The New Extended Flexible Weibull Distribution and Its Applications

Transcription

1 Intenatonal Jounal of Data Scence and Analyss 7; 3(3: do:.648j.jdsa.733. The New Extended Flexble Webull Dstbuton and Its Applcatons Zuba Ahmad, Zawa Hussan Depatment of Statstcs, Quad--Azam Unvesty, Islamabad, Pakstan Emal addess: (Z. Ahmad, (Z. Hussan To cte ths atcle: Zuba Ahmad, Zawa Hussan. The New Extended Flexble Webull Dstbuton and Its Applcatons. Intenatonal Jounal of Data Scence and Analyss. Vol. 3, No. 3, 7, pp do:.648j.jdsa.733. Receved: June 7, 7; Accepted: July, 7; Publshed: Septembe 6, 7 Abstact: The pesent atcle consdes a new functon to popose a new lfetme dstbuton. The new dstbuton s ntoduced by mxng up a lnea system of the two logathms of cumulatve hazad functons. The poposed model s called new extended flexble Webull dstbuton and s able to model lfetme wth bathtub shaped falue ates and offes geate flexblty. Theefoe, t can be qute valuable to use an altenatve model to othe exstng lfetme dstbutons, whee, modelng of eal data sets wth bathtub shaped falue ates ae of nteest. A bef descpton of the statstcal popetes along wth estmaton of the paametes though maxmum lkelhood pocedue ae dscussed. The potentalty of the poposed model s showed by dscussng two eal data sets. Fo these data sets, the poposed model outclasses the Flexble Webull Extenson, Invese Flexble Webull Extenson and Modfed Webull dstbutons. Keywods: Bathtub Shaped Falue Rates, Ode Statstcs, Moment Geneatng Functon, Maxmum Lkelhood Estmaton. Intoducton In the pactce of analyzng eal phenomena of natue one fequently uses the Raylegh, Exponental, o Webull dstbutons. These models possess numeous desable popetes and nce ntepetatons of ts paametes enablng them to be utlzed fequently. Between these lfetme models, Webull dstbuton s the most pomnent dstbuton fo modelng eal phenomena of natue. The Webull model was ognally ntoduced by Webull [9], a Swedsh physcst, and utlzed t to epesent the dstbuton of beakng stength of mateals. Webull model also has the shape and scale paametes, offeng chaactestcs of both the exponental and Raylegh dstbutons. In ecent past yeas, the Webull model becomng vey popula n modelng lfetme data, because n the pesence of censong whch makes t much ease to handle, at least numecally. The cumulatve dstbuton functon (CDF of the Webull model s gven by. z α G( z = e, z, α, >. ( The Webull model s vey useful n modelng eal phenomena exhbtng monotonc falue ates. But, the Webull model s nappopate to use n modelng data havng non-monotonc falue ates. Among non-monotonc falue ate functon, the bathtub shaped falue ate s vey useful and has a numbe of applcatons n the lteatue. Fo example, n bo-analyss the human motalty ate and n elablty engneeng the lfecycle of electonc components s obseved to have a bathtub shaped falue ate functon. Due to pactcal utlty n bo and elablty dscplnes, numeous genealzatons of Webull dstbuton have been poposed n the lteatue amng to mpove ts chaactestcs and to model eal wold scenao wth nonmonotonc falue ate functons. These genealzatons, ncludng a statstcal model wth bathtub falue ate studed by Xe and La [], Sahan and Zandn [7], Beta-Webull (BW dstbuton of Famoye et al. [], Kumaaswamy Webull (KW dstbuton poposed by Codeo et al. [], Genealzed modfed Webull (GMW dstbuton poposed by Caasco et al. [9], Exponentated modfed Webull extenson (EMWEx dstbuton ntoduced by Sahan and Apaloo [6], Flexble Webull (FWEx dstbuton of Bebbngton et al. [8], Genealzed Flexble Webull Extenson (GFWEx dstbuton studed by Ahmad and Iqbal [], othe extensons of Webull model poposed by Ahmad and Hussan (7 ae [], [3], [4], [5] and [6], espectvely. Fo a concse evew of these dstbutons one may call to Pham and La [5] and Muthy et al. [4]. These dstbutons

2 9 Zuba Ahmad and Zawa Hussan: The New Extended Flexble Webull Dstbuton and Its Applcatons have numeous applcatons ncludng elablty analyss, clncal studes, appled statstcs and lfe testng expements etc. Guvch et al. [] poposed a new class of agng dstbutons defned by the CDF gven by. F ( z G z = e, z, >. ( Whee F ( z s monotoncally nceasng functon of z. It s a vey useful method to mx two suvval functons and ceate a new functon as: ( η S( z = η S z + S z, Whee < η, η <, ths method of suggestng new functons s known as mxtue of dstbutons, o S( z = η S z + ψs z, η, ψ>. (3 One may also geneate a new functon by collaboatng two cumulatve hazad functons as: H( z = H z + H z, (4 In tem of cumulatve hazad functon (CHF, the CDF can be wtten as whee a G z = e H ( z, z>, (5 H z fulfls the condtons stated below H z s nonnegatve and nceasng functon of z b lm? H( z and lm H( z. z z The pobablty densty functon (PDF assocatng to (5 has the followng expesson. = g z h z e H z, z >. The modfed Webull dstbutons pesented by Xe and La [], Sahan and Zandn [7] and Almalk and Yaun [7] belongs to the class defned n (5. Hee n (5, the H( z s bounded. On the othe hand, n ths atcle, effot have been made to poduce a new functon annoyng to elax the bounday condtons. Hence, n ths atcle log H( z s used athe than H( z. Because, t mght be moe exctng to use log H( z athe H( z n ode to advance a vey flexble model. Hence, one may wte (4 as H( z = H z H z, (6 z + z H( z e =. (8 By substtutng (8 n (5, one can easly get the CDF of the new extended flexble Webull (NEx-FW dstbuton. The suggested model s capable of modelng data wth bathtub falue ate. The pesent atcle s desgned as: Secton, offes the defnton and gaphcal dsplay of the new model. Secton 3, contans the basc statatcal popetes. Secton 4, 5 and 6, deves the moment geneatng functon, pobablty geneatng functon and factoal moment geneatng functon of the NEx- FW dstbuton. Secton 7 and 8, contans the estmaton of the paametes and densty functons of the ode statstcs. Secton 9, offes the analyss to eal data sets. Fnally, secton, contans concludng emaks.. New Extended Flexble Webull Dstbuton The CDF of the NEx-FW dstbuton s gven by ( z + z e G ( z;,, =? e, z,,, >. (9 The pobablty dstbuton functon (PDF coespondng to (9 s gven by + ( z z z + z e,, = + g z; z z? e e. The suvval functon (SF of the NEx-FW dstbuton s wth HF ( S z;,, =? e e ( z + z, ( z + z h z;,, = z + z? e. ( The fgue dsplays the HF of the NEx-FW dstbuton fo dffeent values of paametes. The expesson povded n (6 can be e-wte as logh( z = logh ( z + logh ( z. (7 A mxtue of the two logathm of cumulatve hazad functons, such asz and z ae poposed to ntoduce a new vey flexble lfetme model. So, the expesson gven n (7, can be wtten n the followng fom. Fgue. HF of the New Extended Flexble Webull dstbuton, fo dffeent values of paametes.

3 Intenatonal Jounal of Data Scence and Analyss 7; 3(3: Basc Popetes Ths secton of the pape coves the basc statstcal popetes of the NEx-FW dstbuton. 3.. Quantle and Medan The expesson fo the dstbuton s gven by q q th q quantle z q of the NEx-FW { } z + z log log q =. ( Usngq =.5, n (, one can easly obtan the medan of the NEx-FW dstbuton. Also, puttng q =.5, and q =.75, n (, one may get the st and 3 d quatles of the NEx-FW dstbuton, espectvely. 3.. Geneaton of Random Numbes The fomula fo geneatng andom numbes fom NEx- FW dstbuton can be deved as whee u U (, ( ;, G z, = u. Afte smplfcaton, ths yelds { } z + z log log u =. The expesson fo geneatng andom numbes fom the poposed dstbuton s not closed fom. Theefoe, the andom numbes fom the poposed model can be geneated usng compute softwae Moments If Z~ NEx-FW ( z deved as ( ; µ = z z,, dz ;,,, then the + ( z z z + z e? th moments of Z s µ = z z + z e e dz + ( z + z µ = z ( z z? e dz! + = ( j + + j ( + z ( µ = z z z e dz! j! + = j= j ( + ( j+ ( + + ( + + z j z µ = z e dz + z e dz.! j! = j= ( Usng the defnton of gamma functon (Zwllnge [] n the followng fom, z z tx Γ z = x t e dt, z, x >. Usng the above defnton of gamma functon n (, and fnally, one may get j j j Γ Γ + µ = +. j! j! + j = j= ( ( ( ( + + (3 4. Moment Geneatng Functon If Z~ NEx-FW ( z functon of Z s deved as ;,,, then the moment geneatng tz Mz ( t = e g ( z;,, dz t Mz ( t = z g ( z;,, dz! = t Mz ( t = µ. (4! = By usng (3, n (4, one may have the poof of the NEx- FW dstbuton. 5. Pobablty Geneatng Functon The pobablty geneatng functon (PGF of NEx-FW dstbuton s deved G z ( g ( z;,, = dz

4 Zuba Ahmad and Zawa Hussan: The New Extended Flexble Webull Dstbuton and Its Applcatons ( log G ( = z g ( z;,, dz! = = ( log G ( = µ. (5! On substtutng (3, n (5, one may get the expesson fo the PGF of NEx-FW dstbuton. 6. Factoal Moment Geneatng Functon The factoal moment geneatng functon (FMGF of NEx-FW dstbuton can be deved as H ( + δ = ( + δ z g ( z ;,, dz ( + δ log H ( + δ = z g ( z;,, dz! H = = ( + δ log + δ = µ. (6! By substtutng (3, n (6, esult n poof of the FMGF of NEx-FW dstbuton. 7. Estmaton Ths secton of the atcle, concen wth estmaton of the model paametes though maxmum lkelhood (ML pocedue. LetZ, Z,, Zk ae andomly sampled fom NEx- FW dstbuton wth paametes (,,. Then, the coespondng log-lkelhood functon of ths sample s lnl = log z + z + z?+? z e. k k k z + z ( ( (7 = = = By attanng the patal devatves of the expesson n (7 on paamete, and then equatng to zeo, one may have k k k dlnl z = + d z z z + z z. z e (8 ( = + = = k k k dlnl z = + d z z z + z z. z e ( (9 = + = = ( z log ( z + z k k k dlnl = + d z z z + z z log log. z e z z ( ( = + = = It s peceved that, the expessons gven n (8-( do not hold soluton n closed foms; so, the estmates of the unknown paametes can be obtaned numecally by usng the teatng pocedue. The SANN algothm n R language s used to estmate the paametes numecally. 8. Ode Statstc Consde a sample say Z, Z,,Z k selected andomly fom NEx-FW dstbuton wth paametes (,,, havng odeed values Z: k, Z: k,, Zk: k. Let Z( :k epesents the smallest ofz: k, Z: k,, Zk: k, smlaly, Z( :k epesents the second smallest ofz: k, Z: k,, Zk: k and Z ( k: k epesents the th k smallest of{ Z: k, Z: k,, Zk: k }. Then, the PDF of Z ( k :, k s gven by k g: k ( z = g ( z G( z G( z. beta (, k + So, the PDF of smallest ode statstc s g ( z + z z + z e z k z z e e = +. : k Also, the PDF of lagest ode statstc s ( z + z ( z + z + k z k k k k z k e e = + k k gk: k z k zk zk e e e. 9. Applcatons In ths secton, two eal lfe applcaton ae pesented. The esult of the goodness of ft of the suggested model s compaed wth thee othe well-known competng lfetme models. The nvestgatve tools such as Akake s Infomaton

5 Intenatonal Jounal of Data Scence and Analyss 7; 3(3: 8-3 Cteon (AIC, Hannan-Qunn nfomaton cteon (HQIC, Came-von-Msses (CM test statstcs, Andeson Dalng (AD test statstc, Consstent Akake s Infomaton Cteon (CAIC, Bayesan nfomaton cteon (BIC, Kolmogoov Smnov (K-S test statstc and log lkelhood l (., z ae consdeed. On the bass these measues, t s showed that the new model povdes geate flexblty. 9.. Example The fst data set obtaned fom Tah et al. [8], denotes the falue tmes of 84 Acaft Wndsheld. The falue tmes ae as:.4,.866,.385, 3.443,.3,.876,.48, 3.467, Table. Goodness of ft esults fo NEx-FW, FWEx, IFWEx and MW..39,.899,.6, 3.478,.557,.9,.65, 3.578,.943,.9,.63, 3.595,.7,.94,.646, 3.699,.4,.98,.66, 3.779,.48,.,.688, 3.94,.8,.38,.8,3, 4.35,.8,.85,.89, 4.,.33,.89,.9, 4.67,.43,.97,.934, 4.4,.48,.35,.96, 4.55,.55,.54,.964, 4.78,.56,.9, 3., 4.35,.568,.94, 3.3, 4.376,.65,.3, 3.4, 4.449,.69,.4, 3.7, 4.485,.65,.9, 3.66, 4.57,.65,.3, 3.344, 4.6,.757,.34, and The fnal esults, afte applyng the poposed dstbuton along wth the competng models ae povded n example, ae summazed n table &. Dst. Max. Lkelhood Estmates AD CM KS logl NEx-FW FWEx IFWEx MW ˆ =.5, ˆ =.4, ˆ = ˆ =.37, ˆ = ˆ =.643, ˆ = ˆ =.798, ˆ σ =.44, ˆ α = Table. Goodness of ft esults fo NEx-FW, FWEx, IFWEx and MW. Dst. AIC BIC CAIC HQIC NEx-FW FWEx IFWEx MW Example The second data set denotes the falue tmes of a sample of 3 devces taken fom Khan and Jan [3]. The tmes ae.75,.3,.47,.3,.8,.3,.65,., 3.,.73,.6, 3., 3.,., 3., 3., 3.,.,.6,.93,.88,.47,.8,.43, 3.,.3, 3.,.8,.45 and.66. The fnal goodness of ft esult of the poposed model along wth the competng models, ae summazed n table 3 & 4. Table 3. Goodness of ft esults fo NEx-FW, FWEx, IFWEx and MW. Dst. Max. Lkelhood Estmates AD CM KS logl NEx-FW FWEx IFWEx MW ˆ =.3, ˆ =.4, ˆ = ˆ =.383, ˆ = ˆ =.9, ˆ = ˆ =4.797, ˆ =.8, ˆ α = Table 4. Goodness of ft esults fo NEx-FW, FWEx, IFWEx and MW. Dst. AIC BIC CAIC HQIC NEx-FW FWEx IFWEx MW Concluson In ths pape, a new lfetme dstbuton enttled New Extended Flexble Webull Dstbuton s ntoduced by takng nto account a lnea system of the two logathms of cumulatve hazad functons. The suggested model offes geate dstbuton flexblty and s able to model lfetme data wth bathtub shaped falue ates. A concse explanaton of the mathematcal popetes of the poposed model, wth estmaton of paametes usng maxmum lkelhood pocedue ae dscussed. The poposed modal s llustated by means of analyzng two eal data sets, and the goodness ft of the poposed model s compaed wth that of thee othe exstng lfetme dstbutons. Analyzng these two data sets, t s obseved that the new model povdes best ft than the

