Comparison of the Bayesian and Maximum Likelihood Estimation for Weibull Distribution

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1 Joural of Mahemacs ad Sascs 6 (2): 1-14, 21 ISSN Scece Publcaos Comarso of he Bayesa ad Maxmum Lkelhood Esmao for Webull Dsrbuo Al Omar Mohammed Ahmed, Hadeel Salm Al-Kuub ad Noor Akma Ibrahm Dearme of Mahemacs, Faculy of Scece, Uversy Pura Malaysa Isue for Mahemacal Research, Uversy Pura Malaysa, 434, UPM, Serdag, Selagor, Malaysa Absrac: Problem saeme: The Webull dsrbuo has bee wdely used esecally he modelg of lfeme eve daa. I rovdes a sascal model whch has a wde varey of alcaos may areas, ad he ma advaage s s ably he coex of lfeme eve, o rovde reasoably accurae falure aalyss ad falure forecass esecally wh exremely small samles. The coveoal maxmum lkelhood mehod s he usual way o esmae he arameers of a dsrbuo. Bayesa aroach has receved much aeo ad coeo wh oher esmao mehods. I hs sudy we exlore ad comare he erformace of he maxmum lkelhood esmae wh he Bayesa esmae for he Webull dsrbuo. Aroach: The maxmum lkelhood esmao, Bayesa usg Jeffrey ror ad he exeso of Jeffrey ror formao for esmag he arameers of Webull dsrbuo of lfe me are reseed. We exlore he erformace of hese esmaors umercally uder varyg codos. Through he smulao sudy comarso are made o he erformace of hese esmaors wh resec o he Mea Square Error (MSE) ad Mea Perceage Error (MPE). Resuls: For all he varyg samle sze, several secfc values of he scale arameer of he Webull dsrbuo ad for he values secfy for he exeso of Jeffrey ror, he esmaors of he maxmum lkelhood mehod resul smaller MSE ad MPE comared o Bayesa majory of he cases. Neverheless all cases for boh mehods he MSE ad MPE decrease as samle sze creases. Cocluso: Based o he resuls of hs smulao sudy he Bayesa aroach used he esmag of Webull arameers s foud o be o sueror comared o he coveoal maxmum lkelhood mehod wh resec o MSE ad MPE values. Key words: Exeso of Jeffrey ror formao, Webull dsrbuo, Bayes mehod INTRODUCTION The Webull dsrbuo s foud o be useful modelg ad aalyzg lfe me daa he felds, medce, bology, egeerg sceces ad ohers. Hossa ad Zmmer (23) have dscussed some comarave esmao of Webull arameers usg comlee ad cesored samles. Besdes, because of s useful alcaos, a array of mehod have bee roosed for esmag arameers of he Webull dsrbuo. The mos-used mehods whch are cosdered o be he radoal mehods are maxmum lkelhood ad he mome esmao (Cohe ad Whe, 1982). The effcecy of he maxmum lkelhood esmao mehod makes oular ad he mome esmao mehod s comuaoally easy ad rovdes exlc esmaors of he arameers. Bayes esmaor of hree arameers of he Webull dsrbuo comarso wh he oseror sadard devao esmaes couerars cum umercal examle were obaed ad gve by Sha ad Sloa (1988). They roosed a mehod based o he rmary formao wh weghed Bayes. Also, usg record sascs from Webull model Bayesa ad o-bayesa aroaches, Solma e al. (26) carred ou he comarso of he esmaes. Kaar ad Peoolu (28) dd comarave sudy for he locao ad scale arameers of he Webull dsrbuo wh a gve shae arameer. I hs sudy we roose Bayes wh Jeffrey ror ad exeso of Jeffrey ror formao for he Webull arameers esmao. Solad e al. (1969) roduce Bayesa aalyss of he Webull Process wh ukow scale ad shae arameers. The res of he sudy s arraged as follows. I Mehods, maxmum lkelhood esmao, Bayes Corresodg Auhor: Al Omar Mohammed Ahmed, Isue for Mahemacal Research, Uversy Pura Malaysa, 434, UPM, Serdag, Selagor, Malaysa 1

