstrbuto ad t has b obsrvd that th trasutd Gubl ca b usd to odl clat data. Also Aryal ad Tsokos 0 prstd a w gralzato o Wbull dstrbuto calld th trasutd Wbull dstrbuto. Mahoud ad Ala 00 troducd w gralzato o th lar potal dstrbuto s gralzd lar potal dstrbuto. Ths dstrbuto s portat sc t cotas as spcal sub-odls so wdly wll kow dstrbutos. It also provds or lblty to aalyz copl ral data sts. Rctly Aryal 03 proposd ad studd th varous structural proprts o th trasutd Log- Logstc dstrbuto ad Muhaad kha ad kg 03 troducd th trasutd odd Wbull dstrbuto whch tds rct dvlopt o trasutd Wbull dstrbuto by Aryal t al. 0. ad thy studd th athatcal proprts ad au lklhood stato o th ukow paratrs. Elbatal 03 proposd a uctoal coposto o th cuulatv dstrbuto ucto o o probablty dstrbuto wth th vrs cuulatv dstrbuto ucto o aothr s calld th trasutato ap.h usd th quadratc rak trasutato ap QRTM ordr to grat a lbl aly o probablty dstrbutos takg odd vrs wbull dstrbuto as th bas valu dstrbuto by troducg a w paratr that would or or dstrbutoal lblty. It wll b show that th aalytcal rsults ar applcabl to odl ral world data. Elbatal ad Aryal 03 prstd th trasutd addtv Wbull dstrbuto that tds th addtv Wbull dstrbuto ad so othr dstrbutos thy usd th quadratc rak trasutato ap QRTM proposd by Shaw & Buckly 007 ordr to grat th R

4 300 Mohad A. Hussa trasutd addtv Wbull dstrbuto. Th rst o th papr s orgazd as ollows. I Scto w dostrat trasutd probablty dstrbuto th hazard rat ad rlablty uctos o TEG dstrbuto. I Scto 3 w studd th statstcal proprts clud quatl uctos ots ot gratg ucto. Th u au ad da ordr statstcs odls ar dscussd Scto 4. I scto 5 w studd th last squar stato. Fally I Scto 6 w dostrat th au lklhood stats o th ukow paratrs.. Trasutd Epotatd Gaa Dstrbuto I ths scto w studd th trasutd potatd gaa TEG dstrbuto. Now usg. ad. w hav th cd o trasutd potatd gaa dstrbuto FTEG. whr ad ar th shap paratrs rprstg th drt pattrs o th trasutd potatd gaa dstrbuto ad ar postv ad s th trasutd paratr. Th rstrctos quato. o th valus o ad ar always th sa. Th probablty dsty ucto pd o th trasutd potatd gaa dstrbuto s gv by TEG. Fgurs ad llustrat th graphcal bhavor o th pd ad cd o trasutd potatd gaa dstrbuto or slctd valus o th paratrs. Fgur th pd ucto or drt valus o th paratrs

5 Trasutd potatd gaa dstrbuto 30 Fgur cd ucto or drt valus o th paratrs Th rlablty ucto RF o th trasutd potatd gaa dstrbuto s dotd by R TEG also kow as th survvor ucto ad s dd as RTEG FTEG.3 It s portat to ot that R TEG F TEG. O o th charactrstc rlablty aalyss s th hazard rat ucto HF dd by h TEG TEG FTEG..4 It s portat to ot that th uts or h TEG s th probablty o alur pr ut o t dstac or cycls. Ths alur rats ar dd wth drt chocs o paratrs. Th cuulatv hazard ucto o th trasutd potatd gaa dstrbuto s dotd by H TEG ad s dd as HTEG l.5 It s portat to ot that th uts or H TEG s th cuulatv probablty o alur pr ut o t dstac or cycls. w ca show that. For all choc o paratrs th dstrbuto has th dcrasg pattrs o cuulatv stataous alur rats. Fgurs 3 ad 4 llustrat th graphcal bhavor o th rlablty ucto ad hazard rat ucto th trasutd potatd gaa dstrbuto or slctd valus o th paratrs.

