Estimation of Reliability in Multicomponent Stress-Strength Based on Generalized Rayleigh Distribution

Transcription

1 Journal of Modern Appled Statstcal Methods Volue 13 Issue 1 Artcle Estaton of Relablty n Multcoponent Stress-Strength Based on Generalzed Raylegh Dstrbuton Gadde Srnvasa Rao Unversty of Dodoa, Dodoa, Tanzana, gaddesrao@yahoo.co Follow ths and addtonal wors at: Part of the Appled Statstcs Coons, Socal and Behavoral Scences Coons, and the Statstcal Theory Coons Recoended Ctaton Rao, Gadde Srnvasa (014) "Estaton of Relablty n Multcoponent Stress-Strength Based on Generalzed Raylegh Dstrbuton," Journal of Modern Appled Statstcal Methods: Vol. 13 : Iss. 1, Artcle 4. DOI: 10.37/jas/ Avalable at: Ths Regular Artcle s brought to you for free and open access by the Open Access Journals at DgtalCoons@WayneState. It has been accepted for ncluson n Journal of Modern Appled Statstcal Methods by an authorzed edtor of DgtalCoons@WayneState.

2 Journal of Modern Appled Statstcal Methods May 014, Vol. 13, No. 1, Copyrght 014 JMASM, Inc. ISSN Estaton of Relablty n Multcoponent Stress-Strength Based on Generalzed Raylegh Dstrbuton Gadde Srnvasa Rao Unversty of Dodoa Dodoa, Tanzana A ultcoponent syste of coponents havng strengths followng - ndependently and dentcally dstrbuted rando varables x 1, x,, x and each coponent experencng a rando stress Y s consdered. The syste s regarded as alve only f at least s out of (s < ) strengths exceed the stress. The relablty of such a syste s obtaned when strength and stress varates are gven by a generalzed Raylegh dstrbuton wth dfferent shape paraeters. Relablty s estated usng the axu lelhood (ML) ethod of estaton n saples drawn fro strength and stress dstrbutons; the relablty estators are copared asyptotcally. Monte-Carlo sulaton s used to copare relablty estates for the sall saples and real data sets llustrate the procedure. Keywords: Generalzed Raylegh dstrbuton, relablty estaton, stress-strength, ML estaton, confdence ntervals Introducton Surles and Padgett (1998, 001) ntroduced the two-paraeter Burr Type X dstrbuton and naed t the generalzed Raylegh dstrbuton. Note that the twoparaeter generalzed Raylegh dstrbuton s a partcular eber of the generalzed Webull dstrbuton, orgnally proposed by Mudholar and Srvastava (1993). The two-paraeter Burr Type X dstrbuton s referred to as the generalzed Raylegh dstrbuton (GRD). For > 0 and > 0, the twoparaeter GRD has the densty functon; 1 ( x) ( x) f ( x;, ) xe 1 e for x 0 (1) Dr. G. Srnvasa Rao s a Professor of Statstcs. Eal at: gaddesrao@yahoo.co. 367

3 ESTIMATION OF RELIABILITY IN STRESS-STRENGTH and the dstrbuton functon s gven by ( x) F( x;, ) 1 e for x 0. () Here and are the shape and scale paraeters respectvely. The GRD has been studed extensvely by Kundu and Raqab (005) and Raqab and Kundu (005). The two-paraeter GRD s denoted by GR( )., Surles and Padgett (001) showed that the two-paraeter GR dstrbuton can be used effectvely n odelng strength as well as general lfete data. Ths artcle studes relablty n a ultcoponent stress-strength based on X, Y, two ndependent rando varables, where X and Y fallow generalzed Raylegh dstrbutons wth shape paraeters and respectvely and wth coon scale paraeter. Let the rando saples y, x1, x,... x be ndependent, G(y) be the contnuous dstrbuton functon of Y and F(x) be the coon contnuous dstrbuton functon of x1, x,... x. The relablty n a ultcoponent stressstrength odel developed by Bhattacharyya and Johnson (1974) s R =P at least s of the( x, x,... x )exceed Y s, 1 s [1 G( y)] [ G( y)] df( y), (3) where x1, x,... x are ndependently and dentcally dstrbuted (d) wth coon dstrbuton functon F(x), ths syste s subjected to coon rando stress Y. The probablty n (3) s called relablty n a ultcoponent stress-strength odel (Bhattacharyya & Johnson, 1974). The survval probablty of sngle coponent stress-strength versons have been consdered by several authors assung varous lfete dstrbutons for the stress-strength rando varates (Ens & Gesser, 1971; Downtown, 1973; Awad & Gharraf, 1986; McCool, 1991; Nand & Ach, 1994; Surles & Padgett, 1998; Raqab & Kundu, 005; Kundu & Gupta, 005, 006; Raqab, et al., 008; Kundu & Raqab, 009). Relablty n a ultcoponent stress-strength was developed by Bhattacharyya and Johnson (1974) and Pandey and Borhan Uddn (1985) and the references theren cover the study of estatng P( Y X ) n any standard dstrbutons assgned to one or both of stress, strength varates. Recently Srnvasa Rao and Kanta (010) 368

