A Note on Bayesian Analysis of Error Function Distribution under Different Loss Functions

Transcription

1 Iteratoal Joural of Probablty ad Statstcs 0, (5: DOI: 0.593/j.jps A Note o Bayesa Aalyss of Error Fucto Dstrbuto uder Dfferet Loss Fuctos Navd Feroze,*, Muhammad Aslam Departmet of Mathematcs ad Statstcs, Allama Iqbal Ope Uversty, Islamabad, Paksta Departmet of Statstcs, Quad--Azam Uversty, Islamabad, Paksta Abstract The Bayesa aalyss of the scale parameter of error fucto dstrbuto has bee cosdered ths paper. A class of formatve ad o-formatve prors has bee assumed to derve the correspodg posteror dstrbutos. The Bayes estmators ad assocated rsks have calculated uder dfferet loss fuctos. The Bayesa credble tervals have bee costructed uder each pror. The performace of the Bayes estmators have bee evaluated ad compared uder a comprehesve smulato study. The purpose s to fd out the combato of a loss fucto ad a pror havg the mmum Bayes rsk ad hece producg the best results. The study depcts that order to estmate the sad parameter use of etropy loss fucto uder formatve prors ca be preferred. Keywords Squared Error Loss Fucto (S, Quadratc Loss Fucto (QLF, Etropy Loss Fucto (, Precautoary Loss Fucto (, Credble Itervals. Itroducto The error fucto dstrbuto s oe of the most wdely used dstrbutos statstcs. Estmatg ts parameter usg Bayesa ferece s etremely useful. Eberly ad Casella[] dscussed the costructo of Bayesa credble tervals usg ao Blackwellzed costructo whch offers smallest stadard error of estmate. Korsgaard et al.[] cosdered the multvarate ormal dstrbuto ad cocetrated o the model where resduals assocated wth labltes of the bary trats have bee assumed to be depedet. A Bayesa aalyss usg Gbbs samplg has bee outled for the model where ths assumpto has bee relaed. Wag[3] proposed a crtero to choose a loss fucto Bayesa aalyss. Lag[4] troduced ad derved De mpster EM -Algorthm for the two-compoet ormal mture models to obta the teratve computato estmates, also used data augmetato ad geeral Gbbs sampler to get the sample from posteror dstrbuto uder cojugate pror. Wag[5]developed the ew method, called matr-varate graphcal models (MGGMs, whch volves smultaeously modelg varable ad sample depedeces wth the matr-ormal dstrbuto. Kha ad Islam[6] evaluated the mateace performace of the system whe tme s cotuous ad cosder half-ormal falure lfetme model as well as repar tme model. * Correspodg author: avdferoz@hotmal.com (Navd Feroze Publshed ole at Copyrght 0 Scetfc & Academc Publshg. All ghts eserved However, error fucto dstrbuto has rarely receved the atteto of the aalysts. But t s always of terest to study the behavour ad propertes of the estmators for the parameters of the ew/deprved dstrbutos. So, the problem of estmato of the parameter of the error fucto dstrbuto uder a Bayesa framework has bee addressed ths paper. A class of prors have bee assumed uder varous loss fuctos to estmate the parameter of the dstrbuto.. Model ad Lkelhood Fucto The probablty desty fucto of error fucto dstrbuto s: f e π ;, > 0 It s a specal case of ormal dstrbuto wth mea zero ad cosderg σ where σ s the stadard devato of the ormal dstrbuto. The lkelhood fucto for a radom sample of sze s L e 3. Bayesa Aalyss uder the Assumpto of Uform Pror The uform pror s assumed to be: p

2 54 Navd Feroze et al.: A Note o Bayesa Aalyss of Error Fucto Dstrbuto uder Dfferet Loss Fuctos The posteror dstrbuto uder the assumpto of uform pror s: p e ; > 0 The Bayes estmators ad rsks uder S, QLF, ad are respectvely preseted the followg. ( S ( QLF S ( ( ( QLF Ε l l 4. Bayesa Aalyss uder the Assumpto of Jeffreys Pror The Jeffreys pror s defed as: pj I here; p j The posteror dstrbuto uder Jeffreys pror s: p( e ; > 0 The Bayes estmators ad rsks uder S, QLF, ad are respectvely gve the followg. ( S ( QLF S ( ( QLF Ε l l

3 Iteratoal Joural of Probablty ad Statstcs 0, (5: ( 5. Bayesa Aalyss uder the Assumpto of Mawell Pror The Mawell pror s assumed to be: p /a e 0 >, a > 0 Where a s hyper-parameter The posteror dstrbuto uder Mawell pror s: 3 a a p( e 3 ; > 0 The Bayes estmators ad rsks uder S, QLF, ad are respectvely show the followg. ( S 4 3 a a ( QLF 3 a S a ( QLF ( 3 3 a ( Ε( l l 3 a a a 6. Bayesa Aalyss uder the Assumpto of aylegh Pror The aylegh pror s assumed to be: p /b e > 0, b > 0 Where b s hyper-parameter. The posteror dstrbuto uder aylegh pror s: b b p e > 0 The Bayes estmators ad rsks uder S, QLF, ad are respectvely preseted the followg. ( S 3 b 3 3 QLF b S b b ;

4 56 Navd Feroze et al.: A Note o Bayesa Aalyss of Error Fucto Dstrbuto uder Dfferet Loss Fuctos ( ( QLF b Ε l l b 3 b b 7. Bayesa Aalyss uder the Assumpto of Ch Pror The ch pror s assumed to be: p k / e > 0, k > 0 Where k s hyper-parameter. The posteror dstrbuto uder ch pror s: k k p( e k ; > 0 The Bayes estmators ad rsks uder S, QLF, ad are respectvely derved the followg. k ( S k k k k S ( QLF ( QLF k ( k k k ( k k k k Ε l l k k k k 8. Bayesa Aalyss uder the Assumpto of Normal Pror Cosderg Locato Parameter to be Zero The ormal pror s assumed to be: k /h p e > 0 ; h > 0 Where h s hyper-parameter. The posteror dstrbuto uder ormal pror s: h h p e > 0 The Bayes estmators ad rsks uder S, QLF, ad are respectvely preseted the followg. ;

5 Iteratoal Joural of Probablty ad Statstcs 0, (5: ( S h ( QLF h S h h ( h Ε l l ( QLF h 9. Bayesa Credble Itervals uder Dfferet Prors The Bayesa credble tervals, as dscussed by Saleem ad aza[7], uder uform, Jeffreys, Mawell, aylegh, ch ad ormal prors are respectvely costructed the followg. U α α,, J α α,, a a M 3α 3 α,, b b α α,, C k α k α,, h h N α α,, 0. Smulato Study Smulato study has bee carred out usg, 00, 00 ad for (,,3, 4,5,6 however, the results are preseted for ;. Whle, the,,3, 4,5,6. dscussos have bee made for all Dfferet values of the hyper-parameters have bee used ad the results for the values gvg better covergece ad the mmum rsks have bee preseted. I order to have more precse estmates, the results have bee replcated suffcetly. The rsks assocated wth Bayes estmates have bee uderled the tables. Smlarly, the dffereces betwee lower ad upper lmts of credble tervals have bee uderled Ta ble. Bayes estmates ad rsks uder uform pror Loss Fuctos S QLF

