ESTIMATION OF RELIABILITY IN MULTICOMPONENT STRESS-STRENGTH BASED ON EXPONENTIATED HALF LOGISTIC DISTRIBUTION

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1 Joural of Stattc: Advac Thor ad Applcato Volu 9 Nubr 03 Pag 9-35 ESTIMATION OF RELIABILITY IN MULTICOMPONENT STRESS-STRENGTH BASED ON EXPONENTIATED HALF LOGISTIC DISTRIBUTION G. SRINIVASA RAO ad CH. RAMESH NAIDU Dpartt of Stattc Dlla Uvrt Dlla P. O. Bo: 49 Ethopa -al: gaddrao@ahoo.co Abtract I th papr w codr th tato of ultcopot tr-trgth rlablt. Th t rgardd a alv ol f at lat out of ( < ) trgth cd th tr. Th rlablt of uch a t obtad wh trgth tr varat ar fro potatd half logtc dtrbuto wth dffrt d paratr. Th rlablt tatd b ug th au llhood (ML) thod of tato wh apl draw fro trgth ad tr dtrbuto. Th rlablt tator ar copard aptotcall. Th all apl coparo of th rlablt tat ad through Mot Carlo ulato. Ug ral data t w llutrat th procdur.. Itroducto A rado varabl Y ad to b potatd rado varabl wth ba dtrbuto F (). f Y ha dtrbuto fucto F (). Gupta X [ ] t al. [6] Mudholar ad Srvatava [] Mudholar t al. [3]. Th cla 00 Mathatc Subct Clafcato: 6N05 6F30. Kword ad phra: potatd half logtc dtrbuto rlablt tato tr-trgth ML tato cofdc trval. Rcvd Octobr Sctfc Advac Publhr X

2 0 G. SRINIVASA RAO ad CH. RAMESH NAIDU [ ] of dtrbuto F (). ca b dfd a th potatd cla of X dtrbuto wth ba dtrbuto F (.) whr a potv ral ubr. O lar l Gupta ad Kudu [7] propod a w odl calld gralzd potal dtrbuto or potatd potal dtrbuto. I th papr w tc to th trolog of Mudholar ad Srvatava [] a th potatd half logtc dtrbuto wth ba dtrbuto codr a half logtc dtrbuto. Half logtc odl obtad a th dtrbuto of abolut tadard logtc varat a probablt odl of rct org (Balarha []. Th probablt dt fucto cuulatv dtrbuto fucto ad hazard fucto wth cal paratr ar gv b X f ( ; ) ( ) ; 0 () ( ; ) F ; 0 h( ; ) ; 0. () (3) A th odl fr fro a hap paratr wth IFR atur t would b or uful rlablt tud ad urvval aal. Th odl paralll to half oral dtrbuto alo. If a potv ral ubr th cuulatv dtrbuto fucto of potatd half logtc dtrbuto gv b ( ; ) [ ( ; )] F F ; 0 (4) ad th probablt dt fucto (pdf) of potatd half logtc dtrbuto (EHLD) wth > 0 ad > 0 gv b

3 ESTIMATION OF RELIABILITY IN MULTICOMPONENT f ( ; ) ( ) ( ) ; 0. (5) Th hazard fucto gv b h( ; ) ( ) ( )( ) ( ) [ ] ; 0. (6) Hr ad ar th hap ad cal paratr rpctvl. Th hap of pdf ad hazard fucto of potatd half logtc dtrbuto wth ad ar gv Fgur ad. Th two-paratr EHLD wll b dotd b EHLD ( ). Th purpo of th papr to tud th rlablt a ultcopot tr-trgth bad o X Y bg two dpdt rado varabl whr X ~ EHLD( ) ad Y ~ EHLD( ). A ultcopot t wth copot ha trgth followg -dpdtl ad dtcall dtrbutd rado varabl X X X ad ach copot prc a rado tr Y. Th t rgardd a alv ol f at lat out of ( < ) trgth cd th tr. Lt th rado apl Y X X X b dpdt G ( ) b th cotuou dtrbuto fucto of Y ad F ( ) b th coo cotuou dtrbuto fucto of X X X.

4 G. SRINIVASA RAO ad CH. RAMESH NAIDU Fgur. Probablt dt fucto of potatd half logtc dtrbuto wth ad 5.0. Fgur. Hazard fucto of potatd half logtc dtrbuto wth ad 5.0.

