Burr-XII Distribution Parametric Estimation and Estimation of Reliability of Multicomponent Stress-Strength

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1 Burr-XII Distributio Paraetric Estiatio ad Estiatio of Reliability of Multicopoet Stress-Stregth G. Sriivasa Rao ad Muhaad Asla ad Debasis Kudu 3 Departet of Statistic School of M & C S, Dilla Uiversity, Dilla, Po. Box: 49, Ethiopia. Departet of Statistic Fora Christia College Uiversity, Lahore, Paista. Departet of Matheatics ad Statistic Idia Istitute of Techology Kapur, Pi 0806, Idia E-ail: gaddesrao@yahoo.co; Eail: asla_ravia@hotail.co; 3 Eail: udu@iit.ac.i Abstract: I this paper, we estiate the ulticopoet stress-stregth reliability by assuig the Burr-XII distributio. The research ethodology adopted here is to estiate the paraeters by usig axiu lielihood estiatio. The reliability is estiated usig the axiu lielihood ethod of estiatio ad results are copared usig the Mote-Carlo siulatio for sall saples. By usig real data sets we well illustrate the procedure. Key Words: Burr-XII distributio, reliability estiatio, stress-stregth, ML estiatio, cofidece itervals.. Itroductio The Burr-XII distributio, which was origially derived by Burr (94 ad received ore attetio by the researchers due to its broad applicatios i differet fields icludig the area of reliability, failure tie odelig ad acceptace saplig pla. Reader ca fid the applicatios i various fields fro Ali ad Jahee (00 ad Burr (94. The two paraeters Burr-XII distributio has the followig desity fuctio ( ( + f( x;, = x + x ; for x > 0 ( ad the distributio fuctio ( F( x;, = + x ; for x > 0. ( Here > 0ad > 0 are the shape paraeters respectively. It is iportat to ote that whe =, the Burr-XII reduces to the log-logistic distributio. Wu ad Yu (005 show that the shape paraeter plays a vital role for the Burr-XII. Now owards two paraeters Burr-XII distributio with shape paraeters ad will be deoted by BD (,. Let the rado saples Y, X, X,... X be idepedet, G(y be the cotiuous distributio fuctio of Y ad F(x be the coo cotiuous distributio fuctio of X, X,... X. The reliability i a ulticopoet stress-stregth odel developed by Bhattacharyya ad Johso (974 is give by

2 [ ] R =P atleastsofthe( X, X,... X exceedy i i = [ F( y][ F( y] dg( y, i= s i (3 where X, X,... Xare idetically idepedetly distributed (iid with coo distributio fuctio F( x ad subjected to the coo rado stress Y. The probability i (3 is called reliability i a ulticopoet stress-stregth odel [Bhattacharyya ad Johso (974]. The survival probabilities of a sigle copoet stress-stregth versio have bee cosidered by several authors for diffet distributios. Eis ad Geisser (97, Dowtow (973, Awad ad Gharraf (986, McCool (99, Nadi ad Aich (994, Surles ad Padgett (998, Raqab ad Kudu (005, Kudu ad Gupta (005& 006, Raqab et al. (008, Kudu ad Raqab (009. The reliability i a ulticopoet stress-stregth was developed by Bhattacharyya ad Johso (974, Padey ad Borha (985. Padey ad Uddi (99 assued the paraeters ot ivolved i reliability as ow usig the Bayesia estiatio. Recetly Rao ad Kata (00 studied estiatio of reliability i ulticopoet stress-stregth for the log-logistic distributio ad Rao (0 developed a estiatio of reliability i ulticopoet stressstregth based o geeralized expoetial distributio. The ai of this paper is to study the reliability i a ulticopoet stress-stregth based o X, Y beig two idepedet rado variable where X ~ BD(, ady~ BD(,. We will use the paraetric estiatio ad estiatio reliability. Suppose a syste, with idetical copoet fuctios if at least s( s copoets siultaeously operate. I its operatig eviroet, the syste is subjected to a stress Y which is a rado variable with distributio fuctio G (.. The stregths of the copoet that is the iiu stresses to cause failure, are idepedetly ad idetically distributed rado variables with distributio fuctio F (.. The reliability of syste ca be obtaied by equatio (3. The estiatio of survival probability whe the stress ad stregth variates follow two paraeters Burr-XII distributio is ot paid uch attetio. Therefore, a attept is ade here to study the estiatio of reliability i ulticopoet stress-stregth odel with referece to two paraeters Burr-XII distributio. The rest of the paper is orgaized as follows. I Sectio, we discussed the research ethodology ad procedure for expressio of R. The asyptotic distributio ad cofidece

