New Extended Weibull Distribution

Transcription

1 Crculaton n Computer Scence Vol No6 pp: (4-9) July 7 New Extended Webull Dstrbuton Zubar Ahmad Research Scholar: Department of Statstcs Quad--Aam Unversty 45 Islamabad 44 Pastan Zawar ussan Assstant Professor: Department of Statstcs Quad--Aam Unversty 45 Islamabad 44 Pastan ABSTRACT Ths artcle consders a new functon to propose a new lfetme model The new model s ntroduced by utlng e lnear scheme of e two logarms of cumulatve haard functons The new model s named as new extended Webull dstrbuton and s able to model data w unmodal or modfed unmodal shaped falure rates A bref explanaton of e maematcal propertes of e proposed model s provded The model parameters wll be estmated by deployng e maxmum lelhood meod To llustrate e usefulness of e proposed model an example wll be dscussed Keywords Modfed unmodal falure rates; factoral moment generatng functon; Order statstcs; Maxmum lelhood estmates INTRODUCTION In e practce of relablty modelng e probablty dstrbutons are most frequently used as tme to falure dstrbutons In same perspectve e qualty of e relablty model sgnfcantly depends on e success n choosng a sutable probablty model of e phenomenon under consderaton Durng e last couple of decades a specfc group of e probablty models such as exponental Pareto Log-normal Gamma Raylegh and Webull dstrbutons were frequently used for modelng lfetme data owever n practce t s observed at most of ese dstrbutons are not qute flexble enough to accommodate dfferent phenomena of nature For s reason e researchers have wored on e expanson of ese dstrbutons to ntroduce more and more flexble and most suted model for modelng data n practce Among ese dstrbutons e Webull model s e most wdely used dstrbuton as t offers e characterstcs of bo e exponental and Raylegh dstrbutons Therefore between e lfetme dstrbutons e Webull dstrbuton ntroduced by Walodd Webull s one of e most useful model for modelng lfetme data w monotonc falure rates The cumulatve dstrbuton functon (CDF) of e two parameters Webull dstrbuton s gven n () G( ) e () The haard functon (F) of e Webull dstrbuton s gven by h ( ) () Alough e Webull model s one of e most frequently used lfetme models for modelng lfetme data havng monotonc falure rates owever e Webull model s ncapable to model w non-monotone falure rates whch are qute common n relablty and bomedcal analyss In more complex scenaros such as dseases whose mortalty reaches a pea after some fnte perod and en declnes gradually Ths s usually termed a unmodal falure rate Also some dseases such as breast cancer s observed to have modfed unmodal falure rates In e case of electronc ones e falure rate s often observed to have non-monotonc Ths usually taes e form of ncreased falure rate early ( wearn ) and late ( wear-out ) n e component lfetme Ths s usually nown as batub-shaped falure rate Many dfferent versons of Webull dstrbuton have arsen n e lterature to mprove ts characteratons A very small amount of ese generalatons ncludng Modfed Webull (MW) dstrbuton due to La et al [] Beta Webull (BW) dstrbuton by Famoye et al [] Exponentated Flexble Webull Extenson (EFWEx) Dstrbuton ntroduced by El- Gohary et al [] generaled power Webull (GPW) dstrbuton due to Nuln and aghgh [4] flexble Webull extenson (FWEx) studed by Bebbngton et al [8] Kumaraswamy Webull (Ku-W) dstrbuton by Cordero et al [9] Transmuted Webull (TW) dstrbuton due to Aryal and Tsoos [6] Beta modfed Webull (BMW) dstrbuton studed proposed by Slva et al [6] On transmuted flexble Webull extenson (OTFWEX) dstrbuton of Ahmad and ussan [] new flexble Webull (NFW) dstrbuton proposed by Ahmad and ussan [] Flexble Webull (FW) dstrbuton of Ahmad and ussan [] and generaled flexble Webull extenson (GFWEx) dstrbuton propose by Ahmad and Iqbal [4] It s a very useful meod to combne two survval functons and generate a new functon as: S( ) S ( ) S ( ) Where functons s charactered as a mxture of dstrbutons or s meod of ntroducng new S S S > () We may also ntroduce a new functon by mxng up two cumulatve haard functons as: ( ) (4) In term of cumulatve haard functon (CF) e cumulatve dstrbuton functon (CDF) can be expressed as where below G e > (5) must ustfes e two condtons specfed Copyrght 7 Zubar Ahmad et al Ths s an open-access artcle dstrbuted under e terms of e Creatve Commons Attrbuton Lcense 4 whch permts unrestrcted use dstrbuton and reproducton n any medum provded e orgnal auor and source are credted

