Volume83No17December2013whereGxisthecdfofthebasedistributionObservethatat0wehavethedistributionofthebaserandomvariableManyauthorsdealingwiththegeneralizationofsomewellknowndistributionsArya ID: 198350
Download Pdf The PPT/PDF document "InternationalJournalofComputerApplicatio..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
InternationalJournalofComputerApplications(09758887) Volume83-No.17,December2013whereG(x)isthecdfofthebasedistribution.Observethatat=0wehavethedistributionofthebaserandomvariable.Manyauthorsdealingwiththegeneralizationofsomewell-knowndistributions.AryalandTsokos(2009)denedthetransmutedgeneralizedextremevaluedistributionandtheystudiedsomeba-sicmathematicalcharacteristicsoftransmutedGumbelprobabilitydistributionandithasbeenobservedthatthetransmutedGumbelcanbeusedtomodelclimatedata.AlsoAryalandTsokos(2011)presentedanewgeneralizationofWeibulldistributioncalledthetransmutedWeibulldistribution.Recently,Aryal(2013)proposedandstudiedthevariousstructuralpropertiesofthetransmutedLog-Logisticdistribution.andMuhammadkhanandKing(2013)in-troducedthetransmutedmodiedWeibulldistributionwhichex-tendedrecentdevelopmentontransmutedWeibulldistributionbyAryaletal.(2011).andtheystudiedthemathematicalpropertiesandmaximumlikelihoodestimationoftheunknownparameters.Inthepresentstudywewillprovidemathematicalformulationofthetransmutedgeneralizedlinearexponentialdistribution(TGLED)distributionandsomeofitsproperties.MerovciandElbatal(2013)introduceanewlifetimedistributionbytransmutedandcompoundingLindleyandgeometricdistribu-tionsnamedtransmutedLindleygeometricdistribution.Theyde-riveexpansionsformomentsandforthemomentgeneratingfunc-tion.Theestimationofparametersisapproachedbythemethodofmaximumlikelihood,alsotheinformationmatrixisderived.AnapplicationofthetransmutedLindleygeometricdistributiontorealdata.Elbatal(2013)proposedafunctionalcompositionofthecu-mulativedistributionfunctionofoneprobabilitydistributionwiththeinversecumulativedistributionfunctionofanotheriscalledthetransmutationmap.Heusedthequadraticranktransmutationmap(QRTM)inordertogenerateaexiblefamilyofprobabilitydis-tributionstakingmodiedinverseweibulldistributionasthebasevaluedistributionbyintroducinganewparameterthatwouldoffermoredistributionalexibility.Itwillbeshownthattheanalyticalresultsareapplicabletomodelrealworlddata.ElbatalandAryal(2013)presentedthetransmutedadditiveWeibulldistribution,thatextendstheadditiveWeibulldistributionandsomeotherdistribu-tionstheyusedthequadraticranktransmutationmap(QRTM)pro-posedbyShaw&Buckley(2007)inordertogeneratethetrans-mutedadditiveWeibulldistribution.Variousstructuralpropertiesofthenewdistributionincludingtheexplicitexpressionsforthemo-ments,randomnumbergenerationandorderstatisticsarederived.Maximumlikelihoodestimationoftheunknownparametersofthenewmodelforcompletesampleisalsodiscussed.Itwillbeshownthattheanalyticalresultsareapplicabletomodelrealworlddata.Therestofthepaperisorganizedasfollows.InSection2wedemonstratetransmutedprobabilitydistribution,andwepresenttheexibilityofthesubjectdistributionandsomespecialsub-models.ThereliabilityfunctionsofthesubjectmodelaregiveninSection3.InSection4westudiedthestatisticalpropertiesincludequantilefunctions,moments,momentgeneratingfunction.Theminimum,maximumandmedianorderstatisticsmodelsarediscussedinSec-tion5.Finally,InSection6wedemonstratethemaximumlikeli-hoodestimatesandtheasymptoticcondenceintervalsoftheun-knownparameters.Finally,somelifetimedatasetsareusedtoillus-tratethatthegeneralizedlinearexponentialdistribution(TGLED)canusedforthedataunderanalysis,comparingwithsomeknowndistributions.2.TRANSMUTEDGENERALIZEDLINEAREXPONENTIALDISTRIBUTIONInthissectionwestudiedthetransmutedgeneralizedlinearexpo-nentialdistribution(TGLED)andthesub-modelsofthisdistribu-tion.Nowusing??and??wehavethecdfoftransmutedgeneral-izedlinearexponentialdistributionFTGLE=(1+)1�e�( x+ 2x2)�1�e�( x+ 2x2)2;=h1�e�( x+ 2x2)ih1+e�( x+ 2x2)i(6)where ,arethescaleparameters,isshapeparameterrepre-sentingthedifferentpatternsofthetransmutedgeneralizedlinearexponentialdistributionandisthetransmutedparameter.There-strictionsinequation(6)onthevaluesof ;;andarealwaysthesame.Theprobabilitydensityfunction(pdf)ofthetransmutedgeneralizedlinearexponentialdistributionisgivenbyfTGIE(x)=( +x)(( x+ 2x2)�1e�( x+ 2x2)h1�+2e�( x+ 2x2)i:(7)Thetransmutedgeneralizedlinearexponentialdistributionisveryexiblemodelthatapproachestodifferentdistributionswhenitsparametersarechanged.(i)If=0wegetthegeneralizedlinearexponentialdistributionGLED( ;;).(ii)If =1 ;=0wegetthetransmutedWeibulldistributionTWD(;;).(iii)If==0; =1 wegettheWeibulldistributionW(;):(iv)If=1wegetthetransmutedlinearexponentialdistributionTLED(; ;)(v)If=1; =0wegetthetransmutedRayleighdistributionTRD(;)(vi)If=1; ==0wegettheRayleighdistributionRD().Figure1showsbothcasesoftheshapeparameter�1and1: 30 InternationalJournalofComputerApplications(09758887) Volume83-No.17,December2013 Fig.1.EffectofshapeparameterontheTGLEDPDF3.RELIABILITYANALYSISThetransmutedgeneralizedlinearexponentialdistributioncanbeausefulcharacterizationoflifetimedataanalysis.Thereliabil-ityfunction(RF)ofthetransmutedgeneralizedlinearexponentialdistributionisdenotedbyRTGLED(x)alsoknownasthesurvivorfunctionandisdenedasRTGLED(x)=1�FTGLED(x)=1�h1�e�( x+ 2x2)ih1+e�( x+ 2x2)i:(8)Figure2(a)and(b)representtheCDFandRFrespectivelyfordifferentvaluesofshapeparameter: Fig.2(a)CDFFig.2(b)RFItisimportanttonotethatRTGLED(x)+FTGLED(x)=1.Oneofthecharacteristicinreliabilityanalysisisthehazardratefunction(HRF)denedbyhTGIE(x) =fTGLED(x) 1�FTGLED(x)=( +x)(( x+ 2x2)�1e�( x+ 2x2)) 1�h1�e�( x+ 2x2)ih1+e�( x+ 2x2)i:h1�+2e�( x+ 2x2)i 1�h1�e�( x+ 2x2)ih1+e�( x+ 2x2)i:(9)ItisimportanttonotethattheunitsforhTGIE(x)istheprobabilityoffailureperunitoftime,distanceorcycles.Thesefailureratesaredenedwithdifferentchoicesofparameters.Thecumulativehazardfunctionofthetransmutedgeneralizedinvertedexponentialdistri-butionisdenotedbyHTGIE(x)andisdenedasHTGIE(x)=�lnh1�e�( x+ 2x2)ih1+e�( x+ 2x2)i(10)ItisimportanttonotethattheunitsforHTGIE(x)isthecumulativeprobabilityoffailureperunitoftime,distanceorcycles.wecanshowthat.Forallchoiceofparametersthedistributionhasthedecreasingpatternsofcumulativeinstantaneousfailurerates.Figure3(a),(b)and(c)representthehazardratefunction(HRF)withdifferentvaluesofshapeparameter:31 InternationalJournalofComputerApplications(09758887) Volume83-No.17,December2013 Fig.3(a) Fig.3(b) Fig.3(c)Fig.(3)Effectofshapeparameteronthehazardratefunction(HRF)oftheTGLED.THEOREM1.Thehazardratefunctionofthetransmutedgen-eralizedlinearexponentialdistributionhasthefollowingproper-ties(i)If=0wegetthefailurerateissameastheGLED( ;;):(ii)If =1 ;=0wegetthethefailurerateissameastheTWD(;;):(iii)If==0; =1 wegetthethefailurerateissameastheW(;):(iv)If=1wegetthethefailurerateissameastheTLED(; ;):PROOF.Thehazardfunction(HF)ofthetransmutedgeneral-izedlinearexponentialdistributionisgiveninequation(9)hasthespecialcaseswithdifferentchoiceofparameters:(i)If=0wegetthefailurerateissameastheGLED( ;;)hGLED=( +x)(( x+ 2x2)�1(ii)If =1 ;=0wegetthethefailurerateissameastheTWD(;;):hTWD(x)=x (x )�1e�(x )h1�+2e�(x )i 1�1�e�(x )1+e�(x )::(iii)If==0; =1 wegetthethefailurerateissameastheW(;)hWD(x)=x (x )�1e�(x ) 1�1�e�(x )(iv)If=1wegetthethefailurerateissameastheTLED(; ;)hTLED(x)=( +x)e�( x+ 2x2)h1�+2e�( x+ 2x2)i 1�h1�e�( x+ 2x2)ih1+e�( x+ 2x2)i: 4.STATISTICALPROPERTIESThissectionisdevotedtostudystatisticalpropertiesforthetrans-mutedgeneralizedlinearexponential,specicallyQuantilefunc-tion,median,moments,momentgeneratingfunction.4.1QuantileandMedianThequantilexqoftheTGLED( ;;,;x)isrealsolutionofthefollowingequationxq=� +r 2+2h�ln(�1)+p (+1)2�4q 2i1 (11)Theaboveequationhasnoclosedformsolutioninxq,sowehavetouseanumericaltechniquesuchasaNewton-Raphsonmethodtogetthequantile.Ifweputq=0:5inequation(11)onegetsthemedianofTGLED( ;;,;x)32 InternationalJournalofComputerApplications(09758887) Volume83-No.17,December2013=Ci;j1Xr=0tr r!�(r�i 2�j +1)"1�+1 2r�i 2�j +1#(19)whichcompletestheproof. 5.ORDERSTATISTICSTheorderstatisticshavemanyapplicationsinreliabilityandlifetesting.Theorderstatisticsariseinthestudyofreliabilityofasys-tem.LetX1;X2;:::;XnbeasimplerandomsamplefromTGLED( ;;,;x)withcumulativedistributionfunctionandproba-bilitydensityfunctionasin(6)and(7),respectively.LetX(1:n)X(2:n):::X(n:n)denotetheorderstatisticsobtainedfromthissample.Inreliabilityliterature,X(i:n)denotethelifetimeofan(n�i+1)�out�of�nsystemwhichconsistsofnindependentandidenticallycomponents.ThenthepdfofX(i:n);1inisgivenbyfi::n(x)=1 (i;n�i+1)[F(x;)]i�1[1�F(x;)]n�if(x;)(20)where=( ;;,;)also,thejointpdfofX(i:n);X(j:n)and1ijnisfi:j:n(xi;xj)=C[F(xi)]i�1[F(xj)�F(xi)]j�i�1[1�F(xj)]n�jf(xi)f(xj)(21)whereCC=n! (i�1)!(j�i�1)!(n�j)!WedenedtherstorderstatisticsX(1)=Min(X1;X2;:::;Xn),thethelastorderstatisticsasX(n)=Max(X1;X2;:::;Xn)andmedianorderXm+1.5.1DistributionofMinimum,MaximumandMedianLetX1;X2;:::;Xnbeindependentlyidenticallydistributedorderrandomvariablesfromthetransmutedgeneralizedlinearexponen-tialdistributionhavingrst,lastandmedianorderprobabilityden-sityfunctionaregivenbythefollowingf1:n(x)=n[1�F(x;)]n�1f(x;)=n1�h1�e�( x(1)+ 2x2(1))ih1+e�( x(1)+ 2x2(1))in�1( +x(1))( x(1)+ 2x2(1))�1e�( x(1)+ 2x2(1))h1�+2e�( x(1)+ 2x2(1))i(22)fn:n(x)=nF(x(n);)n�1f(x(n));)=nnh1�e�( x(n)+ 2x2(n))ih1+e�( x(n)+ 2x2(n))ion�1( +x(n))( x(n)+ 2x2(n))�1e�( x(n)+ 2x2(n))h1�+2e�( x(n)+ 2x2(n))i(23)andfm+1:n(ex)=(2m+1)! m!m!