5 ESTIMATION OF RELIABILITY IN MULTICOMPONENT 3 Th rlablt a ultcopot tr-trgth odl dvlopd b Bhattachara ad Joho [3] gv b R P [at lat of th ( X X X ) cd Y ] [ F ( ) ] [ F ( ) ] dg( ) (7) whr X X X ar dtcall dpdtl dtrbutd (d) wth coo dtrbuto fucto F ( ) ad ubctd to th coo rado tr Y. Th probablt (7) calld rlablt a ultcopot tr-trgth odl [ Bhattachara ad Joho [3]]. Th rlablt tato of a gl copot tr-trgth vro hav b codrd b vral author aug varou lft dtrbuto for th tr-trgth rado varat uch a E ad Gr [5] Dowtow [4] Awad ad Gharraf [] McCool [] Nad ad Ach [4] Surl ad Padgtt [] Raqab ad Kudu [9] Kudu ad Gupta [8 9] Raqab t al. [0] Kudu ad Raqab [0]. Th rlablt a ultcopot tr-trgth wa dvlopd b Bhattachara ad Joho [3] Pad ad Udd [5] ad th rfrc thr covr th tud of tatg P ( Y < X ) for a tadard dtrbuto agd to o or both of tr trgth varat. Rctl Rao ad Kata [8] tudd tato of rlablt ultcopot tr-trgth for th log-logtc dtrbuto ad Rao [7] dvlopd a tato of rlablt ultcopot trtrgth bad o gralzd potal dtrbuto. Suppo a t wth dtcal copot fucto f at lat ( ) copot ultaoul oprat. I t opratg vrot th t ubctd to a tr Y whch a rado varabl wth dtrbuto fucto G (.). Th trgth of th copot that th u tr to cau falur ar dpdtl ad dtcall dtrbutd rado varabl wth dtrbuto fucto F ().. Th th t rlablt whch th probablt that th t do

6 G. SRINIVASA RAO ad CH. RAMESH NAIDU 4 ot fal th fucto R gv (7). Th tato of urvval probablt a ultcopot tr-trgth t wh th tr ad trgth varat follow a EHLD ot pad uch attto. Thrfor a attpt ad hr to tud th tato of rlablt ultcopot tr-trgth odl wth rfrc to EHLD. Th rt of th papr orgazd a follow. I Scto w drv th pro for R ad dvlop a procdur for tatg t. Mor pcfcall w obta th au llhood tat (MLE) of th paratr. Th MLE ar plod to obta th aptotc dtrbuto ad cofdc trval for. R Th all apl coparo ad through Mot Carlo ulato ar Scto 3. Fall th cocluo ad cot ar provdd Scto 4.. Mau Llhood Etator of R Lt ( ) ~ EHLD X ad ( ) ~ EHLD Y wth uow hap paratr ad coo cal paratr whr X ad Y ar dpdtl dtrbutd. Th rlablt ultcopot trtrgth for potatd half logtc dtrbuto ug (7) w gt R 0 ( ) d [ ] [ ] t dt t t t whr 0 ( ) [ ] λ λ λ f 0 t z dz z z

7 ESTIMATION OF RELIABILITY IN MULTICOMPONENT 5 λ ( λ ). Aftr th plfcato w gt! R λ ( λ ) ( )! c ad ar tgr. (8) 0 Th probablt (8) calld rlablt a ultcopot trtrgth odl. If ad ar ot ow t car to tat ad to tat R. I th papr w tat ad b ML thod. Th tat ar ubttutd λ to gt a tat of R b ug Equato (8). Th thor of thod of tato plad blow. It wll ow that th thod of au llhood tato (MLE) ha varac proprt. I th drcto w hav propod ML tator for th rlablt of ultcopot tr-trgth odl b codrg th tator of th paratr of tr ad trgth dtrbuto b ML thod of tato EHLD. Sc w ar trtd tatg a probablt whch a fucto of ad ol w wll cof our attto to tat ol ad aug that ow (wthout lo of gralt w ta ). Lt X < < X < < X; Y < Y < Y b two ordrd rado apl of z rpctvl fro trgth ad tr varat ach followg EHLD wth hap paratr ad rpctvl ad coo cal paratr. Th log-llhood fucto of th obrvd apl L( ) l l ( ) l ( ) l ( ) ( ) l l. ( ) ( ) (9)

8 G. SRINIVASA RAO ad CH. RAMESH NAIDU 6 Th MLE ad of ad rpctvl ar th oluto of olar quato 0 l 0 L (0) 0 l 0 L () ( ) L 0 0. () Fro (0) () ad () w obta l (3) l (4) ad ca b obtad a th oluto of olar quato ( ) 0 g ( ) l

