Inference on Stress-Strength Reliability for Weighted Weibull Distribution

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1 Arca Joural of Mathatcs a Statstcs 03, 3(4: 0-6 DOI: 0.593/j.ajs Ifrc o Strss-Strgth Rlablty for Wght Wbull Dstrbuto Hay M. Sal Dpartt of Statstcs, Faculty of Corc, Al-Azhr Uvrsty, Egypt & Qass Uvrsty, Couty Collg Buraah, Sau Araba Abstract Ths papr als wth th stato of th rlablty R p( Y < X = wh X a Y ar pt varabls strbut as Wght Wbull Dstrbuto. Dffrt thos for statg th rlablty ar obta such as Mau Lklhoo Estators, Last Squar Estators a Baysa Estators whch ar bas o o-foratv a foratv pror strbutos. A coparso of th stats obta s prfor as wll. Fally a urcal vstgato s carr out to stuy th proprts of th w stators. Kywors Rlablty, Strss-Strgth, Mau Lklhoo Estator; Baysa Estator, Wght Wbull Dstrbutos. Itroucto It has bco a fact that provg th qualty of proucts ps aly o th o gog a for ths proucts. I ato, stayg th arkt has bco also assocat wth th rlablty of such proucts. For ths rasos, copas to t spcfc asur to copt worl w arkts such as, ts ablty to hgh qualty, copttv prcs, a rlabl goos o t. I ths rgar s, aufacturs us foratv asssst for thr proucts to achv rlablty whch s closly rlat to urablty, accssblty a survval. Falur ca b cos r a rao varabl sc t s so ffcult to tll prcsly wh a spcfc prouct wll fal ur us coto. Ur oral coto, asur of rlablty for vc bcos too ffcult a t rqurs vry log t. So, acclrat lf tstg ucs falurs a th falur ata at th acclrat cotos ar us to stat th rlablty at oral opratg cotos wh th rlablty of a copot s "hgh" a falur ata of th copot ay ot b attaabl urg ts pct lf. Th probl of R = p Y < X arss th statg th rlab lty ( stuato of chacal rlablty of a syst wth strgth X a strss Y, a R s a tr of syst rlablty. Th syst fals f a oly f, at ay t th strss cs th strgth. May authors hav asur ffrt chocs for strss a strgth strbutos. Johso (988[0] suarz * Corrspog author:.haysal@yahoo.co (Hay M. Sal Publsh ol at ajs Copyrght 03 Sctfc & Acac Publshg. All Rghts Rsrv so of ths chocs. Awa & Gharraf (986[5] cosr th cas wh X a Y ar pt a hav Burr Typ XII rao varabls. Thy obta a au lklhoo stator, u varac ubas stator, a a Baysa stator for R. Thy us a urcal procur for valuatg thr Baysa stator. Ala & Rooh (00[3] stu strss a strgth havg potal strbuto but (003[4] hav stu th R = p Y < X a ffrt way. Thy hav st probl of ( up th rqur paratrc valus of th assu strbutos as a rplact for fg P (Y<X for a gv st of strbuto, For th a, thy assu potal strgth a potal strss. Kotz t al. (003[] prst a rvw of all thos a rsults o th strss-strgth ol th last four cas. Ab-Elfattah & Maouh (004[] obta th thr stators of R for Loa strbuto wth kow scal paratr, ths stators ar au lklhoo stator, uforly Mu varac ubas stato (UMVUE a Bays stator. Mokhls (005[4] stu th cas wh X a Y ar pt rao varabls Burr typ III, h obta th au lklhoo stator (MLE, Mu varac ubas stato (UMVUE a Baysa stats of R, h copar btw th stators by usg Mot Carlo sulato. Latr Kha & Isla (007[] alt wth probl for powr fucto strbuto. Th Wbull strbuto s o of th ost wly us strbutos th rlablty a survval stus. Baklz (0[7] cosr th strss-strgth rlablty bas o rcor valus fro th Wbull strbuto. H obta th Bays stator bas o squar rror loss a th au lklhoo stator of th rlablty R. Akbar t al (0 [] focus o th frc for th strss- strgth