3 iterval of equatio (3 is obtaied usig the MLE. The results of sall saple coparisos ade through Mote Carlo siulatios are i Sectio 3. Soe fidigs are discussed i Sectio 4.. Research Methodology for Maxiu Lielihood Estiator of R Let X ~ BD(, ad Y~ BD(, are idepedetly distributed with uow shape paraeters adad coo shape paraeter respectively. The reliability i ulticopoet stress-stregth for two paraeter Burr-XII distributio usig (3 we get ( ( ( ( i i ( + R = + y + y y + y dy i i= s 0 i+ ν ( ν ( i t t dt i, where t = ( + y, ν = i= s 0 = i= s ( ν ( ν i, i. = Β + + i After the siplificatio we get! R = ν ( ν + j sice ad iare itegers. i= s i! (4 j= i The probability i (4 is called reliability i a ulticopoet stress-stregth odel. It is iportat to ote that the MLE (axiu lielihood estiatio of R depeds o the MLE of ad. Therefore, we eed to fid the MLE of the later before the MLE of first oe. Siilarly, to derive the MLE of ad we eed the MLE of. Let us assue that X < X<...<X is rado saple fro BD(, ad Y<Y <... < Yis a rado saple fro BD(,, the the log-lielihood fuctio (LLF of these observed saple is L(,, = ( + l + l + l + ( lxi + lyj i= j= ( xi ( yj (5 ( + l + ( + l + i= j= The MLE of, ˆ ˆ ˆ ad ; say, ad respectively, ca be obtaied as the solutio of L + x lx y ly = 0 + l + l ( + ( + = 0 = = = + = + i i j j xi yj i j i ( xi (6 j ( yj 3

4 L = 0 l ( + x i = 0, (7 i= L = 0 l ( + y j = 0 (8 j= Fro (7 ad (8, we obtai ˆ ˆ ( ad ( l + x l + y ( i ( j i= j= Puttig the values of ˆ ˆ ( ad ( ito (6, we obtai + x lx i i j j lxi lyj ( ( i j i j l x + xi l ( ( yj y + = = = = = = x lx y ly = y ly i i j j 0 ( (0 i= + xi j= ( + yj Therefore, ˆ ca be obtaied usig the followig o-liear equatio ( h( = ( Where h( = lx ( i j ( i= + xi j= ( + yj x lx i i j j + ( ( i= xi j= + x + ( ( yj + y + l l = = ly y ly Because ˆ is a fixed poit solutio of the o-liear (, therefore, it ca be obtaied by usig a siple iterative procedure as h( = +, (3 ( j ( j (9 ( where ( j th is the j iteratio of. ˆ It should be oted that durig the siulatio process whe the differece betwee ( j ad ( j + is sufficietly sall the stop the iterative process. Oce we obtai ˆ, the paraeters ˆ ˆ ad ca be obtaied fro (9 as ˆ ˆ ˆ ˆ ( ad ( respectively. The MLE of R becoes! ˆ ( j where =. i= s i! j= i ˆ (4 Rˆ ˆ ˆ ˆ = ν ν + ν 4