9 ESTIMATION OF RELIABILITY IN MULTICOMPONENT 7 l 0. (5) Thrfor pl tratv oluto of olar quato ( ). 0 g Oc w obta ad ca b obtad fro (3) ad (4) rpctvl. Thrfor th MLE of R bco ( ) ( ). whr!! 0 λ λ λ R (6) To obta th aptotc cofdc trval for R w procd a follow: Th aptotc varac of th MLE gv b ( ) ( ). ad L E V L E V (7) Th aptotc varac (AV) of a tat of R whch a fucto of two dpdt tattc (a) gv b Rao [6]. ( ) ( ) ( ). R V R V R AV (8) Thu fro Equato (8) th aptotc varac of R ca b obtad. To avod th dffcult of drvato of R w obta th drvatv of R for ( ) ( 3) ad ( 4) paratl th ar gv b

10 8 G. SRINIVASA RAO ad CH. RAMESH NAIDU R 3 3 λ R 3 3 λ ad. ( 3 ) ( 3 λ λ ) R 4 ( 7 λ ) ( 3 λ )( 4 λ ) [ ] ad R 4 ( 7 λ ). [( 3 λ )( 4 λ )] 9 Thu ( λ AV R 3 ). 4 ( 3 λ ) [ ] 44 ( ) ( λ λ 7 AV R 4 ). 4 ( 3 λ )( 4 λ ) [ ] A R R AV ( R ) d N ( 0 ) ad th aptotc 95% cofdc trval for R gv b R.96 AV ( R ). Th aptotc 95% cofdc trval for R 3 gv b 3 λ R 3.96 [( 3 λ )] whr λ Th aptotc 95% cofdc trval for R 4 gv b λ( λ 7) R 4.96 ( 3 λ )( 4 λ ) [ ]. whr λ 3. Sulato Stud. I th cto w prt o rult bad o Mot Carlo ulato to copar th prforac of th R ug for dffrt apl z rado apl of z 0(5)35 ach fro tr populato trgth populato ar gratd for ( ) (3.0.0)

11 ESTIMATION OF RELIABILITY IN MULTICOMPONENT 9 (.5.0) (.0.0) (.5.0) (.0.0) (.5.0) (.5.5) ad (.5 3.0) o l of Bhattachara ad Joho [3]. Th ML tator of ad ar th ubttutd ν to gt th rlablt a ultcopot tr-trgth for ( ) ( 3) ( 4). Th avrag ba ad avrag a quar rror (MSE) of th rlablt tat ovr th 3000 rplcato ar gv Tabl ad. Avrag cofdc lgth ad covrag probablt of th ulatd 95% cofdc trval of R ar gv Tabl 3 ad 4. Th tru valu of rlablt ultcopot tr-trgth wth th gv cobato of ( ) for ( ) ( 3) ar ad for ( ) ( 4) ar Thu th tru valu of rlablt ultcopot tr- trgth cra a cra for a fd whra rlablt ultcopot tr-trgth dcra a cra for a fd both th ca of ( ). Thrfor th tru valu of rlablt cra a ν dcra ad vc vra. Th avrag ba ad avrag MSE ar dcra a apl z cra for both ( ). It vrf th cotc proprt of th MLE of R. Alo th ba gatv both tuato of ( ). Whra aog th paratr th abolut ba ad MSE ar cra a cra for a fd both th ca of ( ) ad th abolut ba ad MSE ar dcra a cra for a fd both th ca of ( ). Th avrag lgth of th cofdc trval alo dcra a th apl z cra. Th covrag probablt ot vr clo to th oal valu 0.95 all ca ot of th cobato. Ovrall th prforac of th cofdc trval qut good for all cobato of paratr. Whra aog th paratr w obrvd th a phoo for avrag lgth ad avrag covrag probablt that w obrvd ca of avrag ba ad MSE.

12 Tabl. Avrag ba of th ulatd tat of R ( ) ( ) ( ) (3.5.5) (3.0.5) (.5.5) (.0.5) (.5.5) (.5.0) (.5.5) (.5 3.0) (.5 3.5) (0 0) (5 5) ( 3) (0 0) (5 5) (30 30) (35 35) (0 0) (5 5) ( 4) (0 0) (5 5) (30 30) (35 35)

13 Tabl. Avrag MSE of th ulatd tat of R ( ) ( ) ( ) (3.5.5) (3.0.5) (.5.5) (.0.5) (.5.5) (.5.0) (.5.5) (.5 3.0) (.5 3.5) (0 0) (5 5) ( 3) (0 0) (5 5) (30 30) (35 35) (0 0) (5 5) ( 4) (0 0) (5 5) (30 30) (35 35)

14 Tabl 3. Avrag cofdc lgth of th ulatd 95% cofdc trval of R ( ) ( ) ( ) (3.5.5) (3.0.5) (.5.5) (.0.5) (.5.5) (.5.0) (.5.5) (.5 3.0) (.5 3.5) (0 0) (5 5) ( 3) (0 0) (5 5) (30 30) (35 35) (0 0) (5 5) ( 4) (0 0) (5 5) (30 30) (35 35)