2 Arca Joural of Mathatcs a Statstcs 03, 3(4: 0-6 rlablty R wh X a Y ar two pt Wbull strbutos wth th sa shap paratr, but havg th ffrt scal paratrs. Thy obta Th au lklhoo a th approat au lklhoo stators of R. Azzal (985[6] suggst a tho of obtag wght strbutos fro ptly tcally strbut rao varabls. H us th sty fucto of o rao varabl a th strbuto fucto of th othr rao varabl as follows: FX ( = fy ( FY ( (. P( X > X Rctly, Gupta & Kuu, (009[9] prst a w class of wght potal strbutos. Makhoo (0[3] stu th stato of Strss-Strgth rlablty wh X a Y ar two wght potal strbutos wth ffrt paratrs. H obta th MLE of R bas o o spl trato procur, a h carr out Baysa stators of paratrs wth ral ata. Sa a t al. (00[5] propos Th wght Wbull ol bas o a a of Azzal (985[6]. Thy stu basc proprts of th strbuto clug ots, gratg fucto, hazar rat fucto a stato of paratrs. Th proposg ol whch Saa t all (00[5] to valuat th paratrs of Wght Wbull strbuto s cosr hr. Th, th cuulatv strbuto fucto s: + FX ( { p( } { p( ( = + } + (. ;,, λ, > 0 a th probablty sty fucto s: whr + fx (. ;,, λ, > 0 hc, λ ar shap paratrs a s scal paratr. Now, lt λ = as th sa us ar at Sa a t all (00[5]. Ths papr cossts of thr sctos corrspog to sctos, 3, a 4, rspctvly. scto provs a approach for schg Strss-Strss rlablty syst. scto 3, scusss th au lklhoo Estators, Last Squar Estators a Baysa stators of th rlablty R. Fally a u rcal vstgato wll b carr out to stuy th proprts of th w stators. ( = λ p( λ p ( λ. Syst Rlablty ( + ( + ( + + = + Lt X b th strgth of a syst a Y b th strss actg o t. X a Y ar th rao varabls fro Wght Wbull strbutos wth paratrs (, a (, rspctvly. Thrfor, th probablty sty fuctos of X a Y ar, rspctvly: ( + f X = p( p( (. ;,, > 0 ( + f Y y = p( p( (. ;,, y > 0 ar shap paratrs a s scal whr a paratr. Th, th rlablty fucto R s: R = P( y < = FY ( f ( 0 + { p( } { p( ( = + } p( p( = c + ( ( + + c + + = + c a c. (3. 3. Pot Estator of th Rlablty R 3.. Ma u Lklhoo Est ator

3 Hay M. Sal: Ifrc o Strss-Strgth Rlablty for Wght Wbull Dstrbuto Lt X, X,, X Y, Y,, Y a strbutos wth paratrs (, a (, th abov sapls ar rspctvly gv as: b th two pt rao sapls tak fro th Wght Wbull rspctvly, th, th lklhoo a log-lklhoo fucto bas o L = y = = y ( (, = = ( ( ( = l + l + l + l + + l = = ( y + + y+ + = = = = Th rvatvs of (, ;, y ar, rspctvly: l l l l. l wth rspct to a = + 0, ˆˆ = + = y y = + 0, ˆˆ = (4.3 + y = l l y l l y y + = + + y = (5.3 y = = = = Ufortuatly, thr s o a valu for (,, (3.3, (4.3 a (5.3, so wth that, Nwto Raphso tho a tratv approach to solv ths quatos urcal aalyss s cosr. It s a tratv tho for solvg quatos f ( t = 0 whr f (t s assu to hav a cotuous rvatv f (t. Gv a fucto f (t a ts rvatv f (t, a frst guss t 0 s tal. Th, a approato of t ( s f t0 t0 a a approato of t f ( t0 f ( t th s t a so o for ubr of tratos r or f tr+ tr τ whr t s th r r stat. f ( t Now, rplac tˆ whch gvs fro trato wth th paratr t aftr trato procss for statg th paratr t quatos (3.3, (4.3 a (5.3. Sc, Mau Lklhoo Estators ar o ot chag, So t bcos: Rˆ MLE = c + c + (6.3 ˆˆˆˆˆˆˆˆ ( ˆˆˆˆˆˆ + ( + ( + ( + ( + + whr ˆ ˆ + ˆ + c = a c ˆ ˆ = ˆ Last S quar Est ators Suppos that (. s a lar rlato btw th two varabls a tak th logarth of th two ss as follows: y (.3 (.3 (3.3