4 GADDE SRINIVASA RAO studed estaton of relablty n ultcoponent stress-strength for log-logstc dstrbuton. Suppose a syste, wth dentcal coponents, functons f s(1 s ) or ore of the coponents sultaneously operate. In ts operatng envronent, the syste s subjected to a stress Y whch s a rando varable wth dstrbuton functon G(.). The strengths of the coponents, that s the nu stresses to cause falure, are ndependent and dentcally dstrbuted rando varables wth dstrbuton functon F(.). Then the syste relablty, whch s the probablty that the syste does not fal, s the functon R gven n (3). The estaton of survval probablty n a ultcoponent stress-strength syste when the stress, strength varates are followng Raylegh dstrbuton s not pad uch attenton. Therefore, ths artcle studes the estaton of relablty n ultcoponent stress-strength odel wth reference to Raylegh dstrbuton. Maxu Lelhood Estator of R Let X ~ GR(, ) and Y ~ GR(, ) wth unnown shape paraeters, and coon scale paraeter, where X and Y are ndependently dstrbuted. The relablty n ultcoponent stress- strength for generalzed Raylegh dstrbuton usng (3) results n: 1 ( y) ( y) ( y) ( y) Rs, 1 1 e 1 e ye 1 e dy s 0 s 1 1 [1 ] [ ] t t t dt where t 1 e y s z [1 z] dz f z t, 0 s (, 1). After splfcaton ths reduces to ( )! Rs, ( j) ( )! s j0 1 (4) 369

5 ESTIMATION OF RELIABILITY IN STRESS-STRENGTH because and are ntegers. The probablty n (4) s called relablty n a ultcoponent stress-strength odel. If and are not nown, t s necessary to estate and to estate R. In ths artcle and are estated usng the ML ethod. The estates are substtuted n to obtan an estate of R usng equaton (4). It s nown that the ethod of Maxu Lelhood Estaton (MLE) has nvarance property. In ths drecton, ths artcle proposes the ML estator for the relablty of a ultcoponent stress-strength odel by consderng the estators of the paraeters of stress, strength dstrbutons by ML ethod of estaton n a generalzed Raylegh dstrbuton. x x... x ; y y... y be two ordered rando saples of Let 1 n 1 sze n, respectvely on strength, stress varates each followng GRD wth shape paraeters and, coon scale paraeter. The log-lelhood functon of the observed saple s L(,, ) ( n )ln nln ln n n ( n )ln x y j ln x ln y j 1 j1 1 j1 n ( x ) ( y ) e e j ( 1) ln 1 ( 1) ln 1 1 j1 (5) The MLEs of, and, for exaple, ˆ, ˆ and ˆ, respectvely can be obtaned as the teratve soluton of 1 n L n 0 ln 1 ( x ) e 0 (6) j1 L ( y ) 0 ln 1 j e 0 (7) 370

6 GADDE SRINIVASA RAO L ( n) 0 ( 1) j1 ( y j ) j y j ye ( 1) 0 1 e n n ( x ) xe x yj ( x ) 1 j1 1 1 e (8) and then fro (6), (7) and (8) ˆ ˆ n 1 j1 n ln 1 ln 1 ˆ ( x ) e ˆ ( y j ) e (9) (10) where ˆ can be obtaned as the soluton of non-lnear equaton g( ) 0 ( ) ( ) j y n x xe ye j n ( x ) ( y j ) n 1 1 e j1 1 e n ( x) ( y) e e ln 1 ln n x y j 0 ( x) ( yj) 11 e j11 e (11) Therefore, ˆ s sple teratve soluton of non-lnear equaton g( ) 0. Once ˆ s nown, ˆ and ˆ can be obtaned fro (9) and (10) respectvely. Therefore, the MLE of R becoes! ˆ R ˆ ˆ ˆ ( j) where. ( )! ˆ s j0 1 (1) 371