2 J. Mah. & Sa., 6 (2): 1-14, 21 esmaor wh he Jeffrey ror ad he ew exeso of Jeffrey ror formao are reseed. I Resuls, smulao sudy s dscussed ad he resuls are reseed ad followed by he cocluso MATERIALS AND METHODS Maxmum lkelhood esmao: We roduce he coce of maxmum lkelhood esmao wh Webull dsrbuo. We have se of radom lfeme 1,, ad vecors of ukow arameers = ( 1,, ) ad = ( 1,, ) he Le (,, ) s he lkelhood fuco: L(;,) f ( ;,) 1 The h elemes of he score vecor wh resec o ad : L(,,) U() L(,,) U() Le (,, ) be he se of radom lfeme from Webull dsrbuo wh arameers ad. The robably desy fuco of Webull dsrbuo s gve: 1 f(;,) ex( ) The lkelhood fuco s: L(;,) f ( ;,) 1 1 ex l L(;, ) U() l( ) 1 1 l( ) Le U() equal o zero, he he maxmum lkelhood esmaor s: 1 ˆ M (1) The shae arameer s ake o be cosa. Bayes esmao: Le 1,, be a radom samle of sze wh dsrbuo fuco F(,,) ad robably desy fuco f(,,). I he Webull case, we assumed ha he robably desy fuco of he lfeme s gve by: 1 f(;,) ex( ) Jeffrey ror formao: We fd Jeffery ror by akg g(q) I( ), where: g( ) k I( ) he g( ) k wh k a cosa We ca foud Bayes esmaor wh Jeffrey ror of aramerc dsrbuo by usg codoal dsrbuo, deed o jo robably desy fuco ad margal robably desy fuco, so he codoal dsrbuo s gve by: H( 1,..., ;,) ( 1,..., ;, ) (,..., ) 1 The jo robably desy fuco f( 1,,,,)) s gve by: The score vecors are: l L(;, ) U( ) H(,..., ;,) f (;,)g( ) 1 1 k 1 ( 1) l L 1 1

3 The margal robably desy fuco of ( 1,, ) s gve by: (,..., ) H(,..., )d 1 1 k ( 1)! ex ( 1) l 1 ( ) 1 The he codoal robably desy fuco of gve he daa ( 1,, ) by: H( 1,..., ;, ) ( 1,..., ;,) (,..., ) 1 ( ) 1 1 ex( 1 ( 1)! By usg squared error loss fuco 2 ( ˆ ) c( ˆ ), he Rsk fuco s: R( ˆ ) E ( ˆ ) Le ( ˆ ) (,..., ;,)d 1 ( ) 2 ˆ ( 1)! cˆ 2c ex( )d ( ) R( ˆ ), he he Bayes esmaor s: ˆ 1 1 ( ) ˆ BJ ex( ) ( 1)! he scale arameer for Jeffrey ror s: ˆ BJ 1 1 J. Mah. & Sa., 6 (2): 1-14, 21 (2) Exeso of Jeffery Pror Iformao: The exeso of Jeffrey ror s akg g()[(i()] c, cr + c he g(q) k k s a cosa. 2c 12 The same way above, we ca fd he Bayes esmaor wh exeso of Jeffrey ror deed o codoal robably desy fuco as followg: The jo robably desy fuco s gve by: H(,...,,, ) f (,,)g( ) 1 1 c k 1 2c ex ( 1) l 1 The margal robably desy fuco of ( 1,, ) s gve by: P(,..., ) H(,... ;,)d 1 1 c k ex ( 1) l (2c 2)! 1 2c1 ( ) 1 The codoal robably desy fuco of gve by: H( 1,..., ;, ) ( 1,..., ;,) (,..., ) The Rsk fuco: R( ˆ ) E ( ˆ ) Le 1 2c c ( ) ex( ) (2c 2)! ( ˆ ) (,..., ;,)d 1 2c1 1 ( ) ex( ) ˆ2 ˆ 1 c 2c d ( ) 2c (2c 2)! R( ˆ ), he he Bayes esmaor s: ˆ 2c1 ( 1 ) 1 ˆ 1 BE ex d 2c1 (2c 2)!

