PARAMETER ESTIMATION FOR TWO WEIBULL POPULATIONS UNDER JOINT TYPE II CENSORED SCHEME

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1 Sept 04 Vol 5 No 04 Intenatonal Jounal of Engneeng Appled Scences 0-04 EAAS & ARF All ghts eseed wwweaas-ounalog ISSN PARAMETER ESTIMATION FOR TWO WEIBULL POPULATIONS UNDER JOINT TYPE II CENSORED SCHEME SK ASHOUR OE ABO-KASEM Depatment of Mathematcal Statstcs Insttute of Statstcal Studes & Reseach Cao Unesty Egypt Depatment of Statstcs Faculty of Commece Zagazg Unesty Egypt E-mal: ashousam@hotmalcom usama_eaky84@yahoocom ABSTRACT In ths pape maxmum lkelhood estmaton hae been obtaned fo two Webull populatons unde ont type II censoed scheme whch genealze esults of Balakshnan Rasoul (008 Moeoe appoxmate confdence egon ae also dscussed compaed wth two Bootstap confdence egons A numecal llustaton fo these new esults s gen Keywods: Webull dstbuton; Jont type-ii censong; Maxmum lkelhood estmaton; Appoxmate confdence; Bootstap nteals; Coeage pobabltes INTRODUCTION Thee ae aous types of censoed data to be dealt wth n the analyss of lfetme expements see Lawless (003 Almost all of these types of data ae concened wth the onesample poblems But thee ae stuatons n whch the expemente plans to compae dffeent populatons In such poblems the ont censong scheme has been suggested n the lteatue As mentoned by Rasoul Balakshnan (00 a ont censong scheme s qute useful n conductng compaate lfetme test of poducts comng fom dffeent unts wthn the same faclty Moe pecsely suppose that the poducts ae beng poduced by two lnes unde the same faclty Two ndependent samples of szes m n ae selected fom these lnes put smultaneously on a lfe testng expement Then to sae tme money the expemente follows a ont censong scheme temnates the lfe testng when a cetan numbe of falues (say occu Suppose that X X m the lfetmes of m specmens of poduct A ae d om aables fom dstbuton functon Fx ( densty functon f ( x Y Y n the lfetmes of n specmens of poduct B ae d om aables fom dstbuton functon G(x densty functon g(x Futhe suppose W ( W ( W ( N denote the ode statstcs of the N m n om aables { X X m ; Y Y n} Then unde the ont type-ii censong scheme the obseable data consst of ( ZW W = (W W W wth ( N whee ( ( ( beng a pe-fxed ntege Z = (Z Z wth z o 0 accodng as w s fom an X- o Y- falue Lettng M X-falues n W Z denote the numbe of N ( Z M (e the numbe of Y-falues n W the lkelhood of (Z W s gen by Balakshnan Rasoul (008 as: Z Z m m n n [{ ( } { ( } ]{ ( } { ( } L C f w g w F w G w ( whee F F GG ae the sual functons mn!! of the two populatons C ( m m!( n n! Balakshnan Rasoul (008 deeloped lkelhood nfeence fo the paametes of two exponental populatons unde ont type-ii censong They deeloped nfeental methods based on maxmum lkelhood estmates (MLE compaed the pefomance wth those based on some othe appoaches such as Bootstap Shafay et al (03 deed the Bayesan nfeence fo the unknown paametes of two exponental populatons unde ont type II censong they 3

2 Sept 04 Vol 5 No 04 Intenatonal Jounal of Engneeng Appled Scences 0-04 EAAS & ARF All ghts eseed wwweaas-ounalog ISSN deeloped wth the use of squaed-eo lneaexponental geneal entopy loss functons The poblem of pedctng the futue falue tmes both pont nteal pedcton based on the obseed ont type-ii censoed data s obtaned; see also Rasoul Balakshnan (00 fo a genealzaton of the esults to pogesse type-ii censong Balakshnan Feng (04 genealzed Balakshnan Rasoul (008 Shafay et al (03 woks by consdeed a ontly type II censoed sample asng fom h ndependent exponental populatons Fnally Ashou Abo- Kasem (04 deed Bayesan non-bayesan estmatos fo two genealzed exponental populatons unde ont type II censoed scheme Succeedng secton deals wth the computatonal pocedue to obtan the MLEs of The asymptotc aance coaance matx appoxmate confdence egon based on the asymptotc nomalty of the maxmum lkelhood estmatos hae been obtaned n secton 3 Whle secton 4 s descbes the aous bootstap confdence nteals All estmatos ae not n nce closed foms theefoe numecal examples ae consdeed to llustate the poposed estmatos n secton 5 Last secton ncludes a bef concluson MAXIMUM LIKELIHOOD ESTIMATORS Suppose that the two populatons ae Webull dstbuton wth densty dstbuton functons as x x f ( x exp x F ( x exp 0 x 0 fo espectely whee ae the shape paametes ae the scale paametes In ths case the lkelhood functon n ( becomes m n L ( w z C z exp( exp( u u q q ( m m ( n n exp ( z exp ( whee w w w u q w Theefoe to obtan the MLE s of we fnd the fst deates of the natual logathm of the lkelhood functon ( wth espect to equatng them to zeo we get the followng fou equatons ln L m z ln u z u ln u ( m m ln 0 ln L n ( z ln q ( z q ln q ( n n ln 0 ln L m z u m m ( 0 ln L n ( ( z q n n 0 w w w whee u q w (3 By solng (3 we get the followng MLEs of fo as m ( m m ( w z ( w ( m m ( w ln ( w z ( w ln ( w n ( n n ( w ( z ( w z ln( w ( n n ( w ln ( w ( z ( w ln ( w ( z ln( w whch can be soled by usng an teate numecal method ( m m w z ( w m ( n n w ( z ( w (4 n Not that fo we obtan MLEs based on a ontly type-ii censoed sample fom two exponental populatons whch ntoduced by Balakshnan Rasoul (008 Remak: Fom the MLEs n (4 t s edent that when m z 0 o o do not exst espectely Hence the MLEs n (4 ae only condtonal MLEs condtoned on m 3 APPROXIMATE CONFIDENCE INTERVALS The appoxmate asymptotc aancecoaance matx fo fo can be obtaned by netng the nfomaton matx wth the elements that ae negate of the expected 3

3 Sept 04 Vol 5 No 04 Intenatonal Jounal of Engneeng Appled Scences 0-04 EAAS & ARF All ghts eseed wwweaas-ounalog ISSN alues of the second ode deates of logathms of the lkelhood functons Cohen (965 concluded that the appoxmate aance coaance matx may be obtaned by eplacng expected alues by the MLEs Now the Fshe nfomaton matx assocated wth s defned as: ln L ln L 0 0 ln L ln L 0 0 I( E ln L ln L 0 0 ln L ln L 0 0 whee ln L m z u ln u ( m m ln ln L n ( z q ln q ( n n ln ln L m z u ( m m ln L n ( z q ( n n ln L m z u ln u ( m m ln ln L n ( z q ln q ( n n ln (5 Usng the asymptotc nomalty of the MLEs we can expess the appoxmate 00( % confdence nteals fo fo Suppose that s the MLE of the paamete ecto ( Denote the Fshe nfomaton matx coespondng to by I lm n ni Then s asymptotcally nomal dstbuted (see Seflng (980 e n ( ~ N (0 In patcula let S ( n whee ( ae the ( elements n the matx ni I s the estmato of I Theefoe asymptotc nomalty confdence nteals of wth confdence leel 00( % ae gen by z S z S whee z ( denotes the uppe ( pecentage pont of the stad nomal dstbuton Also an appoxmate 00( % smultaneous confdence nteal (SCI fo ( usng the Bonfeon method can be obtaned as z S z S (3 4 (3 4 4 BOOTSTRAP INTERVALS In ths secton we pesent seeal bootstap methods to constuct confdence nteals fo fo z Studentzed-t nteal (Boot-t Pecentle nteal (Boot-p (see Efon (98 Efon Tbshan (994 fo detals a Bootstap Pecentle Inteal Pocedue (Boot-p The bootstap pecentle method defnes the lowe uppe bounds of the confdence nteals ust usng the 00 th 00( th quantles of the empcal bootstap dstbuton of espectely In patcula: ( Compute the MLE ( of ( based on two Webull populatons usng ont type II censoed sample ( w z ( Use ( to geneate a bootstap ont type II censoed sample ( w z compute the bootstap estmate of ( say ( based on ths bootstap sample (3 Repeat step B tmes to hae ( ( ( B ( ( ( B (4 Aange ( ( ( B ( ( ( B n ascendng ode obtan [] [] [ ] B [] [] [ B ] (5 A two-sded 00( % pecentle bootstap confdence nteal fo ( say [ ] s gen by [ L U ] L U 33

4 Sept 04 Vol 5 No 04 ([ B ] ([ B( ] L U Intenatonal Jounal of Engneeng Appled Scences 0-04 EAAS & ARF All ghts eseed wwweaas-ounalog ( ( ([ B ] ([ B( ] L U b Studentzed-t Inteal Pocedue (Boot-t The Boot-t confdence nteals estmatos ae computed accodng to the followng steps: ( Same as the steps n (a (3 Compute the t statstc T ( S T ( S whee S S ae the bootstap esons (4 Repeat steps 3 B tmes obtan ( ( ( B ( ( ( B T T T T T T ( ( ( B (5 Aange T T T T T T ( ( ( B n B ascendng ode obtan T T T T T T [] [] [ B ] [] [] [ ] (6 A two-sded 00( % bootstap-t confdence nteal fo ( say [ ] [ ] s gen by tl tu ([ B ] ([ B( ] tl tu T S T S T S T S ([ B ] ([ B( ] In secton 5 we wll hae a smulaton study n ode to ealuate the pefomance of the thee confdence nteals 5 NUMERICAL ILLUSTRATION It clea that thee ae no explct solutons fo obtanng new estmatos Theefoe atfcal data numecal soluton compute facltes ae needed The man obect of ths secton s to llustate numecally most of the new theoetcal esult obtaned n the peous two sectons 5 Illustate Example In ths sub secton we pesent Poschan s Data (963 whch ges falue tmes (n hous of the a-condtonng systems of Boeng 70 et aplanes It s obseed that the falue dstbuton of the a-condtonng system fo each of the planes was well appoxmated by exponental dstbuton whee m = 4 n = 7 we an the aous ont censong schemes on ths dataset wth as 0 30 The data ae pesented n table Table : Falue tmes of a-condtonng systems n two aplanes Plane Plane Table pesents the ontly type-ii censoed data that hae been obtaned fom the two samples n table wth = 0 30 ISSN Table : Jontly type-ii censoed data obseed fom table wth = 30 w z w z w z We then computed the MLEs of the estmates of the stad deatons fo the choces of = 0 30 these ae pesented n table 3 Table 3: The MLEs the estmates of the stad deatons based on ontly type-ii censoed data fom table MLEs SD ( ( 0 ( ( ( ( Table 4 pesents the 95% appoxmate Boot-p Boot-t confdence nteals fo coespondng to case = 0 30 Fom these esults we obsee that Boot-p Boot-t confdence nteals ae satsfactoy compaed to the appoxmate confdence 34

5 Sept 04 Vol 5 No 04 Intenatonal Jounal of Engneeng Appled Scences 0-04 EAAS & ARF All ghts eseed wwweaas-ounalog ISSN Table 4: The 95% appoxmate Bootstap-p Bootstap-t confdence nteals fo = 0 CI fo CI fo CI fo CI fo Appoxmate ( ( ( ( Boot-p ( ( ( ( Boot-t ( ( (0 556 (0 039 = 30 CI fo CI fo CI fo CI fo Appoxmate ( ( ( ( Boot-p ( ( ( ( Boot-t ( ( (0 0 ( Monte Calo smulaton A smulaton study was conducted n ode to ealuate the pefomance of MLEs also all the confdence nteals dscussed n the pecedng sectons We consdeed dffeent sample szes fo the two populatons as m = n = dffeent choces fo = We