7 ESTIMATION OF RELIABILITY IN STRESS-STRENGTH The asyptotc confdence nterval for R, s calculated as: Frst, the asyptotc varance of the MLE s gven by 1 1 L V( ˆ) and V( ˆ L E ) E n (13) The asyptotc varance (AV) of an estate of R whch a functon of two ndependent statstcs, for exaple,, s gven by Rao (1973). ˆ Rs, ˆ Rs, AV(R ˆ )=V( ) V( ) (14) Fro the asyptotc optu propertes of MLEs (Kendall & Stuart, 1979) and of lnear unbased estators (Davd, 1981), t s nown that MLEs are asyptotcally equally effcent havng the Craer-Rao lower bound as ther asyptotc varance as gven n (13). Thus, fro Equaton (14), the asyptotc varance of Rˆs, can be obtaned. To avod the dffculty of dervaton of R, the dervatves of R are obtaned for (s,)=(1,3) and (,4) separately, they are gven by R1,3 3 R1,3 3ˆ and. (3 ˆ ) (3 ˆ ) R,4 1(7 ˆ ) R,4 1 ˆ (7 ˆ ) and. (3 ˆ )(4 ˆ ) (3 ˆ )(4 ˆ ) Thus, ˆ 9 ˆ 1 1 AV( R1,3 )= 4 (3 ˆ ) n and AV( Rˆ )= ˆ ˆ. (3 ˆ )(4 ˆ ) n 144 (7 ) 1 1,4 4 As n,, Rˆ s, Rs, d N(0,1), AV( Rˆ ) 37

8 GADDE SRINIVASA RAO and the asyptotc 100(1 )% confdence nterval for R s gven by Rˆ Z AV( Rˆ ) s, (1 ) s,. The asyptotc 100(1 )% confdence nterval for R 1,3 s gven by Rˆ Z 1,3 (1 /) 3 ˆ 1 1 (3 ˆ ) n, where ˆ ˆ / ˆ. The asyptotc 100(1 )% confdence nterval for R,4 s gven by Rˆ Z 1 ˆ (7 ˆ ) 1 1, where ˆ ˆ / ˆ. (3 ˆ )(4 ˆ ) n,4 (1 /) Z where (1 /) s the (1 / ) th percentle of the standard noral dstrbuton. Sulaton Study and Data Analyss Sulaton Study Results based on Monte Carlo sulatons to copare the perforance of the R usng dfferent saple szes are presented. 3,000 rando saple of sze 10(5)35 each fro stress populaton, strength populaton were generated for (, ) = (3.0,1.0), (.5,1.0), (.0,1.0), (1.5,1.0), (1.0,1.0), (1.5,.0),(1.5,.5) and (1.5,3.0) on lnes of Bhattacharyya and Johnson (1974). The ML estators of and were then substtuted n to obtan the relablty n a ultcoponent stressstrength for (s, ) = (1, 3), (, 4). The average bas and average ean square error (MSE) of the relablty estates over the 3,000 replcatons are gven n Tables 1 and. Average confdence length and coverage probablty of the sulated 95% confdence ntervals of R are gven n Tables 3 and 4. The true value of relablty n ultcoponent stress- strength wth the gven cobnatons of (, ) for (s, ) = (1, 3) are 0.563, 0.600, 0.643, 0.69, 0.750, 0.800, 0.833, 0.857, and for (s, ) = (, 4) are 0.355, 0.400, 0.454, 0.519, 0.600, 0.674, 0.75, 373

9 ESTIMATION OF RELIABILITY IN STRESS-STRENGTH 0.76, Thus the true value of relablty n ultcoponent stress- strength ncreases as ncreases for a fxed whereas relablty n ultcoponent stress- strength decreases as ncreases for a fxed n both the cases of (s, ). Therefore, the true value of relablty s ncreases as decreases and vce-versa. The average bas and average MSE are decreases as saple sze ncreases for both (s, ). It verfes the consstency property of the MLE of R. Also the bas s negatve n both stuatons of (s, ). Whereas, aong the paraeters the absolute bas and MSE are ncreases as ncreases for a fxed n both the cases of (s, ) and the absolute bas and MSE are decreases as ncreases for a fxed n both the cases of (s, ). The average length of the confdence nterval also decreases as the saple sze ncreases. The coverage probablty s close to the nonal value n all cases but slghtly less than 0.95 n ost of the cobnatons. Overall, the perforance of the confdence nterval s good for all cobnatons of paraeters. Whereas, aong the paraeters observed, the sae phenoenon for average length and average coverage probablty were observed n the case of average bas and MSE. Table 1. Average bas of the sulated estates of R (, ) (s,) (n,) (3.5,1.5) (3.0,1.5) (.5,1.5) (.0,1.5) (1.5,1.5) (1.5,.0) (1.5,.5) (1.5,3.0) (1.5,3.5) (10,10) (15,15) (1,3) (0,0) (5,5) (30,30) (35,35) (10,10) (15,15) (,4) (0,0) (5,5) (30,30) (35,35)