(F(ex))m(1�F(ex))mf(ex)=(2m+1)! m!m!nh1�e�( x(m+1)+ 2x2(m+1))ih1+e�( x(m+1)+ 2x2(m+1))iomn1�h1�e�( x(m+1)+ 2x2(m+1))i1+e�e�( x(m+1)+ 2x2(m+1))m( +x(m+1))( x(m+1)+ 2x2(m+1))�1e�( x(m+1)+ 2x2(m+1))ih1�+2e�( x(m+1)+ 2x2(m+1))i(24)5.2JointDistributionoftheithandjthorderStatisticsThejointdistributionofthetheithandjthorderStatisticsfromtransmutedgeneralizedlinearexponentialdistributionisfi:j:n(xi;xj)=C[F(xi)]i�1[F(xj)�F(xi)]j�i�1[1�F(xj)]n�jf(xi)f(xj)=C1�h(i)1+h(i) i�11�h(j)1+h(j)�1�h(i)1+h(i) j�i�11�1�h(j)1+h(j) n�j( +x(i))( x(i)+ 2x2(i))�1h(i)1�+2h(i)( +x(j))( x(j)+ 2x2(j))�1h(j)1�+2h(j)(25)whereh(i)=e�( x(i)+ 2x2(i))34 InternationalJournalofComputerApplications(09758887) Volume83-No.17,December2013 TGLED( ;;;) n MSE(b ) MSE(b) MSE(b) MSE(b) 15 0.0185 0.1548 0.0305 0.2507 25 0.0148 0.1498 0.0224 0.2276 35 0.0117 0.0129 0.0152 0.1252 (0:15;0:35;0:65;0:3) 45 0.0114 0.0155 0.0135 0.0865 55 0.0162 0.0379 0.0125 0.0643 65 0.0052 0.0208 0.0065 0.0316 75 0.0093 0.0137 0.0059 0.0138 15 0.0340 0.0760 0.4058 0.0847 25 0.0216 0.0391 0.2918 0.0584 35 0.0212 0.0332 0.1807 0.0452 (0:3;0:6;2;0:7) 45 0.0176 0.0292 0.1662 0.0338 55 0.0098 0.0191 0.0899 0.0260 65 0.0017 0.0063 0.0337 0.0067 75 0.0012 0.0056 0.0246 0.0087 15 0.1521 0.4083 0.5003 0.1452 25 0.0335 0.0693 0.4104 0.1335 35 0.0326 0.0497 0.3951 0.0841 (0:3;0:9;3:5;0:8) 45 0.0227 0.0393 0.3170 0.0618 55 0.0180 0.0600 0.2866 0.0417 65 0.0240 0.0546 0.2468 0.0390 75 0.0232 0.0485 0.1470 0.0163 b,b,andb.Using(33),weapproximate100(1� )condenceintervalsfor ;;andaredeterminedrespectivelyasb z 2q dV ;bz 2q cV,andbz 2q dVwherez istheupper100 thepercentileofthestandardnormaldistribution.Thefollowingtablerepresentsthemeansquareerror(MSEs)oftheMLEs.Table1ThemeansquareerrorsoftheMLEs.WenoticedfromtheaboveTable1thatallMSEsdecreaseasthesamplesizeincreases,whiletheyincreasewithincreasingofthetrueparameter.7.APPLICATIONSInthissectiontworealdatasetsareconsideredtoseewhichoneofdistributionsismoreappropriatetothedatasetforsomeMLEsofparameters.Suchasthetransmutedgeneralizedlinearexponentialdistribution(TGLED),theLinearexponentialdistribution,Trans-mutedLinearexponentialdistribution,TransmutedRaylightdistri-bution,Raylightdistribution(LED,TLED,TRD,RD).Totestthegoodness-of-tofselecteddistributionsineachexample,wecal-culatedtheKol-mogorovSmirnov(K-S)distanceteststatisticanditscorrespondicorrespondingp-value.EXAMPLE1.ConsiderthedatagivenbyAbouammohetal.(1994)whichrepresent40patientssufferingfromleukemiafromoneoftheMinistryofHealthHospitalsinSaudiArabia.Theor-deredlifetimes(indays)aregiveninTable2.Table2Lifetimesof40patientssufferingfromleukemia.11518125541844146151673974378980786592498310241062106311651191122212221251127712901357136914081455147815491578157815991603160516961735179918151852 Table3.TheK-Sdistanceteststatisticandcorrespondingp-values.ModelingdistributionK-Stestp-valueTGLED0.35544140.000049LED0.2136240.044205TLED0.20948590.051105TRD0.2072880.055130RD0.1842780.116136Fig.4isprovidedtocomparetheempiricalreliabilityfunctionsagainstthetheoreticalreliabilityfunctionsofthemodelingdistri-butions.Fig.4.EmpiricalandestimatedsurvivalfunctionsoftheLED,RD,TRD,TGLEDandTLEDmodelsfor(Leukemiadata)EXAMPLE2.Thelifetimesof50devicesthatwereprovidedby(Aarset,1987)aregiveninTable4.Table4Lifetimesof50devices. 