4 Arca Joural of Mathatcs a Statstcs 03, 3(4: ( l ( l ( l ( l F + = + + +, So, f ths fucto s z wth rspct to (, w up wth th Last Squar Estators. Ths tho s kow as Last Squar tho (or tals, s Flah t al (0[8]. Now, usg th a rak tho, w ca stat F ( fro ths rlato ar th rak falur ts. + ( = + (7.3,,, F, whr Lt y = l(, so, (7.3 bcos straght l quato. Thrfor, th last squar stators of fro zg th followg quato: a co Q ( ( ( (, y l l l + = = Q, wth rspct to a, rspctvly, ar: Q (, ( + = y l ( + + l ( l ( = + ( + ( + ( ( ( = + + = + +, + ( + ( + = Q (, ( ( ( l l l y + = + + = + ( ( ( ( l l + + = =. ( + + = Q, wth rspct to s: Q (, ( ( ( l l l y y y + = + + = + ( + y ( + y y ( + = + = ( y + y + = (8.3 Th frst partal rvatvs of ( Slarly, th frst partal rvatvs of ( (9.3 (0.3 (.3

5 4 Hay M. Sal: Ifrc o Strss-Strgth Rlablty for Wght Wbull Dstrbuto If both ( Q,, ( Q, a ( Q, qual zro, th, th Last Squar Estators of, a wll rsult atly. Howvr, ths procss s too ffcult to b o wthout urcal soluto. So, w aapt th prvous tchqu of, Nwto Raphso tho Bays Estator of Rlablty R Lt X, X,, X Y, Y,, Y a strbutos wth paratrs (, a (, wth paratrs ( λ, θ a ( τ whr ( λ θ, ε, τ b two pt rao sapls, raw fro th Wght Wbull rspctvly, a assocat gaa pror strbutos for ε, ar ploy rspctvly, so that th pror strbuto for a ar: a π ( λ θ, > 0 (.3 π ε τ, > 0 (3.3 (, ar pt. A th jot sty fucto for ata a th paratrs a s: ( ; (, θ τ ata λ + ε = + = = Th, th postror sty fucto for a bas o ata s: ( ata;,, ata = ( 0 0 (4.3 ( ata;, Now, by usg Postror Mo Mtho to obta th Baysa stators of paratrs as follows: Th log postror strbuto for th sapl prors ( λ, θ a ( τ ε, rspctvly s: X, X,, X (5.3 tak fro Wght Wbull strbuto wth Gaa ( ata ( ( λ ( ( ε ( l, l + + l + + l ( l ( l ( ( θ τ l C = = = whr C os ot pt o th paratrs, a th paratrs λ θ, ε Th frst rvatvs of l(, ;, y wth rspct to l (, λ Lt h Th, (, a τ ar f. a ar, rspctvly: (6.3 ata = + θ + (7.3 + = g.l( l, ata + ε = + g λ = + θ + + g.l( = l l ( τ + (8.3 = = = + ε, a h= + l l ( τ +. ( + = g = = = h λ =. g (9.3 h + ε = ( l ( + g g = = (0.3

6 Arca Joural of Mathatcs a Statstcs 03, 3(4: h h = = g g = (.3 whr ( b.l = g =. So, th Hssa atr of th rvatvs I s : l ( (, ata l, ata I = ( ( (.3 l, ata l, ata Th objct of ths work s stat ˆ δ = (, urcally to f ˆ δ ˆ + = δ + I ϕ whr ϕ = [ h h ]. Soluto ca b covrg usg Nwto Raphso algorth to stat, whch ar rplac by ˆ ˆ, gla fro trato. 4. Sulato Stuy Ta bl (. Wh R MLE R BN-IF R B-IF MSE MLE MSE BN-IF MSE B-IF Ta bl (. Wh R MLE R BN-IF R B-IF MSE MLE MSE BN-IF MSE B-IF