2 Crculaton n Computer Scence Vol No6 pp: (4-9) July 7 wwwccsarchveorg s dfferentable non-negatve and ncreasng functon of lm ( ) and lm ( ) The probablty densty functon (PDF) assocatng to (5) has e form g h e The modfed Webull extensons proposed by Xe and La [7] Sarhan and Zandn [5] Lemonte et al [] and Almal and Yuan [5] belongs to e class specfed n (5) ere n (5) e () s bounded owever s artcle suggests a new functon amng to relax e boundary condtons so log () s used nstead of () Because t would be more nterestng to carry w log () raer an () to develop a new flexble model ence e expresson n (4) can be wrtten as ( ) (6) The SF of e NEx-W dstrbuton s S ; e w haard functon (F) e h; e () The possble shapes of e Fs of NEx-W dstrbuton are provded for ree combnatons of e parameters n fgure & The shapes n fgure & fgure show at e F of NEx-W dstrbuton can be unmodal or modfed unmodal shaped falure rates: The quantty mentoned n (6) can be wrtten n e form as log ( ) log ( ) log ( ) (7) ere a mxture of e two logarm of cumulatve haard functons taen as and to ntroduce a new flexble lfetme model So e quantty gven n (7) can be wrtten n e form gven by (8) ( ) e By usng (8) n (5) one mght get e CDF of e new extended Webull (NEx-W) dstrbuton The new model s capable of modelng lfe tme data w unmodal or modfed unmodal falure rates The rest of paper s structured n e followng form: Secton ntroduces e NEx-W dstrbuton The basc maematcal propertes of e new model are dscussed n Secton Secton 4 and 5 contans e moment generatng functon and factoral moment generatng functon of e new model Denstes of e order statstcs are calculated n secton 6 The estmaton of e parameters are dscussed n secton 7 Secton 8 offers analyss to a real data set Fnally concluson remars are provded n secton 9 NEW EXTENDED WEIBULL DISTRIBUTION The CDF of e NEx-W dstrbuton s defned by e followng expresson e G ; e (9) Fgure : F of e NEx-W dstrbuton for a choce values of parameters The densty functon assocatng to (9) s gven by e g ; e e Fgure : F of e NEx-W dstrbuton for a choce values of parameters 5

3 Crculaton n Computer Scence Vol No6 pp: (4-9) July 7 wwwccsarchveorg BASIC PROPERTIES Ths secton of e artcle contans e basc maematcal propertes of e NEx-W dstrbuton Quantle and Medan The expresson for e dstrbuton s gven by q quantle q of e NEx-W q log log q () q Usng q=5 n () we have e medan of e NEx-W dstrbuton Also by settng q=5 and q=75 n () we obtan e st and rd quartles of e NEx-W dstrbuton respectvely Generaton of Random Numbers The formula for generatng random numbers from NEx-W dstrbuton s gven by log log R R U Moments ; en e If Z NEx-W obtaned as r r g ; d r moments of Z s r e r e e d r r e d! r e d r! r!! r ( ) e d r( ) ( ) e d!! Fnally e followng expresson s observed r r!! r ( r + r!! ( () 4 MOMENT GENERATING FUNCTION ; en e formula for moment If Z NEx-W generatng functon (MGF) of Z s derved as t M t E e t ; M t e g d M t r t r r! () r Usng () n () we have e MGF of NEx-W dstrbuton 5 FACTORIAL MOMENT GENERATING FUNCTION ; en e factoral moment If Z NEx-W generatng functon (FMGF) of Z s derved as E E e ln ln e g ; d r r ln r (4) r! Usng () n (4) we get (FMGF) of NEx-W dstrbuton 6 ORDER STATISTICS Let Z Z Z are ndependently and dentcally dstrbuted (d) random varables taen from NEx-W ; n such a way at Z Z : : So e densty functon of Z : = s g : g G Beta n G 5 Where Φ And e ont densty of s : : g :: C G G G Where G g g! C!!! 6