10 GADDE SRINIVASA RAO Table. Average MSE of the sulated estates of R (, ) (s,) (n,) (3.5,1.5) (3.0,1.5) (.5,1.5) (.0,1.5) (1.5,1.5) (1.5,.0) (1.5,.5) (1.5,3.0) (1.5,3.5) (10,10) (15,15) (1,3) (0,0) (5,5) (30,30) (35,35) (10,10) (15,15) (,4) (0,0) (5,5) (30,30) (35,35) Table 3. Average confdence length of the sulated 95% confdence ntervals of R (, ) (s,) (n,) (3.5,1.5) (3.0,1.5) (.5,1.5) (.0,1.5) (1.5,1.5) (1.5,.0) (1.5,.5) (1.5,3.0) (1.5,3.5) (10,10) (15,15) (1,3) (0,0) (5,5) (30,30) (35,35) (10,10) (15,15) (,4) (0,0) (5,5) (30,30) (35,35)

11 ESTIMATION OF RELIABILITY IN STRESS-STRENGTH Table 4. Average coverage probablty of the sulated 95% confdence ntervals of R (, ) (s,) (n,) (3.5,1.5) (3.0,1.5) (.5,1.5) (.0,1.5) (1.5,1.5) (1.5,.0) (1.5,.5) (1.5,3.0) (1.5,3.5) (1,3) (,4) (10,10) (15,15) (0,0) (5,5) (30,30) (35,35) (10,10) (15,15) (0,0) (5,5) (30,30) (35,35) Data Analyss Strength data, whch was orgnally reported by Badar and Prest (198), represents the strength easured n GPA for sngle carbon fbers and pregnated 1,000-carbon fber tows. Sngle fbers were tested under tenson at gauge lengths of 0 (Data Set I) and 10 (Data Set II), wth saple szes n = 69 and = 63 respectvely (see Data sets I and II). Data Set I (gauge lengths of 0 ). 1.31, 1.314, 1.479, 1.55, 1.700, 1.803, 1.861, 1.865, 1.944, 1.958, 1.966, 1.997,.006,.01,.07,.055,.063,.098,.140,.179,.4,.40,.53,.70,.7,.74,.301,.301,.359,.38,.38,.46,.434,.435,.478,.490,.511,.514,.535,.554,.566,.570,.586,.69,.633,.64,.648,.684,.697,.76,.770,.773,.800,.809,.818,.81,.848,.880,.809,.818,.81,.848,.880,.954, 3.01, 3.067, 3.084, 3.090, 3.096, 3.18, 3.33, 3.433, 3.585, Data Set II (gauge lengths of 10 ) ,.13,.03,.8,.57,.350,.361,.396,.397,.445,.454,.474,.518,.5,.55,.53,.575,.614,.616,.618,.64,.659,.675,.738,.740,.856,.917,.98,.937,.937,.977,.996, 3.030, 3.15, 3.139, 3.145, 3.0, 3.3, 3.35, 3.43, 3.64, 3.7, 3.94, 3.33, 3.346, 3.377, 3.408, 3.435, 3.493, 3.501, 3.537, 3.554, 3.56, 3.68, 3.85, 3.871, 3.886, 3.971, 4.04, 4.07, 4.5, 4.395,