6 30 Mohad A. Hussa Fgur 3 th rlablty ucto RF or drt valus o th paratrs 3. Statstcal Proprts Ths scto s dvotd to studyg statstcal proprts o th TEG dstrbuto spccally quatl ucto ots ad ot gratg ucto 3.. Quatl Fucto Th qth quatl ro. as q o th trasutd potatd gaa dstrbuto ca b obtad q w sulat th TEG dstrbuto by solvg th olar quato u 3. whr u has th uor U0; dstrbuto.

7 Trasutd potatd gaa dstrbuto 303 Fgur 4 th hazard rat HF ucto or drt valus o th paratrs 3.. Mots I ths subscto w dscuss th th r ot ad ot gratg ucto or TEG dstrbuto. Mots ar cssary ad portat ay statstcal aalyss spcally applcatos. It ca b usd to study th ost portat aturs ad charactrstcs o a dstrbuto.g. tdcy dsprso skwss ad kurtoss. I X has TEG th th th r ot o X s gv by th ollowg. 0 0 r r r 3. Basd o th rst our ots o th TEG dstrbuto th asurs o skwss A ad kurtoss k o th TEG dstrbuto ca obtad as A ad k

8 304 Mohad A. Hussa ad th ot gratg ucto o X t M X has th ollowg or X t t M Ordr Statstcs I act th ordr statstcs hav ay applcatos rlablty ad l tstg. Th ordr statstcs ars th study o rlablty o a syst. Lt X... X X b a spl rado sapl ro TEG wth cuulatv dstrbuto ucto ad probablty dsty ucto as. ad. rspctvly. Lt X X... X dot th ordr statstcs obtad ro ths sapl. I rlablty ltratur X dot th lt o a out-o- syst whch cossts o dpdt ad dtcally copots. Th th pd o X s gv by F F 4. whr. Substtutg. ad. to 4. w gt H H H H H H 4. whr H also th ot pd o X X ad s F F F F C 4.3 whr C!!!! C

9 Trasutd potatd gaa dstrbuto 305 W dd th rst ordr statstcs X M X... X th last ordr statstcs as X X Ma X X... X. thus th Dstrbuto o th u ad th au ad F H H H H H H H F H 5. Last Squars ad Wghtd Last Squars Estators I ths scto w provd th rgrsso basd thod stators o th ukow paratrs o th trasutd potatd gaa dstrbuto whch was orgally suggstd by Swa Vkatraa ad Wlso 988 to stat th paratrs o bta dstrbutos. It ca b usd so othr cass also. Suppos Y...Y s a rado sapl o sz ro a dstrbuto ucto G. ad suppos Y ;... dots th ordrd sapl. Th proposd thod uss th dstrbuto o G. For a sapl o sz w hav Y ad E G Y V G Y Cov k G Y G Y ; or k k s Johso Kotz ad Balakrsha 995. Usg th pctatos ad th varacs two varats o th last squars thods ca b usd.

10 306 Mohad A. Hussa Mthod Last Squars Estators. Obta th stators by zg G Y wth rspct to th ukow paratrs. Thror cas o TEG dstrbuto th last squars stators o ad say ˆ ˆ ad ˆ rspctvly ca b obtad by zg wth rspct to ad. Mthod Wghtd Last Squars Estators. Th wghtd last squars stators ca b obtad by zg wg Y wth rspct to th ukow paratrs whr w V G Y. Thror cas o TEG dstrbuto th wghtd last squars stators o ad ca b obtad by zg w wth rspct to th ukow paratrs oly. ˆ ˆ ad ˆ 6. Mau Lklhood Estato I ths scto w dtr th au lklhood stats MLEs o th paratrs o th TEG dstrbuto ro coplt sapls oly. Lt X X... X b a rado sapl o sz roteg.th lklhood ucto or th vctor o paratrs ca b wrtt as L

11 Trasutd potatd gaa dstrbuto Takg th log-lklhood ucto or th vctor o paratrs w gt log log log log log L. log 5. Th log-lklhood ca b azd thr drctly or by solvg th olar lklhood quatos obtad by drtatg 5.. Th copots o th scor vctor ar gv by log log L log 5.3 log L 5.4 ad log L 5.5 W ca d th stats o th ukow paratrs by au lklhood thod by sttg ths abov o-lar quatos to zro ad solv th sultaously. Thror w us athatcal packag to gt th MLE o th ukow paratrs.