4 Crculaton n Computer Scence Vol No6 pp: (4-9) July 7 wwwccsarchveorg ere e denstes for e medan order statstcs as Zm f = m+ st order statstcs as Z Z Z Z and for order mn Z max Z Z Z statstcs as 6 Dstrbuton of Mnmum and Maxmum Order Statstcs Let Z Z Z sampled randomly from NEx-W ; w CDF provded n (5) Then e densty functon of e mnmum and maxmum order statstcs s derved n (6) and (7) respectvely g: g G e g: e e Also densty for e maxmum order statstcs s : g g G e g : e e (6) e e 7 7 ESTIMATION Ths secton determnes e maxmum lelhood estmates of e model parameters 7 Maxmum lelhood estmaton Let Z Z ; en e lelhood functon of s sample s ln Z are observed randomly from NEx-W L log e 8 Fndng e partal dervatves of e expresson n (8) on parameter and en equatng e result equal to ero dln L d e (9) dln L d log log e log () d L e d ln () From (9)-() t s well-clear at ese equatons are not n closed forms and cannot be solved manually and statstcal software can be utled to solve em numercally by deployng e teratve technques such as e Newton- Raphson algorm The SANN algorm s used n R language to obtan e numercal estmates of e parameters 7 Asymptotc Confdence Lmts From (9)-() shows at ese equaton are not n a closed form ence t s qute tough to derve e exact dstrbuton of e MLE s Therefore t s reasonable to obtan e asymptotc confdence bounds of e unnown parameters The most wdely and sound good procedure s to assume at ˆ ˆ ˆ are approxmately normally e MLE s dstrbuted havng mean and covarance matrx Σ All of e second order dervatves for e densty functon of NEx-W dstrbuton exst ence we have ˆ ˆ ~ N Σ ˆ w Where V V V Σ EV V V V V V lnl lnl lnl V = V = V = lnl lnl lnl V = V = V = lnl lnl lnl V = V = V = As contans of unnown parameters erefore to have an estmate of e unnown parameters are replaced by er correspondng MLE s gven by Vˆ Vˆ Vˆ Σˆ Vˆ V ˆ Vˆ Vˆ Vˆ Vˆ () 7

5 Crculaton n Computer Scence Vol No6 pp: (4-9) July 7 wwwccsarchveorg Usng () approxmately and can be found respectvely as V V % confdence lmts for ˆ Z ˆ ˆ Z ˆ ˆ Z ˆ ere Z represents e upper standard normal dstrbuton V percentle of e 8 APPLICATIONS To llustrate e sgnfcant mprovement of e suggested model an example s presented The goodness of ft result of e suggested model s compared w at of four oer wellnown lfetme models such as Webull Flexble Webull extenson (FWEx) generaled power Webull (GPW) and Kumaraswamy generaled power Webull (Ku-GPW) dstrbutons The nvestgatve measures ncludng Aae s Informaton Crteron (AIC) corrected Aae nformaton crteron (CAIC) annan-qunn nformaton crteron (QIC) l On decdng ese measures t and log lelhood s proved at e newly developed model provdes greater dstrbutonal flexblty Example: The data set taen Bader and Prest [7] contanng e sngle fbers of mm w a sample of se Table : The fbers data of mm Mn st Quartle Medan Mean rd Qu Max Table : The fbers data of mm The selected crtera of e NEx-W W FWExD GPW and Ku-GPW dstrbutons are summared n table & 4 Table : Goodness of ft results for NEx-W W FWEx GPW and Ku-GPW Dst Max Lelhood Estmates logl NEx-W W FWEx GPW Ku-GPW ˆ =54 ˆ =7 ˆ =65 ˆ =549 ˆ =5 ˆ = 7 ˆ = 96 ˆ =5 ˆ =98 ˆ =796 â =47 ˆb =47 ˆ =67 ˆ =5 ˆ = Table 4: Goodness of ft results for NEx-W W FWEx GPW and Ku-GPW Dst AIC CAIC QIC NEx-W W FWEx GPW Ku-GPW CONCLUSIONS In s paper a new functon s suggested to ntroduce a new model whch s called e new extended Webull dstrbuton Some of e statatcal propertes and e estmaton rough maxmum lelhood meod for e model parameters are dscussed The plots of e haard functon are setched to show e flexblty of e new model The proposed model s able to model lfetme data w unmodal or modfed unmodal falure rates At e end an applcaton of e suggested model to a real data set s presented to llustrate at e NEx-W dstrbuton can be used qute effectvely an alternatve to oer lfetme dstrbutons It s hoped at e new extenson wll be a good canddate model to model lfetme data REFERENCES [] Ahmad Z and ussan Z (7) Flexble Webull dstrbuton Journal of Computer and Maematcal Scences 8(6) 5-6 [] Ahmad Z and ussan Z (7) New Flexble Webull dstrbuton Internatonal Journal of Advanced Research n Electrcal Electroncs and Instrumental Engneerng; 6 (5) [] Ahmad Z and ussan Z (7) On Transmuted Flexble Webull Extenson Dstrbuton w Applcatons to Dfferent Lfetme Data Sets : [4] Ahmad Z and Iqbal B (7) Generaled Flexble Webull Extenson Dstrbuton Crculaton n Computer Scence Volume (4) [5] Almal S J and Yuan J () A new modfed Webull dstrbuton Relablty Engneerng and System Safety 64 7 [6] Aryal G R and Tsoos C P () Transmuted Webull dstrbuton: A generalaton of e Webull probablty dstrbuton European Journal of Pure and Appled Maematcs [7] Bader M G and Prest A M (98) Statstcal aspects of fbre and bundle streng n hybrd compostes Progress n scence and engneerng of compostes 9-6 [8] Bebbngton M La C D and Zts R (7) A flexble Webull extenson Relablty Engneerng and System Safety [9] Cordero GM Ortega EM and Nadaraah S () The Kumaraswamy Webull dstrbuton w applcaton to falure data Journal of e Franln Insttute 47(8) 8