5 To obtai the asyptotic cofidece iterval for R, we proceed as follows: The asyptotic variace of the MLE is give by L V( ˆ = E = ad L V( ˆ = E = The asyptotic variace (AV of a estiate of R which is a fuctio of two idepedet statistics (say, is give by Rao (973. ˆ R R AV(R ˆ ˆ =V( + V(. (6 Thus fro Equatio (6, the asyptotic variace of ˆR s, ca be obtaied. To avoid the difficulty of derivatio of R, we obtai ˆR s, ad their derivatives for ( =(,3 ad (,4 separately, they are give by (5 ˆ ˆ ν( ˆ ν + 6ˆ ν + R,3 = ad ˆ ˆ ν( ˆ ν + 9ˆ ν + 6 R,4 =. ( + ˆ ν( + ˆ ν(3 + ˆ ν ( + ˆ ν(3 + ˆ ν(4 + ˆ ν Rˆ ˆ,3 6 ˆ ν(3ˆ ν + ˆ ν + R,3 6(3ˆ ν + ˆ ν + = ad = ( + ˆ ν( + ˆ ν(3 + ˆ ν ( + ˆ ν( + ˆ ν(3 + ˆ ν Rˆ [ ] [ ] 4 ˆ ν(3ˆ ν + 8ˆ ν + 6 Rˆ 4(3ˆ ν + 8ˆ ν + 6,4,4 = ad = [( + ˆ ν(3 + ˆ ν(4 + ˆ ν ] [( + ˆ ν(3 + ˆ ν(4 + ˆ ν ] Therefore, AV(R ˆ = AV(R ˆ = As,, ˆ ˆ + ˆ + ad ( ( (3 36 ν (3ν ν,3 + 4 [ + ˆ ν + ˆ ν + ˆ ν ] { ˆ ν ˆ ν ˆ ν } 4 ( ( (3 (4 [ + ˆ ν + ˆ ν + ˆ ν ],4 4 Rˆ R d N(0,, AV(R ˆ.. where Rˆ ˆ.96 AV(R is the asyptotic 95% cofidece iterval (C.I of syste reliability R ad asyptotic 95% C.I for R,3 is give by ˆ 6 ˆ ν(3ˆ ν + ˆ ν + R,3.96 +, where ˆ ν = ˆ / ˆ. [( + ˆ ν( + ˆ ν(3 + ˆ ν ] The asyptotic 95% cofidece iterval for R,4 is give by 5

6 { ˆ ν ˆ ν ˆ ν } 4 ( ˆ R,4.96 +, where ˆ ν = ˆ / ˆ. [( + ˆ ν(3 + ˆ ν(4 + ˆ ν ] 3. Results ad Data Aalysis 3.. Results fro siulatio study Suppose 3000 rado saple of size 0(530 each fro stress ad stregth populatios are geerated for (, =(3.0,.5, (.5,.5,(.0,.5,(.5,.5,(.5,.0,(.5,.5 ad (.5,3.0 as proposed by of Bhattacharyya ad Johso (974. The MLE of shape paraeter is estiated by iterative ethod ad usig the shape paraeters ad are estiated fro (9. These ML estiators of ad are the substituted i ν to get the ulticopoet reliability for ( = (, 3, (, 4. The average bias ad average ea square error (MSE of the reliability estiates over the 3000 replicatios are give i Tables ad. Average cofidece legth ad coverage probability of the siulated 95% cofidece itervals of R are give i Tables 3 ad 4. The true value of reliability i ulticopoet stress-stregth with the give cobiatios of (, for ( = (, 3 are 0.543, 0.599, 0.668, 0.750, 0.8, 0.869, ad for ( = (, 4 are 0.390, 0.443, 0.50, 0.600, 0.688, 0.75, Thus the true value of reliability i ulticopoet stress-stregth icreases as icreases for a fixed whereas reliability i ulticopoet stress-stregth decreases as icreases for a fixed i both the cases of (. Therefore, the true value of reliability is icreases as ν icreases ad vice versa. The average bias ad average MSE are decreases as saple size icreases for both cases of estiatio i reliability. Also the bias is egative i all the cobiatios of the paraeters i both situatios of (. It proofs the cosistecy property of the MLE of R. Wherea aog the paraeters the absolute bias ad MSE are icreases as icreases for a fixed i both the cases of ( ad the absolute bias ad MSE are decreases as icreases for a fixed i both the cases of (. Fro these table it is clear that the as saple size decrease the legth of C.I is also decreases ad coverage probability i all cases is tha 0.95 which shows the perforace of C.I usig the Burr type XII distributio is quite good for various cobiatios of the paraeters.. 6