4 J. Mah. & Sa., 6 (2): 1-14, 21 he scale arameer for exeso Jeffrey ror s: ˆ BE 1 2c 2 RESULTS (3) I hs smulao sudy, we have chose = 25, 5, 1 o rerese small moderae ad large samle sze, several values of arameer =.5, 1.5 ad =.8, 1.2, wo values of Jeffery exeso c =.4,.8. The umber of relcao used was R = 1. The smulao rogram was wre by usg Malab rogram. Afer he arameer was esmaed, Mea Square Error (MSE) ad Mea Perceage Error (MPE) were calculaed o comare he mehods of esmao, where: 1 1 ˆ 2 ( ) 1 1 MSE( ) MPE( ) R ˆ R Table 1: MSE esmaed arameers of Webull dsrbuo Sze C P MLE Bayes Exeso Table 2: MPE of esmaed arameers of Webull dsrbuo Sze C P MLE Bayes Exeso The resuls of he smulao sudy are summarzed ad abulaed Table 1-2 for he MSE ad he MPE of he hree esmaors for all samle sze ad, values resecvely. I each row of Table 1-2 we have hree values of esmaors ha s he MLE esmaor, Jeffrey ror ad exeso of Jeffrey ror. The bes mehod s he mehod ha gves he smalles value of (MSE) ad (MPE). DISCUSSION I Table 1, whe we comared aramerc esmaors of Webull dsrbuo Maxmum Lkelhood (MLE) ad Bayes usg Jeffery ror ad exeso of Jeffery ror by Mea Square Error (MSE) we fd he bes esmaor s Maxmum Lkelhood (MLE) by 5% bu s clear from he Table 1 whe c =.4 he Maxmum Lkelhood (MLE) s equal o exeso of Jeffrey ror ad whe c =.8 he maxmum lkelhood (MLE) s equal o Bayes usg Jeffrey ror. I Table 2, whe we comared aramerc esmaors of Webull dsrbuo Maxmum Lkelhood (MLE) ad Bayes usg Jeffery ror ad exeso of Jeffery ror by Mea Perceage Error (MPE) we fd he bes esmaor s Maxmum Lkelhood (MLE) by 5% bu s clear from he Table 2 whe c =.4 he Maxmum Lkelhood (MLE) s equal o exeso of Jeffrey ror ad whe c =.8 he Maxmum Lkelhood (MLE) s equal o Bayes usg Jeffrey ror. CONCLUSION The esmaed arameers of Webull dsrbuo obaed from he maxmum lkelhood esmao s he bes comared o Bayes. Bayes usg exeso of Jeffrey ror gves beer resul ha Bayes usg Jeffrey ror. 13

5 J. Mah. & Sa., 6 (2): 1-14, 21 Whe he umber of samle sze creases he Mea Square Error (MSE) ad mea erceage error (MPE) decrease all cases. REFERENCES Cohe, A.C. ad B. Whe, Modfed maxmum lkelhood ad modfed mome esmaors for he hree-arameer Webull dsrbuo. Commu. Sa. Theory Mehods, 11: DOI: 1.18/ Hossa, A.M. ad W.J. Zmmer, 23. Comarso of esmao mehods for Webull arameers: Comlee ad cesored samles. J. Sa. Comu. Smul,73: DOI:1.18/ Kaar, Y.M. ad B. Peoolu, 28.A comarave sudy for he locao ad scale arameers of he webull dsrbuo wh gve shae arameer. Comu. Geosc., 34: DOI: 1.116/j.cageo Sha, S. K. ad J.A. Sloa, Bayes esmao of he arameers ad relably fuco of he 3- arameer Webull dsrbuo. IEEE Tras. Relab., 37: DOI: 1.116/j.js Solad, R.M., Bayesa aalyss of he Webull Process wh ukow scale ad shae arameers. IEEE Tras. Relab. R., 18: DOI: 1.119/TR Solma, A.A., A. H.Abd Ellah ad K.S. Sula, 26. Comarso of esmaes usg record sascs from Webull model: Bayesa ad o-bayesa aroaches. Comu. Sa. Daa Aal., 51: DOI: 1.116/j.csda