12 GADDE SRINIVASA RAO Surles and Padgett (1998, 001) observed that generalzed Raylegh wors well for strength data. Raqab and Kundu (005) analyzed the data by subtractng 1.0 and 1.8 fro the frst and second data set respectvely. The transfored data sets correspond to 0 and 10 gauge lengths are assued to follow GR(, ) and GR(, ) respectvely. The obtaned fnal estates for these two data sets are ˆ =.441, ˆ = 1.416, and ˆ = Also they checed the valdty of the odels usng the Kologorov-Srnov (K-S) tests for each data set. It was observed that for Data Sets I and II, the K-S dstances are 0.09 and 0.1 wth the correspondng p values of and respectvely. It ndcates that the GR odel provdes reasonable ft to the transfored data sets. Based on estates of and the MLE of R becoe ˆR 1,3 = and ˆR,4 = The 95% confdence ntervals for R 1,3 becoe ( , ) and for R,4 becoe ( , ). Conclusons Ths artcle used real data sets to nvestgate ultcoponent stress-strength relablty for a generalzed Raylegh dstrbuton when both stress, strength varates follow the sae populaton. Asyptotc confdence ntervals for ultcoponent stress-strength relablty were estated usng the ML ethod. Sulaton results ndcate that the average bas and average MSE decreases as saple sze ncreases n both cases of (s, ). Aong the paraeters the absolute bas and MSE are ncreases (decreases) as ncreases ( ncreases) n both the cases of (s, ). The length of the confdence nterval also decreases as the saple sze ncreases and coverage probablty s close to the nonal value n all sets of paraeters consdered. References Awad, M. & Gharraf, K. (1986). Estaton of p(y<x) n Burr case: A coparatve study. Councatons n Statstcs - Sulatons & Cop., 15: Badar, M. G. & Prest, A. M. (198). Statstcal aspects of fber and bundle strength n hybrd copostes. In T. Hayash, K. Kawata, and S. Ueawa (eds.), Progress n Scence and Engneerng Copostes, (pp ). Toyo: ICCM-IV. 377

13 ESTIMATION OF RELIABILITY IN STRESS-STRENGTH Bhattacharyya, G. K. & Johnson, R. A. (1974). Estaton of relablty n ultcoponent stress strength odel. JASA, 69: Davd, H. A. (1981). Order Statstcs. New Yor: John Wley and Sons. Downtown, F. (1973). The estaton of p(x>y) n the noral case. Technoetrcs, 15: Ens, P. & Gesser, S. (1971). Estaton of the probablty that Y<X. JASA, 66: Kendall, M. G. & Stuart, A. (1979). The Advanced Theory of Statstcs, (Vol. ). London: Charles Grffn and Copany Lted. Kundu, D. & Gupta, R. D. (005). Estaton of p(y<x) for the generalzed exponental dstrbuton. Metra, 61(3): Kundu, D. & Gupta, R. D. (006). Estaton of p(y<x) for Webull dstrbuton, IEEE Transactons on Relablty, 55(): Kundu, D. & Raqab, M. Z. (005). Generalzed Raylegh dstrbuton: dfferent ethods of estaton. Coputatonal Statstcs and Data Analyss, 49: Kundu, D. and Raqab, M. Z. (009). Estaton of R= p(y<x) for threeparaeter Webull dstrbuton, Statstcs and Probablty Letters, 79: McCool, J. I. (1991). Inference on p(y<x) n the Webull case, Councatons n Statstcs - Sulatons & Cop., 0: Mudholar, G. S. & Srvastava, D. K. (1993). Exponentated Webull faly for analyzng bathtub falure data, IEEE Transactons on Relablty, 4: Nand, S. B. & Ach, A. B. (1994). A note on estaton of p(x>y) for soe dstrbutons useful n lfe- testng, IAPQR Transactons, 19(1): Pandey, M. & Uddn, B. (1985). Estaton of relablty n ultcoponent stress strength odel followng Burr dstrbuton. Proceedngs of the Frst Asan congress on Qualty and Relablty, (pp ). New Delh, Inda. Rao, C. R. (1973). Lnear Statstcal Inference and ts Applcatons. Inda: Wley Eastern Lted. Raqab, M. Z. & Kundu, D. (005). Coparson of dfferent estators of p(y<x) for a scaled Burr type X dstrbuton, Councatons n Statstcs Sulaton and Coputaton, 34(): Raqab, M. Z., Mad, M. T. & Kundu, D. (008). Estaton of p(y<x) for the 3-paraeter generalzed exponental dstrbuton, Councatons n Statstcs - Theory and Methods, 37(18):

14 GADDE SRINIVASA RAO Srnvasa Rao, G. & Kanta, R. R. L. (010). Estaton of relablty n ultcoponent stress strength odel: log-logstc dstrbuton, Electronc Journal of Appled Statstcal Analyss, 3(): Surles, J. G. & Padgett, W. J. (1998). Inference for p(y<x) n the Burr Type X odel, Journal of Appled Statstcal Scences, 7: Surles, J. G. & Padgett, W. J. (001). Inference for relablty and stressstrength for a scaled Burr Type X dstrbuton, Lfete Data Analyss, 7: