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1 Iraoal Joural o Emrgg Tchologs (Scal Issu NCETST-07) 8(): 88-94(07) (Publshd by Rsarch Trd, Wbs: ISSN No. (Pr) : ISSN No. (Ol) : Comarso bw Baysa ad Mamum Lklhood Esmao of Scal Paramr Gralzd Gamma Ty Dsrbuo wh Kow Sha Paramrs udr Dffr Loss Fucos Gaurav Shukla, Vod Kumar, Shva Shukla ad Moh Gr Goswam Darm of Maagm, Ivrs Uvrsy, Barlly Darm of MSCH, CBSH, G.B. Pa Uv. of AG & Tch. Paagar Darm of AS&H, Ivrs Uvrsy, Barlly Darm of CSE, AITS Haldwa ABSTRACT: Wbull dsrbuo ad oal dsrbuos ar wdly usd modlg ad aalyzg of lfm daa. I s of hs, som ohr dsrbuos ar also usful for aalyzg of lfm daa. Th rs ar cosdrs h smao of h scal aramr of hr aramr gralzd gamma y dsrbuo wh kow sha aramrs. Mamum lklhood smao s also dscussd. Bays smaor s obad by usg Jffrys ror udr L loss fuco, Asymmrc Prcauoary Loss fuco ad Squar rror loss fuco. Rlav ffccs of h smaors ar also calculad for h daa s ad s obsrvd ha Bays smaor fars br ha MLE all cass. Kywords: MLE, Bays Esmaor, Loss Fuco, Pror, Posrr I. INTRODUCTION Wbull dsrbuo ad oal dsrbuos ar wdly usd as a lfm dsrbuo aalyzg of lfm daa. Bsds hs, gamma, log ormal, vrs gamma ad h gralzd gamma ar also suggsd as a lfm dsrbuos. Hr w us a hr aramrs gralzd gamma y modl whos robably dsy fuco s gv as f ( ) = k k / k > 0,, k, > 0. () Whr s a scal aramr ad k ad ar sha aramrs. Ths modl cluds oal ( = k =), Wbull = α (k=), gamma (=) ad modl roosd by Sacy ( ) as scal cass. Is uly as a lfm modl has b amd by Shukla ad Kumar (006) by alyg hs modl o varous ss of lfm daa. Pady ad Rao (006) hav obad Bays smaors of scal aramr by usg rcauoary loss fuco whras Shukla ad Kumar (008) hav drvd Bays smaors of scal aramr for dffr rors by usg Ldly aroach for hs dsrbuo. Thy (009) hav also drvd Bays smaors of sha aramrs for dffr rors udr h assumo ha scal aramr s kow. Th mamum lklhood (ML) mhod of smao s qu ffc ad vry oular. I Baysa aroach, a ror dsrbuo for h aramr s cosdrd ad h h osror dsrbuo s obad by codog o h daa ad afr ha h frc s do basd o h osror. Ahmd al.(00) hav cosdrd ML ad Baysa smao of h scal aramr of Wbull dsrbuo wh kow sha ad comard hr rformac udr squard rror loss. Pady.al. (0) hav mad h comarso bw Baysa ad mamum lklhood smao of scal aramr Wbull dsrbuo udr l loss fuco. Sama.al. (05) hav show Baysa smaors of ukow aramrs of a hr aramr gralzd gamma dsrbuo, basd o dffr rors usg dffr loss fucos. Thy also rsd osror mas ad varacs for α udr dffr rors by usg dffr loss fucos. Shukla, Kumar, Shukla ad Goswam 88

2 I hs ar, ML smaor ad Bays smaor of h scal aramr of h hr aramrs gralzd gamma dsrbuo s cosdrd udr l loss fuco, asymmrc rcauoary loss fuco ad squar rror loss fuco wh h assumo ha h sha aramrs ar kow. II. MAXIMUM LIKELIHOOD FUNCTION L (), h Lklhood fuco (L) s gv by L,, 3, b a radom saml of sz from h roosd lf sg modl, whos.d.f. s gv by k k = = = k Logarhm of hs lklhood fuco s k log L = log k log log k + log = = Or, log L = log k log log k + ( k ) log = = (3) Th MLE 0f ca b obad as = k = (4) Whr ad k ar kow. III. BAYESIAN ESTIMATION PRIOR DISTRIBUTION () I Baysa frc, h ror dsrbuo or a ror robably dsrbuo, of calld smly h ror, s a ky ar ad rrss h formao abou a ucra aramr ha s combd wh h robably dsrbuo of w daa o yld h osror dsrbuo, whch ur s usd for fuur frcs ad dcsos volvg. Th drvao of h ror dsrbuo basd o formao ohr ha h curr daa s mossbl or rahr dffcul bcaus h lklhood fuco ad h ror rovd qu dffcul osror forms whch ar mossbl o aalyz aalycally ad ar v vry challgg from h usual umrcal rscv. Morovr, h sasca may b rqurd o mloy as ll subjcv u as ossbl, so ha h cocluso may aar solly basd o samlg modl ad h curr daa. Jffrys roosd a formal rul for obag a o-formav ror. I s rooroal o h squar roo of h drma of h Fshr formao: g ( ) I( ) (5) Whr s k-vcor valud aramr ad I() s h Fshr's formao mar of ordr k k. I arcular, f s a scalar aramr, Jffrys o-formav ror for s g ) I( ). Thus our roblm, w cosdr g ( ) Whr c s a cosa. g ( ) = c ( (6) IV. POSTERIOR DISTRIBUTION Th osror dsrbuo of gv h radom saml wh ad k ar fd s gv as Shukla, Kumar, Shukla ad Goswam 89

3 π ( /,,..., ) = Θ L(,,..., ) g( ) L(,,, ) g( ) d (7) g( ) k π ( /,,..., ) = (8) g( ) d( ) k Whr g () s a ror of. A dscv faur of h Baysa aroach s h roduco of a ror dsy o rrs ror formao abou h ossbl valus of h aramrs of h modl. Thr ar hr dsc Baysa aroachs for slco of ror dsrbuo [Dacos ad Ylvskr (985)]. Th choc of a cov ror dsrbuo whch combs asly wh h lklhood fuco has rcly b smlfd by h cosruco of cojuga famly. Th coc of cojuga famly was roducd frs by Brard (954) ad fully lad by Raffa ad Schlafr (96). Th rsrco o h cojuga famly s o cssary, bu has h advaag ha h osror dsrbuo blogs o h sam famly. If g ( ) = c, h k + π ( /,,..., ) = (9) d( ) k + π k ( /,,..., ) = k+ Whr k (0) V. LOSS FUNCTION Th loss fuco lays a mora rol Baysa frc. A loss fuco s a fuco ha mas a v or valus of o or mor varabls oo a ral umbr uvly rrsg som "cos" assocad wh h v. A omzao roblm sks o mmz a loss fuco. Th smaor havg h las cd loss s usually rfrabl comar o h ohrs. Mos auhors us h sml quadrac (symmrc) loss fuco ad oba h osror ma as h Baysa sma. Howvr, racc, h ral loss fuco s of o symmrc. Vara (975) roducd LINEX (Lar-Eoal) loss fuco, whch s h sml gralzao of squard rror (SE) loss fuco ad ca b usd almos vry suao. VI. LINEX LOSS FUNCTION Th LINEX loss fuco s dfd as follows: L( δ ) = ( aδ ) aδ ˆ Whr = ad a 0 δ () Shukla, Kumar, Shukla ad Goswam 90

4 VII. ESTIMATION UNDER LINEX LOSS To oba h Bays smaor, w mmz h osror cd loss whch s gv as 0 [( aδ ) aδ ] π ( /,..., ) d () Whr ˆ δ = Igrag, w hav a k ( a ˆ ) whr k a ˆ k + ( a ) Whr ρ Solvg ˆ = 0, W oba Bays Esmaor as a ˆ b = a k + VIII. ASYMMETRIC PRECAUTIONARY LOSS FUNCTION (APLF) A vry usful ad sml asymmrc rcauoary loss fuco s ˆ ( ) L ( ˆ, ) = ˆ IX. ESTIMATION UNDER ASYMMETRIC PRECAUTIONARY LOSS To oba h Bays smaor, w mmz h osror cd loss whch s gv as 0 ˆ ( ) ˆ Igrag, w hav π ( /,,..., ) d ˆ k + Whr ˆ( k )( k ) ( k ) Solvg 0 ˆ W oba Bays Esmaor as = (7) (3) (4) (5) (6) Shukla, Kumar, Shukla ad Goswam 9