also chose the paametes ( to be ( Fo these cases we computed the MLEs oot mean squaed eos MSE the 95% appoxmate confdence nteals fo ( the coespondng coeage pobabltes We epeated ths pocess 5000 tmes computed the aeage alues of all the estmates The aeage alue of the MLEs ( ( MSE summazed n table 5 Fom table 6 we obsee that the coeage pobabltes the aeage wdths of 95% CIs ( fo appoxmate confdence nteals ae pesented fo some small modeate lage alues of m n Table 5: The aeage alue of the MLEs ( ( MSE fo small modeate lage alues of m n ( m n MSE MSE MSE MSE ( ( ( ( ( Table 6: Smulated coeage pobabltes (CP the aeage wdths of the 95% confdence nteals of fo some small modeate lage alues of ( nm ( nm (55 CP(% Length CP(% Length CP(% Length CP(% Length

6 Sept 04 Vol 5 No 04 Intenatonal Jounal of Engneeng Appled Scences 0-04 EAAS & ARF All ghts eseed wwweaas-ounalog ISSN (00 (3030 (5050 ( CONCLUSIONS In ths pape the MLEs fo the unknown paametes of two Webull dstbutons has been dscussed based on a ont type- II censoed sample We obtaned the MLEs of the paametes found coespondng Fshe nfomaton matx Also we studed thee appoxmate methods Asymptotc Nomalty Bootstap-t paametc Bootstap pecentle pocedues fo constuctng nteals fo the paametes The MLEs hae then been compaed though a Monte Calo smulaton study a numecal example has also been pesented to llustate all the nfeental esults establshed hee The computatonal esults show that the MLEs hae a modeate bas when the essental sample sze s small een when the sample szes m n ae not small Ths bas also seems to affect the appoxmate confdence nteals based on nomalty as they ae not centeed popely n ths case Howee the bas of the MLEs becomes neglgble when nceases as s edent fom Accodng to the smulaton study when the sample szes of two populatons n m the total numbe of falues ae lage the estmatos bases ae small the confdence nteals hae desable coeage pobabltes Also we obseed that the appoxmate bette than the two bootstap methods often pefom as well as each othe REFERENCES [] Ashou S K Abo-Kasem O E (04 Bayesan non Bayesan estmaton fo two genealzed exponental populatons unde ont type II censoed scheme Pakstan Jounal of Statstcs Opeaton Reseach 0 ( 57-7 [] Balakshnan N Rasoul A (008 Exact lkelhood nfeence fo two exponental populatons unde ont Type-II censong Computatonal Statstcs & Data Analyss [3] Balakshnan N Feng S (04 Exact lkelhood nfeence fo k exponental populatons unde ont type-ii censong Communcatons n Statstcs - Smulaton Computaton Submtted fo publcaton [4] Cohen AC (965 Maxmum Lkelhood Estmaton n the Webull Dstbuton Based on Complete Censoed Samples Technometcs [5] Efon B (98 The Jackknfe the bootstap othe Resamplng Plans SIAM Phladelpha [6] Efon B Tbshan R J (994 An Intoducton to the Bootstap New Yok: Chapman & Hall/CRC Pess [7] Lawless J F (003 Statstcal Models Methods fo Lfe Tme Data nd Edton John Wley New Yok [8] Poschan F (963 Theoetcal explanaton of obseed deceasng falue ate Technometcs [9] Rasoul A Balakshnan N (00 Exact lkelhood nfeence fo two exponental populatons unde ont pogesse type-ii censong Communcatons n Statstcs-Theoy Methods 39 ( 7 9 [0] Seflng R J (980 Appoxmaton Theoems of Mathematcal Statstcs New Yok: Wley [] Shafay A R Balakshnan N Abdel- Aty Y (03 Bayesan nfeence based on a ontly type-ii censoed sample fom two exponental populatons Communcatons n Statstcs - Smulaton Computaton