12 308 Mohad A. Hussa Tabl Th a squar rrors o th MLEs. o th TEG or drt sapl szs MSE MSE MSE W otcd ro th abov Tabl that all MSEs dcras as th sapl sz crass whl thy cras wth crasg o th tru paratr. REFERENCES [] G. R. Aryal. ad C. P. Tsokos O th trasutd tr valu dstrbuto wth applcatos. Nolar Aalyss Thory Mthods ad applcatos [] G. R. Aryal ad C. P. Tsokos Trasutd Wbull dstrbuto A Gralzato o th Wbull Probablty Dstrbuto. Europa Joural o Pur ad Appld Mathatcs [3] G. R. Aryal Trasutd Log-Logstc Dstrbuto. J. Stat. Appl. Pro [4] I. Elbatal Trasutd odd vrswbull Dstrbuto AGralzato o th Modd vrs Wbull Probablty Dstrbuto. Itratoal Joural o Mathatcal Archv [5] I.Elbatal ad G. R. Aryal O th Trasutd AddtvWbull Dstrbuto.Austra Joural o Statstcs

13 Trasutd potatd gaa dstrbuto 309 [6] R. C. Gupta R. D. Gupta ad P. L. Gupta Modlg Falur T Data by Lha Altratvs Cou. Statst.-Thory Mth [7] Ghazadh A Pazra H. ad R. Lot Classcal Estatos o th Epotatd Gaa Dstrbuto Paratrs wth Prsc o K Outlrs. Australa Joural o Basc ad Appld Sccs [8] N. L. Johso S. Kotz ad N. Balakrsha Cotuous uvarat dstrbutos vol.. Wly. Nw York [9] R. Kha ad D. Kuar Lowr Gralzd Ordr Statstcs ro Epotatd Gaa Dstrbuto ad Its Charactrzato. Prob. Stat. Foru [0] M. A.Mahoud ad A. F. Wad Th gralzd lar potal dstrbuto Statst. Probabl. Ltt [] K. S. Muhaad ad K. Robrt Trasutd Modd Wbull Dstrbuto A Gralzato o th Modd Wbull Probablty Dstrbuto. Europa Joural o Pur ad Appld Mathatcs [] F. Navd ad A. Muhaad Baysa Aalyss o Epotatd Gaa Dstrbuto udr Typ II Csord Sapls. Itratoal Joural o Advacd Scc ad Tchology [3] N. Parvz L. Rasoul ad V. Hoss Classcal ad Baysa stato o paratrs o th gralzd potatd gaa dstrbuto. Sctc Rsarch ad Essays [4] K. Saay S. Ush ad K. Dsh Baysa Estato o th Epotatd Gaa Paratr ad Rlablty Fucto Udr Asyptotcs Sytrc Loss Fucto. Rvsta -- Statstcal Joural [5] A. I. Shawky ad R. A. Bakoba Baysa ad No-Baysa Estatos o th Epotatd Gaa Dstrbuto. Appld Mathatcal Sccs [6] A. I. Shawky ad R. A. Bakoba Ordr Statstcs ro Epotatd Gaa Dstrbuto ad Assocatd Irc. It. J. Cotp. Math. Sccs [7] W. Shaw ad I. Buckly Th alchy o probablty dstrbutos byod Gra-Charlr pasos ad a skw- kurtotc- oral dstrbuto ro a rak trasutato ap 007.

14 30 Mohad A. Hussa [8] S. Sgh U. Sgh ad D. Kuar Baysa Estato o th Epotatd Gaa Paratr ad Rlablty Fucto Udr Asytrc Loss Fucto. Rvsta Statstcal Joural [9] J. J. Swa S. Vkatraa ad J. R. Wlso Last-squars stato o dstrbuto uctos Johso's traslato syst. Joural o Statstcal Coputato ad Sulato Rcvd Fbruary 04