Bayesian estimation of the scale parameter and survival function of weighted weibull. distribution under different loss functions using r software

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1 Bys stmto of th sl rmtr survvl futo of wght wbull strbuto ur ffrt loss futos usg r softwr Kwsr Ftm,.P Ahm Uvrsty of Kshmr, rgr Abstrt: I ths r, w roos to obt th Bys stmtors of th sl rmtr of thr rmtr wght wbull strbuto, bs o o-formtv formtv rors usg Etroy loss futo Qurt loss futo. Th rsk futos of ths stmtors hv b stu. A rl lf ml hs b us to omr th rform of th stmts ur ffrt loss futos. Kywors: Wght Wbull strbuto, Jffry s ror Gmm ror, loss futos. Itrouto Th Wbull strbuto s wll kow ommo strbuto hs b owrful robblty strbuto rlblty lyss, whl wght strbutos r us to just th robblts of th vts s obsrv ror. Th Wbull strbuto lso b us s ltrtv to Gmm Log orml strbuto rlblty grg lf tstg. Gut Kuu 9 roos wght otl strbuto by usg th mtho of Azzl 985. Th roos mol b us s ltrtv to Gmm Wbull strbuto. m t l. roos th wght Wbull mol bs o of Azzl 985. Thy stu bs rorts of th strbuto lug momts, grtg futo, hzr rt futo stmto of rmtrs. Hmy M. lm 3 work o Ifr o trss-trgth lblty for Wght Wbull Dstrbuto..Dy t l. 4 suss th rorts mthos of stmto for th wght wbull strbuto. Frh Khorrm 4 osr th Bys sttstl fr for th wght otl strbuto. A w thr-rmtr strbuto, ll th w wght wbull strbuto NWW hs b trou rtly by ulm Nsru 5. Th w wght wbull strbuto hs th robblty sty futo f f,,,, th umultv strbuto futo f of th strbuto s F Wth o sl rmtr two sh rmtrs. Th orrsog survvl futo s gv by F...3

2 th hzr futo s h.4. Pror Loss Futos Th Bys fr rqurs rort ho of rors for th rmtrs. From th Bys vwot, thr s o lr ut wy from whh o olu tht o ror s bttr th th othr. Nvrthlss, vry oft rors r hos org to o s subjtv kowlg blfs. Howvr, f o hs qut formto bout th rmtrs, t s bttr to hoos formtv rors; othrws, t s rfrbl to us o-formtv rors. I ths r w osr both tys of rors: th Jffry, s ror th turl ojugt ror. Th Jffry s ror roos by Jffry, H.964, s gv s: g,. Th ojugt ror ths s wll b th gmm ror, th robblty sty futo s tk s b b g,, b,. Wth th bov rors, w us two ffrt loss futos for th mol... Qurt Loss Futo QLF Th us of qurt loss futo s ommo, for ml wh usg lst squrs thqus. It s oft mor mthmtlly trtbl th othr loss futos bus of th rorts of vrs, s wll s bg symmtr: rror bov th trgt uss th sm loss s th sm mgtu of rror blow th trgt. If th trgt s t, th qurt loss futo C t For som ostt C, th vlu of th ostt mks o ffr to so, b gor by sttg t qul to. Th qurt loss futo lso b f s

3 l, b. Etroy Loss Futo I my rtl stutos, t rs to b mor rlst to rss th loss trms of th rto. I ths s, Clbr Pul 994 ot out tht usful symmtr loss futo s th troy loss futo: L [ log ] whr >, whos mmum ours t. Also, th loss futo L hs b us Dy t l 987 Dy Lu 99, th orgl form hvg =. Thus, L b wrtt s L [ log ] ;. 3. Mmum lklhoo Estmto of th sl Prmtr α Lt us osr rom sml,,..., of sz from th wght wbull fmly. Th th log-lklhoo futo for th gv sml obsrvto s L 3. Th log-lklhoo futo s l L l l l l 3. As sh rmtrs r ssum to b kow, th ML stmtor of sl rmtr s obt by solvg th l L 3.3 MLE 3

4 4. Bys Estmto of sl rmtr ur th Assumto of Jffry s Pror 4. Bys stmtor of α Combg th ror strbuto. th lklhoo futo, th ostror sty of s rv s follows: k k 4. whr k s t of, k Thrfor from 4. w hv k J 4. whh s th sty krl of gmm strbuto hvg rmtrs. o th ostror strbuto of ~ G, 4.. Estmto ur Qurt loss futo By usg qurt loss futo l, th rsk futo s gv by, 4