0.10.21.01.01.01.01.02.03.06.07.011.012.018.018.018.018.018.021.032.036.040.045.046.047.050.055.060.063.063.067.067.067.067.072.075.079.082.082.083.084.084.084.085.085.085.085.085.086.086.0Table5.TheK-Sdistanceteststatisticandcorrespondingp-values.ModelingdistributionK-Stestp-valueTGLED0.28129180.000542LED0.1760990.0793432TLED0.237468930.005777640TRD0.238325170.005878RD0.262372340.0015858Fig.5isprovidedtocomparetheempiricalreliabilityfunctionsagainstthetheoreticalreliabilityfunctionsofthemodelingdistri-butions.36 InternationalJournalofComputerApplications(09758887) Volume83-No.17,December2013Fig.5.EmpiricalandestimatedsurvivalfunctionsoftheLED,RD,TRD,TGLEDandTLEDmodelsfor(Leukemiadata)Lifetimesof40patientssufferingfromLeukemia.BycalculatingK-Stestandcorrespondingp-valuesforTGLEDandsomespecialcasesasLED,RD,TRDandTLEDformen-tionedtwosurvivaldataexampleswecansaythatthedistributionofTGLEDcanbeappropriatetodealwithLifedataunderdifferentlevelsofsignicant.Table3andTable5containsomeoftheresultsthathavebeenobtainedforthetwomentionedpreviousexamples.8.REFERENCES1Aarset,M.V.(1987).Howtoidentifybathtubhazardrate.IEEETransactionsonReliabilityR-36,106108.2Abouammoh,A.,Abdulghani,S.,Qamber,I.(1994).Onpartialorderingsandtestingofnewbetterthanrenewalusedclasses.ReliabilityEngineeringandSystemSafety43,3741.3Aryal,G.R.andTsokos,C.P.(2011).TransmutedWeibulldistri-bution:AGeneralizationoftheWeibullProbabilityDistribution.EuropeanJournalofPureandAppliedMathematics,4(2),89-102.4Aryal.G.R.andTsokos,C.P.(2009).Onthetransmutedextremevaluedistributionwithapplications.NonlinearAnalysis:Theory,Methodsandapplications,Vol.71,1401-1407.5Aryal,G.R.(2013).TransmutedLog-LogisticDistribution.J.Stat.Appl.Pro.2,No.1,11-20.6Elbatal,I.(2013).TransmutedmodiedinverseWeibullDis-tribution:AGeneralizationoftheModiedinverseWeibullProbabilityDistribution.InternationalJournalofMathematicalArchive-4(8),117-129.7Elbatal,I.andAryal.G.R..OntheTransmutedAdditiveWeibullDistribution.AustrianJournalofStatistics(ToAppear).8Lai,C.D.,Xie,M.,Murthy,D.N.P.(2001).Bathtubshapedfail-ureratedistributions.In:Balakrishnan,N.,Rao,C.R.(Eds.),HandbookinReliability,vol.20.69104.9Mahmoud,M.A.WandAlam,F.(2010).Thegeneralizedlinearexponentialdistribution,Statist.Probabil.Lett.80,1005–1014.10Merovci,F.andElbatal,I.(2013).Transmutedlindley-Geometricdistributionanditsapplica-tions.Stat.ME.arXiv:1309:3774V1.(Tosubmitted).11Muhammad,K.S.andRobert,K.(2013).TransmutedModiedWeibullDistribution:AGeneralizationoftheModiedWeibullProbabilityDistribution.EuropeanJournalofPureandAppliedMathematics.6(1),66-88.12Shaw,W.andBuckley,I.(2007).Thealchemyofprobabilitydistributions:beyondGram-Charlierexpansionsandaskew-kurtotic-normaldistributionfromaranktransmutationmap.13Zhang,T,Xie,M,Tang,LandNg,S(2005)ReliabilityandMod-elingofSystemsIntegratedwithFirmwareandhardware,Int.J.Reliab.QualitySafetyEng.12(3),227–239.37