7 6 Hay M. Sal: Ifrc o Strss-Strgth Rlablty for Wght Wbull Dstrbuto Th coputr progras MathCAD (00 s us to obta urcal llustrato for th last thortcal rsults. A coparso btw th thr stators, MLE, Bays bas o o-foratv stator a Bays bas o foratv stator s prfor. 00 sapls grat fro Wght Wbull strbutos wth paratrs (, a (, ar us, rspctvly, a ffrt valus of, a wth varous szs ( 5, 0, 0, 30 for both a tabl. Also, varous szs ( 5, 50, 75, 00 for th tabl. Th th as of ths rplcats ar calculat to obta au lklhoo stator of R (R MLE, Bays stator bas o o-foratv (R BN-IF a Bays stator bas o foratv R B-IF. Th a squar rror of R MLE, R BN-IF a R B-IF ar coput as wll. Fro Tabl ( a (, ot that th a squar rror of R B-IF s sallr tha ach of th a squar rror of R MLE a R BN-IF f th sapl szs, ar s all or larg. But, tabl (, wh th sapl szs ar larg th a squar rror of all stators ar lss tha th othr. For apl, wh th sapl szs = 5 a = 0, w f that th a squar rrors of R MLE a R BN-IF a R B-IF ar (0.055, a 0.00 rspctvly, whl, thy ar (0.0, 0.09 a wh th sapl szs ar = 5 a = 75. Ths bhavor appls o th rst of paratrs,,, λ, θ, ε a τ as wll. Also, th thr stators R B-IF, R MLE a R BN-IF ar cras wth crasg th sapl sz a tak th sapl sz s f, a thy ar cras wth crasg th sapl sz a tak th sapl sz s f. O th othr ha, thy ar cras wth crasg both th sapl szs a. 5. Coclusos I ths papr, th stato probl of th rlablty of a syst R strss-strgth ol wh X a Y ar pt varabls strbut as Wght Wbull Dstrbuto s cosr. Mau Lklhoo Estators, Last Squar Estators a Baysa Estators bas o o-foratv a foratv pror strbutos wr scuss va urcal rsults. REFERENCES [] Ab-Elfattah, A. M., Maouh, R. M. (004. Estato of P(Y < X Loa cas. Th 39 th Aual Cofrc o Statstcs, Coputr Scc a Oprato Rsarch, ISSR, Caro Uvrsty, Egypt, part, [] Akbar A., Rza V. a Mohaa Z. (0. Strss-strgth rlablty of Wbull strbuto bas o progrssvly csor sapls3sort 35 (, [3] Ala, S.N. a Rooh (00: O augtg potal strgth-rlablty, IAPQR Trasactos. 7, -7. [4] Ala, S.N. a Rooh (003: O facg a potal strss a strgth havg powr fucto strbuto. Algarh J. Statstc. 3, [5] Awa, A.M., Gharraf, M.K. (986 Estato of P(Y < X th Burr cas: a coparatv stuy. Cou. Statstc. - Sul. Cop., 5, [6] Azzal, A. (985. A class of strbutos whch clu th oral os, Sca. J. Stat., Vol., [7] Baklz A. (0 Ifrc o ( Y X pr < th Two-Paratr Wbull Mol Bas o Rcors. Itratoal Scholarly Rsarch Ntwork. Volu 0. o:0.540/0/636. [8] Flah, A., Elsalloukh, H., M, a E. a Mlaova, M. (0. Th Epotat Ivrt Wbull Dstrbuto. Appl. Math. If. Sc. 6, No., [9] Gupta, R. D. & Kuu, D. (009. A w class of wght potal strbutos, Statstcs, 43(6, [0] Johso, R.A. (988 Strss-strgth Mols for Rlablty. I Habook of Statstcs. E. Krshaah, P.R. a Rao, C.R., Vol. 7, Elsvr, North Holla, [] Kha, M.A. a Isla, H.M. (007: O facg Raylgh strss wth strgth havg powr fucto strbuto. J. Appl. Statst. Sc. 6 pp [] Kotz, S., Lulsk, Y. a Psky, M. (003. Th Strss-Strgth Mol a Its Gralzatos. Nw Jrsy: Worl Sctfc, Ic. [3] Makhoo, I. (0. Estato of R p( Y < X = for Wght Epotal Dstrbuto. J. Appl Sc. ( [4] Mokhls, N. A. (005 Rlablty of a Strss-Strgth Mol wth Burr Typ III Dstrbutos. Co. Statst. - Thory Mth., 34, [5] Saa, S., Muhaa, Q. & Na, S. (00. A class of Wght Wbull Dstrbuto. Pak. j. stat. opr. rs. Vol. VI No. 00 pp Elctroc copy avalabl at: co/abstract=77489.