5 IN Pr IN 5-5 Ol 5,, Now solvg,, w obt th By s stmtor s Q Estmto ur Etroy Loss Futo By usg troy loss futo log L for som ostt th rsk futo s gv by log, log, l l, l, Now solvg,, w obt th By s stmtor s E Bys stmtor of

6 IN Pr IN 5-5 Ol 6 By usg ostror strbuto futo 4., w fou th survvl futo suh tht J J J J 4.5 Whr hs b f bov 5. Bys Estmto of ur th Assumto of Gmm Pror 5. Bys stmtor of α Combg th ror strbuto. th lklhoo futo, th ostror sty of s rv s follows: b b b k k 5. k b, of k s t whr k Thrfor from 5. w hv

7 IN Pr IN 5-5 Ol 7 G 5. whh s th sty krl of gmm strbuto hvg rmtrs b. o th ostror strbuto of, ~ G 5.. Estmto ur Qurt loss futo By usg qurt loss futo, l th rsk futo s gv by,,, Now solvg,, w obt th By s stmtor s Q Estmto ur Etroy Loss Futo By usg troy loss futo log L for som ostt th rsk futo s gv by log, log,

8 IN Pr IN 5-5 Ol 8 l l, l, Now solvg,, w obt th By s stmtor s E Bys stmtor of By usg ostror strbuto futo 5., w fou th survvl futo suh tht G G G G 5.5 Whr hs b f bov Numrl Eml: I ths sto, w lyz th t st from Bjrkl 96 rrsts th survvl tms, ys of gu gs jt wth ffrt oss of tubrl bll. Th t st ossts of 7 obsrvtos r lst blow:, 5,, 4,4, 3, 3,33, 34, 38, 38, 43,44, 48, 5, 53, 54, 54, 55, 56, 57, 58, 58, 59, 6, 6, 6, 6,6, 6, 63, 65, 65, 67, 68,7, 7, 7, 73, 75, 76,76, 8, 83, 84, 85, 87, 9, 95,96, 98, 99, 9,,, 7, 9, 3, 43, 46, 46, 75, 75,, 33, 58,58, 63, 97, 34, 34, 376.

9 Tbl : Bys Estmts of α ur Jffrys Pror MLE QLF ELF Tbl : Bys Estmts of α ur gmm ror b MLE QLF ELF Blow r gv th tbls omutg rsks of th stmts. Tbl 3: Bys sk of λ ur Jffry s Pror QLF ELF C=.5 C= Tbl 4: Bys sk of λ ur gmm ror b QLF ELF C=.5 C=

10 Coluso: It s obsrv from tbl to 4 th omrso of Bys ostror rsk ur troy loss futo qurt loss futo usg Jffry s ror gmm ror hs b m through whh w olu tht Qurt loss futo s mor rfrbl loss futo mog th rors gmm ror rovs lst ostror rsk wth h loss futo. H w olu tht gmm ror s bst for wght wbull strbuto. frs:. Azzl, A A lss of strbuto whh lus th orml os. v Jourl of ttsts,, Bjrkl T 96: Aqusto of rsst gu gs ft wth ffrt oss of vrult tubrl bll, Am J Hyg 7: Clbr,. Pul, G., 994: A grg roh to Bys stmto for th Wbull strbuto, Mroltros lblty, vol.34, o.5, Frh, Z..M., Khorrm, E., 4: Bys sttstl fr for wght otl strbuto, Commutos ttsts-multo Comutto 43, Gut,. D. & Kuu, D. 9: A w lss of wght otl strbutos, ttsts, 436, Hmy M. lm 3: Ifr o trss-trgth lblty for Wght Wbull Dstrbuto, Amr Jourl of Mthmts ttsts 34: Jffry s, H. 964: A vrt form for th Pror Probblty stmto roblms, Progs of th oyl oty of Loo, rs. A., 86: m,., Muhmm, Q. Nm,.: A lss of Wght Wbull strbuto, Pk. j. stt. or. rs. Vol. VI. No ku Dy, Tujt Dy M.Z.As 4: Wght Wbull strbuto rorts stmto, Jourl of ttstl thory rt.9, -6. ulm Nsru 5: Aothr wght wbull strbuto from Azzl, s fmly, Euro tf Jourl vol. No.9,