5 ˆ b = ( k )( k ) whr X. SQUARED ERROR LOSS FUNCTION (SELF) A commoly usd loss fuco s h squard rror loss fuco (SELF) whch s gv as L ( ˆ, ) = ( ˆ ) 9) (8) XI. ESTIMATION UNDER SQUARE ERROR LOSS To oba h Bays smaor, w mmz h osror cd loss whch s gv as ( 0 ˆ ) π ( /,,..., ) d Igrag, w hav ˆ ˆ + ( k )( k ) ( k ) (0) () ρ Solvg ˆ = 0, () XII. RELATIVE EFFICIENCY W oba Bays Esmaor as ˆ b = k whr Th rlav ffccy of h Bays smaor wh rsc o h ML smaor s gv by Rsk ( ml ) RE = (3) Rsk ( ) Bays XIII. NUMERICAL ILLUSTRATION I hs sudy, w hav grad radom samls of sz 00 from gralzd gamma y dsrbuo by usg R- 3.. sofwar (=, k= ad ha = ) ad comard h rformac of ML ad Bays smaor basd o hm. I Tabl:, w rs MLE, Bays smaors of scal aramr by usg dffr loss fucos for Jffry s Pror, hr corrsodg Rsk fucos ad rlav ffccs. Shukla, Kumar, Shukla ad Goswam 9

6 N=00 a= -3 a= - a= - a= a= a= 3 N= N = 00.. Tabl: MLE Bays R MLE(L) R Bays (L) RE MLE Bays R MLE(APLF) R Bays (APLF) RE MLE Bays R MLE(SELF) R Bays (SELF) RE XIV. CONCLUSION Th rs abl lors ML ad Baysa smao of scal aramr gralzd gamma y dsrbuo udr l loss, asymmrc rcauoary loss ad squar rror lossad dmosras ha h Bays smaor rforms br ha h MLE all cass. Rlav ffccy s mmum cas of l loss fuco wh a = - ad mamum wh a =3. REFERENCES []. Ahmd, A. O. M., Al-Kuub, H.S., ad Ibrahm, N. A. (00). Comarso of h Baysa ad mamum lklhood smao for Wbull dsrbuo. Joural of Mahmacs ad Sascs 6,, []. Coh, A. C. (965). Mamum lklhood smao Wbull dsrbuo basd o coml ad o csord samls. Tchomrcs, 7, ( ). [3]. Jffrys, H. (946). A vara form for h ror robably smao roblms. Procdgs of h Royal Socy of Lodo, Srs -A 86, [4]. Lawlss, J. F. (98). Sascal Modls ad Mhods for Lf Tm Daa. Joh Wly & Sos, Nw York. [5]. Ma, N. R., Schaffr, R. E., ad Sgurwala, N.D. (974). Mhods for Sascal Aalyss of Rlably ad Lf Daa. Joh Wly & Sos, Nw York. [6]. Marz, H. F., ad Wallr, R. A. (98). Baysa Rlably Aalyss: Joh Wly & Sos, Nw Yark. [7]. Naqash, S., Ahamad, S.P. ad Ahamad, A. (05). Baysa Aalyss of Gralzd Gamma Dsrbuo, Joural of Sascs Alcaos & Probably, 4, 3, Shukla, Kumar, Shukla ad Goswam 93

7 [8]. Pady, B.N. (997). Tsmaor of scal aramr of oal dsrbuo usg l loss fuco. Commucao Sascs- Thory ad Mhods, 6, 9, 9-0. [9]. Pady, B.N., Mshra, G.C., ad Srvasava, A.K. (004). Ivara vrso of l loss fuco ad s alcaos oal y II csord daa. Th Algarh Joural of Sascs 4, -. [0]. Pady, H. ad Rao, A. K. (006). Baysa smao of scal aramr of gralzd gamma dsrbuo usg rcauoary loss fuco, Ida Joural of Ald Sascs, 0 (006), -7. []. Shukla, G. ad Kumar, V. (008). Bays Esmaors of h Scal Paramr of A Gralzd Gamma Ty Modl, Joural of Rlably ad Sascal Suds, :, []. Solad, R.M. (968). Baysa aalyss of h Wbull rocss wh ukow scal aramr ad s alcao o accac samlg. IEEE Trasacos o Rlably 7, [3]. Vara, H.A. (975). A Baysa aroach o ral sa assssm. I Suds Baysa Ecoomrcs ad Sascs Hoorof Loard J. Savag, S. E. Fbrg ad A. Zllr, Eds. Norh-Hollad, Amsrdam, [4]. Zllr, A. (986). Baysa smao ad rdco usg asymmrc loss fucos. Joural of Amrca Sascal Assocao 8, 394, Shukla, Kumar, Shukla ad Goswam 94