Bayesian Estimation of the parameters of the Weibull-Weibull Length-Biased mixture distributions using time censored data

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1 Bys Eso of h s of h Wull-Wull gh-bs xu suos usg so S. A. Sh N Bouss I.S.S. Co Uvsy I.N.P.S. Algs Uvsy As I hs h s of h Wull-Wull lgh s xu suos s usg h Gs slg hqu u y I sog sh. wo s of w os oso o-oso. wo ul xls gv fo lluso. Ky wos: Wull suo Wull lgh-s suo Gs slg y I sog Poso No-oso. - Iouo h Wull suo lys o ol lf sg lly sus hs s u o s ly olg oh sg sg flu s. Suos h h lfs of gv sl of s follow h Wull suo h s o' hv h s h of g sl u h o

2 s sl og o s lgh o lf lgh h h sulg suo s o Wull u Wull lgh-s. h o of lgh-s-suos fs vous los ol suh s fly hsoy ss suvvl vs ly o of AIDS u o loo sfuso Gu A 995. h suy of hu fls wllf oulos ws su y Pll o 978. h Wull-lgh s suo WB ws su y Sh Bouss 007. hs ol s o fo olg h w ou ho. I lf sg of hs h h s o h o so of flu h s xu ols ov o o. Suos h o us hs h Wull suo W h oh o hs h Wull-lgh s suo WB h s h ovll suo s h Wull- Wull lgh-s suo whh wll v hough h x y W- WB. Suos h s o vl hvg suh ol h s sy fuo s gv y: f f f 0 - Wh f x > 0 > 0 - s h sy fuo of h Wull suo f x > 0 > 0 Γ -3 h sy fuo of h WB suo s Sh Bouss 007. lg gvs h followg fo of h sy of h x suo: f x x Γ Wh: 0 > 0 0 > -4. hs ol s fuo of fv s wo sh s wo sl s h xg ooo. h W-WB xu suo ovs ff suos og o h vlus of s s h xg ooos. h lly fuo of hs suo s gv y: -5

3 wh: x γ Γ h lly fuos of h Wull h Wull lgh-s ols svly. h oul of hs s s follows: so wo h Gs sl hqu s xl so h s vo o Bys so of h s of h W-WB suo u y I so oso So fou vsgs h s of y I so o-oso h s of ulssf hs v wh w hv sl of h sgl suo fl o f h s ss so o hoogy xu suo us us. I so fv wo ul xls gv fo lluso. - Gs sl h Gs sl s sulo ool o lgoh o g squ of sls fo ol jo sy fuo of wo o o o vls. Suh sls y gl ovg sls fo h gl suos sso wh h jo sy. Gs slg s ll wh h jo suo s o ow xlly whl h full ool suo of h o vl s ow l. h sl qus l sg o fo h o vls h whh s os houghou ouh 997 h o vlu fo h o o vl of s s sl gv vlus of ll h oh vls h o whh h s sl s y Ms Bhg 005. O ll h s hv sl h h oss s s ov. Af suff u- o ov h ff of l sl vlus h s s os o lly su sl fo h jo suo. -- Illusg h Gs sl: Cos ol wh wo o vls o wo s Bys lyss ox l x o h. Suos fuh h h full ool suos l w o f x x

4 x x. h Gs sl os y hoosg so l vlus whh w wll o y 0 0 fo h s. hs y sol vlu of h w ws fo h wo ool suos h followg squ: 0 x x x x.. - x x hs squ of ws osus Mov h us h vlus s o h vlus s -. hs lgoh sly gl o h - s. Fo h h o: 3 x K 3 x K. 3 x K Wh: 3 x K K s h full ool suo of h gv ll h oh s -x - h. -- Ilo of h Gs sl h Gs sl ou usg h BUGS og Bys f Usg Gs Slg whh s fly owlol fo h s h: Wh BUGS w oly o so gl sfo ou h ol of s h sofw wll ous ll h qu uv gl's. 3- Bys so u y I so : Assu h o sl of s s u o s o sv ul fuho h lfs of h fl s follow h W-