6 Crculaton n Computer Scence Vol No6 pp: (4-9) July 7 wwwccsarchveorg [] El-Gohary A El-Bassouny A and El-Morshedy M (5) Exponentated Flexble Webull Extenson Dstrbuton Internatonal Journal of Maematcs and ts Applcatons (A) [] Famoye F Lee C and Olumolade O (5) The beta- Webull dstrbuton Journal of Statstcal Theory and Applcatons 4() 6 [] La C D Xe M and Mury D N P () Modfed Webull dstrbutons IEEE Transactons on Relablty 5() 7 [] Lemonte A J Cordero G M and Ortega E M (4) On e addtve Webull dstrbuton Communcatons n Statstcs-Theory and Meods [4] Nuln M aghgh F (6) A ch-squared test for e generaled power Webull famly for e head-andnec cancer censored data Journal of Maematcal Scences 4 [5] Sarhan A M and Zandn M (9) Modfed Webull Dstrbuton Appled Maematcal Scences - 6 [6] Slva G O Ortega E M and Cordero G M () The beta modfed Webull dstrbuton Lfetme data analyss 6 49 [7] Xe M and La C D (996) Relablty analyss usng an addtve Webull model w batub-shaped falure rate functon Relablty Engneerng and System Safety 5() 87-9 APPENDIX R codes: ere a s used for b s used for s s used for and pm s used for proposed model data=c( ) # Proposed Model (pm) - Probablty densty functon pdf_pm <- functon(parx) { a= par[] b= par[] s= par[] (b*a*(x^(a-))+(*sx^))*(exp((b*x^a)-(sx^)))*(exp(- (exp((b*x^a)-(sx^))))) } # Proposed Model (pm) - Cumulatve dstrbuton functon cdf_pm <- functon(parx) { a= par[] b= par[] s= par[] -(exp(-(exp((b*x^a)-(sx^))))) } setseed() goodnessft(pdf=pdf_pm cdf=cdf_pm starts =c() data = data meod="sann" doman=c(inf)mle=null) AUTOR S PROFILE Zubar Ahmad SO Wal Muhammad research scholar at Quad--Aam Unversty obtaned hs Master degree n Statstcs n 4 at Unversty of Malaand and receved hs MPhl Degree n Statstcs n 7 at Quad--Aam Unversty Islamabad Pastan s research topc n MPhl was On Dfferent Modfcatons of Webull Dstrbuton under e gudance & supervson of Dr Zawar ussan Quad--Aam Unversty Islamabad Pastan e-mal: ferry@gmalcom CCS 7 ISSN Publshed by: CSL Press USA 9