6 58 Navd Feroze et al.: A Note o Bayesa Aalyss of Error Fucto Dstrbuto uder Dfferet Loss Fuctos Ta ble. Bayes estmates ad rsks uder Jeffreys pror Loss Fuctos S QLF Ta ble 3. Bayes estmates ad rsks uder Mawell pror Loss Fuctos S QLF Ta ble 4. Bayes estmates ad rsks uder aylegh pror Loss Fuctos S QLF Ta ble 5. Bayes estmates ad rsks uder ch pror Loss Fuctos S QLF From the above study t ca be see that by creasg the sample sze the estmated value of the parameter coverges to the true value of the parameter ad magtude of rsk assocated wth each estmate decreases. The creasg values of the parameter mpose a egatve mpact o rate of covergece uder each pror; smlarly, the performace of squared error loss fucto ad precautoary loss fucto s badly affected. However, the performace of quadratc loss fucto ad etropy loss fucto s depedet of choce of parametrc value. I comparso of o-formatve prors the uform pror gves the better estmates as the correspodg rsks are smaller for each loss fucto. Whle case of formatve prors the Mawell pror for QLF ad, Ch pror for S ad aylegh pror for provde the best results. Smlarly, estmates uder etropy loss fucto gve the mmum rsks amog all loss fuctos for each pror. It ca also be assessed that the performace of estmates uder formatve prors s better tha those uder o-formatve prors. Some pror elctato techque may further stregthe ths argumet. Hece, the use of Mawell pror uder etropy loss fucto ca be preferred to estmate the parameter of the error fucto dstrbuto usg a Bayesa framework. I case of terval estmato, the credble tervals uder uform pror are aga arrower tha those uder Jeffreys pror. Usg formatve prors, the tervals uder ch pror are havg the mmum wdth. So for Bayesa terval estmato of the parameter of error fucto dstrbuto, the use of ch pror ca be preferred. Ta ble 6. Bayes estmates ad rsks uder ormal pror Loss Fuctos S QLF Ta ble 7. 95% credble tervals uder uform ad Jeffreys prors Uform Pror Jeffreys Pror lower Lmt Upper Lmt lower Lmt Upper Lmt Ta ble 8. 95% credble tervals uder Mawell ad aylegh prors Mawell Pror aylegh Pror lower Lmt Upper Lmt lower Lmt Upper Lmt

7 Iteratoal Joural of Probablty ad Statstcs 0, (5: Ta ble 9. 95% credble tervals uder ch ad ormal prors Ch P ror Normal P ror lower Lmt Upper Lmt lower Lmt Upper Lmt Coclusos ad ecommedatos The study has bee coducted to estmate the parameter of the error fucto dstrbuto usg four dfferet loss fuctos ad uder s formatve ad o-formatve prors. The study dcates that for Bayesa pot estmato, the use of etropy loss fucto uder Mawell pror ca be preferred. Whle for terval estmato, the ch pror ca affectvely be employed. The study ca be eteded by usg more prors ad loss fuctos. Some cesorg procedures ad fte mture of compoets of error fucto dstrbuto ca also be used. EFEENCES [] L. E Eberly, G. Casella, Estmatg Bayesa credble tervals, Joural of Statstcal Plag ad Iferece, vol., pp.5-3, 003. []. I. Korsgaard, S. M. Luda, D. Sorese, et al., Multvarate Bayesa aalyss of Gaussa, rght cesored Gaussa, ordered categorcal ad bary trats usg Gbbs samplg, Geet. Sel. Evol., vol.35, pp , 003. [3] H. Wag, Bayesa aalyss of Gaussa graphcal models for correlated sample, Preprt submtted to Elsever, 0. [4] L. Lag, O smulato methods for two compoet ormal mture models uder Bayesa approach, Uppsala Uverstet, Project eport, 009. [5] L. Wag, A ote o the choce betwee two loss fuctos Bayesa aalyss, Soochow Joural of Mathematcs, vol.3, o.3, pp , 005. [6] M. A. Kha, H. M. Islam, Bayesa aalyss of system avalablty wth half-ormal lfe tme, Qualty Techology & Quattatve Maagemet, vol.9, o., pp , 0. [7] M. Saleem, A. aza, O Bayesa aalyss of the epoetal survval tme assumg the epoetal cesor tme, Paksta Joural of Scece, vol.63, o., pp , 0..