5 WB xu suo. Suos h so o owlg xss ou h s of h xu fo hs so w hoos os o h s whh G os fo h sh sl s o fo h xg ooo. x x 3-3--Eso usg oso : Suos h w - h of h s- llo h fl o h oulo fo whh ws w. hs s h w hv f g h s h lfs of h fl s h foo ou h oulo of h fl s s lssf o of h wo oos o foo ou h suvv s x h h lfs g h. ss h u of fl s fo suoulo wh h ol fl s - suvv s fo whh w hv o foo ou h oulo of og; h llhoo fuo h s s h fo gv y Mhll H 958: f f!! lg h sy of h oo h ov foul ullyg y h o gv 3- gvs h followg jo oso: Γ x!! v η l l l l x 3- I o o l h Gs sl w o x h ool suos fo h oso 3- fo h uow h ool suo of h xg ooo Igog ll s 3- whh ul o h xg ooo "" w g: 3-3

6 hs ool os' hv y s fo h w hv o ov oly h s log-ov fuo wh s o h xg ooo. g h so vv of h logh of h ool suo 3-3 wh s o "" s gv y l s s sly gv h h ool suo 3-3 s log-ov h xg ooo h h g usg h v jo slg ou of Gls Wl 99. As sl s wh ol oso s h ool suo of h xg ooo s o us fo hs suo g ly usg s o u go s of h v jo hqu h ool suo of h sl ' ' Igog ll s 3- whh ul o h sl " " w f: l x 3-4 h ool suo of sl os' hv x s suo howv s fo suggs usg G suo sou λ G wh λ s whh o y solvg h sl o ol x λ λ A wh λ A s h s f y h o of h ool suo s gv 3-4 o h o suo sou whh s hs s g h logh of hs o s gv y: λ λ λ l l l A Dffg l λ A wh s o λ u qug h suls o o suls h h ol vlu of h λ s wh solvs h ol quo: 0

7 Fo h ol sl h ool suo gv 3-4 s g suo G [s sos E.G.004] h ool suo of h sl ' ' l x 3-5 By h s wy h fo of hs ool suggs usg g suo sou λ G usg h s ss s fo w f h h ol vlu of h λ s wh solvs h followg ol quo: 0 h g usg h -jo slg ou usg g suo sou. Fo 3-5 wll u o g suo G h - jo hqu s o h ool suo of h sh ' ' ovg ll s 3- whh ul o h sh w g h followg ool suo: l l x 3-6 S h fo of h ool os' s o y s fo w o g suo sou λ G w hoos λ suh h gvs h xu ws fou h h ol vlu of h λ s wh solvs h followg ol quo: 0 l l h g usg h -jo slg ou usg g suo sou. Fo h ool suo 3-6 s lso os.

8 3--5- h ool suo of h sl ' ' ovg ll s 3- whh ul o h sh w g h followg ool suo: Γ l l x 3-7 Fo s fo h ool suo 3-7 os' hv s fo u suggss usg xol suo s sou. I suls h h ool suo of h sh ox s xol o wh λ wh solvs h ol quo: 0 l l l ψ h g usg h -jo slg ou usg xol suo sou. hs hqu s lso us h ol s s h ool suo of h s lso o s h s. o h full ool suos h Gs sl wll o y vly wg h vlus of h s fo h ool suos og wh sg vlus fo h uow s whh glly hos o vy los o h of h suo h ME vfy hs oy. 3-- Eso u o-oso : Suos h sl of s s l o s h lf s s o so " ". Af h s s so so s wh lf s lss h fl h s whos lf s g h h of o of h s suvv. Suos fuh o h h swhg lls of h fl s uow y osqu h fl s ' ssg o o of h wo suoulos.. h osvos ulssf. h llhoo hs s s gv y:

9 f Wh f gv svly y -4-5 s o fuo s gv y: > f f 0 Du o h olo of h llhoo fuo fo h s of ulssf osvos h xu suo s - o foul s ws o ou y Ds l. 977 y gg s ssg ol oug fo h o " " s of y l vls " j " sg whh oo h xu gv s o h of h os suh h: Ohws j j oo j j 0 Wh j. hs s h f gv o s g fo h oo "j" h h l vl s o h vlu o o f o. h l vls h s uow s whh us s fo h ol o g y gvg h s ol suo. hs foulo wll gly slfy h llhoo fuo whh s gv y: j j j j j f f f lg h sy fuo of h oo h ov llhoo ullyg y h o gv 3- gvs h followg jo oso: Γ l x l l x 3-8 Wh: j j j j j j j j j j Fo hs oso w g h followg ool suos: x

10 l x l x Γ l l x h wo fs ools hv s fos B G svly whl w us g sou o sl fo h h h fouh ool suos xol o o sl fo h ls o usg h jo ho. 4- Nul xls: o llus h ov hos wo ul xls gv low: 4-- Poso : A sl of s 00 ws g fo h W-WB xu suo wh s h s w s usg h g WBUGS ovg ws o usg 0000 os h fs 000 os w s h ol ws oh 9000 os. h suls gv h x fo whh w g: l : h Bys ss of h s of h W-WB xu suo wh v s squ o fo oso. P u vlu Es V Bs MSE P

11 Fo l o h ll h ss los o h u vlus us o g h sl ll h s ovs x h sl of h Wull oulo. 4-- No-oso : A sl of s 80 ws g fo h W-WB xu suo wh s h g sl ws so wo os of. &.4. h s w s usg h Bys ho h s y I so usg h WBUGS g. h suls gv h x fo whh h vs ss squ os w lul h followg suls w o: l : h Bys ss of h s of h W-WB xu suo wh v s squ o fo so o-oso. u..4 P P vlu Es V Bs MSE Es V Bs MSE P Fo hs l h suls gv h x w olu: - h v h squ o s wh sg h s o. - h u of os f ovg ss wh sg h s o. 3- Fo. ll h s ovs x h sl of h

12 Wull oulo. 4- Fo.4 h sl s h xg ooo us Ax h sh s ovs. - Poso : o s MC o.5% 97.5% s sl P[] E [] [] h[] h[] P[] sl: [] sl: [] sl: h[] sl: h[] sl: P[] o

13 [] [] o h[] o o h[] o - y I so No-oso : -- h o s. o s MC o.5% 97.5% s sl P[] [] [] h[] h[]

14 P[] sl: [] sl: [] sl: h[] sl: h[] sl: P[] [] o o

15 h[] o h[] h o s.4 o o s MC o.5% 97.5% s sl P[] [] [] h[] h[] P[] sl: 000 [] sl: [] sl: h[] sl: h[] sl:

16 P[] [] o [] o o h[] o h[] o

17 fs: [] w BUGS : Bys Usg Gs Slg 000 : h: [] Ds A. P. N. M. & u D. B Mxu llhoo fo ol v h EM lgoh wh susso Joul of oyl Ssl Soy B [3] Gls W.. & Wl P. 99. Av jo slg fo Gs slg. Al Sss [4] Gu. C. & A H. O O h lly sus of wgh vs Guss ol. Joul of Ssl Plg If [5] Ms G. & Bhg. M A ouo o h DA- Gs sl fo wo- logs P ol yo. Ps-Colog [6] Mhll W. & H Eso of s of x xolly su flu suos fo so lf s. Bo [7] Pl G. P. & o C Wgh suos s-s slg wh los o wllf oulos hu fls. Bos [8] ouh E. C A f vw of Gs sl IBC sss suy. [9] Sh S. A. & Bouss N. A h Wull lgh-s suo: Pos so. Is vll fo owlo : h:s.sjouls.yea007ls07000.f [0] Sh S. A. & Bouss N. A O h lly of h Wull lgh-s suo. F Es Joul of hol Sss 3-7 Push Pulshg Hous. [] sose.g Bys f fo ulv g suo. Sss Coug.vol.0 ssu 4

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