15 Tabl 4. Avrag covrag probablt of th ulatd 95% cofdc trval of R ( ) ( ) ( ) (3.5.5) (3.0.5) (.5.5) (.0.5) (.5.5) (.5.0) (.5.5) (.5 3.0) (.5 3.5) (0 0) (5 5) ( 3) (0 0) (5 5) (30 30) (35 35) (0 0) (5 5) ( 4) (0 0) (5 5) (30 30) (35 35)

16 34 G. SRINIVASA RAO ad CH. RAMESH NAIDU 4. Cocluo I th papr w hav tudd th ultcopot tr-trgth rlablt for potatd half logtc dtrbuto wh both of tr trgth varat follow th a populato. Alo w hav tatd aptotc cofdc trval for ultcopot tr-trgth rlablt b ug ML tato. Th ulato rult dcat that th avrag ba ad avrag MSE ar dcra a apl z cra both ca of ( ). Aog th paratr th abolut ba ad MSE ar cra (dcra) a cra ( cra) both th ca of ( ). Th lgth of th cofdc trval alo dcra a th apl z cra ad covrag probablt ot vr clo to th oal valu all t of paratr codrd hr. Ug ral data w llutrat th tato proc. Rfrc [] M. Awad ad K. Gharraf Etato of p ( Y < X ) Burr ca: A coparatv tud Cou. Statt.-Sul. & Cop. 5 (986) [] N. Balarha Ordr tattc fro th half logtc dtrbuto Joural of Stattcal Coputato ad Sulato 0 (985) [3] G. K. Bhattachara ad R. A. Joho Etato of rlablt ultcopot tr-trgth odl JASA 69 (974) [4] F. Dowtow Th tato of p ( X > Y ) th oral ca Tchotrc 5 (973) [5] P. E ad S. Gr Etato of th probablt that Y < X JASA 66 (97) [6] R. C. Gupta P. L. Gupta ad R. D. Gupta Modlg falur t data b Lha altratv Cou. Statt.-Thor. Mth. 7(4) (998) [7] R. D. Gupta ad D. Kudu Gralzd potal dtrbuto Autrala ad Nw Zalad Joural of Stattc 4 (999) [8] D. Kudu ad R. D. Gupta Etato of p ( Y < X ) for th gralzd potal dtrbuto Mtra 6(3) (005) [9] D. Kudu ad R. D. Gupta Etato of p ( Y < X ) for Wbull dtrbuto IEEE Traacto o Rlablt 55() (006)

17 ESTIMATION OF RELIABILITY IN MULTICOMPONENT 35 [0] D. Kudu ad M. Z. Raqab Etato of R p( Y < X ) for thr-paratr Wbull dtrbuto Stattc ad Probablt Lttr 79 (009) [] J. I. McCool Ifrc o p ( Y < X ) th Wbull ca Cou. Statt.-Sul. & Cop. 0 (99) [] G. S. Mudholar ad D. K. Srvatava Epotatd Wbull fal for aalzg bathtub falur data IEEE Traacto o Rlablt 4 (993) [3] G. S. Mudholar D. K. Srvatava ad M. Frr Th potatd Wbull fal: A raal of th bu-otor falur data Tchotrc 37 (995) [4] S. B. Nad ad A. B. Ach A ot o tato of p ( X > Y ) for o dtrbuto uful lf- ttg IAPQR Traacto 9() (994) [5] M. Pad ad Md. Borha Udd Etato of rlablt ultcopot tr-trgth odl followg Burr dtrbuto Procdg of th Frt Aa Cogr o Qualt ad Rlablt Nw Dlh Ida (985) [6] C. R. Rao Lar Stattcal Ifrc ad t Applcato Wl Eatr Ltd Ida 973. [7] G. S. Rao Etato of rlablt ultcopot tr-trgth odl bad o gralzd potal dtrbuto Coloba Joural of Stattc 35() (0) [8] G. S. Rao ad R. R. L. Kata Etato of rlablt ultcopot trtrgth odl: Log-logtc dtrbuto Elctroc Joural of Appld Stattcal Aal 3() (00) [9] M. Z. Raqab ad D. Kudu Coparo of dffrt tator of p ( Y < X ) for a cald Burr tp X dtrbuto Cou. Statt.-Sul. & Cop. 34() (005) [0] M. Z. Raqab M. T. Mad ad D. Kudu Etato of p ( Y < X ) for th thrparatr gralzd potal dtrbuto Cou. Statt.-Thor. Mth. 37(8) (008) [] J. G. Surl ad W. J. Padgtt Ifrc for p ( Y < X ) th Burr tp X odl Joural of Appld Stattcal Scc 7 (998) g