6 3 Zuba Ahmad and Zawa Hussan: The New Extended Flexble Webull Dstbuton and Its Applcatons compettve models. It s hoped that the New Extended Flexble Webull dstbuton wll seve as one of the most useful lfetme model and wll attact a wde ange of pactcal applcatons n the feld of bo-medcal and elablty engneeng. Acknowledgements The authos ae so gateful to the edto and anonymous efeees fo a caeful checkng of the detals and fo the helpful comments on an eale veson of the pape that mpoved the pesentaton n the pape. On behalf of coespondng autho of ths atcle, I would lke to thank my espected paents fo allowng me to ealze my own potental. Also, I need to thank my sweet sste, who taught me the value of handwok & educaton. Wthout he, I may neve have gotten to whee I am today. Refeences [] Ahmad, Z. and Iqbal, B. (7. Genealzed Flexble Webull Extenson Dstbuton. Cculaton n Compute Scence, Volume (4, [] Ahmad, Z. and Hussan, Z. (7. New Flexble Webull Dstbuton. Intenatonal Jounal of Advanced Reseach n Electcal, Electoncs and Instumentaton Engneeng, Vol.6, No.3. [3] Ahmad, Z. and Hussan, Z. (7. Modfed New Flexble Webull Dstbuton. Cculaton n Compute Scence, Vol., No.6, pp: ( [4] Ahmad, Z. and Hussan, Z. (7. New Extended Webull Dstbuton. Cculaton n Compute Scence, Vol., No.6, pp: ( [5] Ahmad, Z. and Hussan, Z. (7. Flexble Webull Dstbuton. Jounal of Compute and Mathematcal Scence, Vol.8, No.6, pp: (5-6. [6] Ahmad, Z. and Hussan, Z. (7. Vey Flexble Webull Dstbuton. Mayfeb Jounal of Mathematcs. (Accepted. [7] Almalk, S. J. and Yuan, J. (3. A new modfed Webull dstbuton. Relablty Engneeng and System Safety,, [8] Bebbngton, M., La, C. D. and Ztks, R. (7. A flexble Webull extenson. Relablty Engneeng and System Safety, 9, [9] Caasco M., Otega E. M. and Codeo G. M. (8. A genealzed modfed Webull dstbuton fo lfetme modelng. Computatonal Statstcs and Data Analyss, 53(, Kumaaswamy Webull dstbuton wth applcaton to falue data. Jounal of the Fankln Insttute, 347, [] Famoye, F., Lee, C. and Olumolade, O. (5. The beta- Webull dstbuton. Jounal of Statstcal Theoy and Applcatons, 4(, 36. [] Guvch, M. R., Dbenedetto, A. T. and Rande, S. V. (997. A new statstcal dstbuton fo chaactezng the andom length of bttle mateals. J Mate Sc; 3: [3] Khan, A. H. and Jan, T. R. (6. The new modfed genealzed lnea falue ate dstbuton. J. Stat. Appl. Po. Lett. 3, No., [4] Muthy, D. N. P., Xe, M. and Jang, R. (3. Webull Models. John Wley and Sons, New Yok. [5] Pham, H. and La, C. D. (7. On ecent genealzatons of the Webull dstbuton. IEEE Tansactons on Relablty, 56, [6] Sahan, A. M. and Apaloo, J. (3. Exponentated modfed Webull extenson dstbuton. Relablty Engneeng and System Safety,, [7] Sahan, A. M. and Zandn, M. (9. Modfed Webull dstbuton. Appled Scences,, [8] Tah, M. H., Codeo, G. M., Mansoo, M. and Zuba, M. (5. The Webull-Lomax Dstbuton: Popetes and Applcatons', Hacettepe Jounal of Mathematcs and Statstcs. [9] Webull, W. (939. A statstcal theoy of the stength of mateal. Ingenos Vetenskaps Akademens, Stockholm 5. [] Xe, M. and La, C. D. (996. Relablty analyss usng an addtve Webull model wth bathtub-shaped falue ate functon. Relablty Engneeng and System Safety, 5(, [] Zwllnge, D. (4. Table of ntegals, sees, and poducts. Elseve. Bogaphy Zuba Ahmad SO Wal Muhammad, eseach schola at Quad--Azam Unvesty, obtaned hs Maste degee n Statstcs n 4 at Unvesty of Malakand and eceved hs M. Phl. Degee n Statstcs n 7 at Quad--Azam Unvesty, Islamabad, Pakstan. Hs eseach topc n M. Phl. was On Dffeent Modfcatons of Webull Dstbuton unde the gudance & supevson of D. Zawa Hussan, Quad--Azam Unvesty, Islamabad, Pakstan. [] Codeo, G. M., Otega, E. M. and Nadaajah, S. (. The