7 Wherea aog the paraeters we observed the sae pheoeo for average legth ad average coverage probability that we observed i case of average bias ad MSE. 3.. Data Aalysis I this sub sectio we aalyze two real data sets ad deostrate how the proposed ethods ca be used i practice. Both datasets were discussed by Zier et al. (998 ad Lie et al. (00 for the Burr XII reliability aalysis. Lie et al. (00 studied the validity of the odel for both data sets ad they showed that Burr-XII distributio fits quite well for both the data sets. These data sets are reproduced here for easy referece. (X: 0.9, 0.78, 0.96, 0.3,.78, 3.6, 4.5, 4.67, 4.85, 6.50, 7.35, 8.0, 8.7,.06, 3.75, 3.5, 33.9, 36.7 ad (Y: 0.9,.5,.3, 3., 3.9, 5.0, 6., 7.5, 8.3, 0.4,.,.6, 5.0, 6.3, 9.3,.6, 4.8, 3.5, 38. ad We use the iterative procedure to obtai the MLE of usig ( ad MLEs of ad are obtaied by substitutig MLE of i (9. The fial estiates for real data sets are ˆ = , ˆ = ad ˆ= Base o estiates of ad the MLE of R, becoe ˆR,3= ad ˆR,4= The 95% cofidece itervals for R,3becoe ( , ad for R,4becoe ( , Coclusios I this paper, we have studied the ulticopoet stress-stregth reliability for two paraeters Burr-XII distributio whe both of stres stregth variates follows the sae populatio. Also, we have estiated asyptotic cofidece iterval for ulticopoet stress-stregth reliability. The siulatio results idicates that the average bias ad average MSE are decreases as saple size icreases for both ethods of estiatio i reliability. Aog the paraeters the absolute bias ad MSE are icreases (decreases as icreases ( icreases i both the cases of (. Fro the legth of C.I ad tred i saple show the perforace of the proposed procedure usig the Burr-XII distributio is quite good. Further, the coverage probability is quite close to give value i all sets of paraeters. The real exaple shows that the proposed procedure ca be used i real world to estiate the reliability of ulticopoet stress-stregth usig the Burr- XII distributio. s 7

8 Refereces []. Ali Mousa, M.A.M., Jahee, Z.F. (00. Statistical iferece for the Burr odel based o progressively cesored data. Coputers ad Matheatics with Applicatio 43, []. Awad, M., Gharraf, K. (986. Estiatio of P( Y < X i Burr case: A coparative study, Cou. Statist. - Siula. & Cop., 5, [3]. Bhattacharyya, G.K., Johso, R.A. (974. Estiatio of reliability i ulticopoet stress-stregth odel, JASA, 69, [4]. Burr, I.W. (94. Cuulative frequecy fuctio Aals of Matheatical Statistic 3, 5-3. [5]. Dowtow, F. (973. The estiatio of P( X > Y i the oral case, Techoetric 5, [6]. Ei P., Geisser, S. (97. Estiatio of the probability that Y < X, JASA, 66, [7]. Kudu, D., Gupta, R.D. (005. Estiatio of P( Y < X for the geeralized expoetial distributio, Metria, 6 (3, [8]. Kudu, D., Gupta, R.D. (006. Estiatio of P( Y < X for Weibull distributio, IEEE Trasactios o Reliability, 55 (, [9]. Kudu, D., Raqab, M.Z. (009. Estiatio of R= P( Y < X for three-paraeter Weibull distributio, Statistics ad Probability Letter 79, [0]. Lio, Y.L., Tsai, T.R., Wu, S.J. (00. Acceptace saplig plas fro trucated life tests based o the Burr type XII percetile Joural of the Chiese Istitute of Idustrial Egieer 7:4, []. McCool, J. I. (99. Iferece o P( Y < X i the Weibull case, Cou. Statist. - Siula. & Cop., 0, []. Nadi, S.B., Aich, A.B. (994. A ote o estiatio of P( X > Y for soe distributios useful i life- testig, IAPQR Trasactio 9(, [3]. Nelso, W. (98. Applied Life Data Aalysi Joh Wiley ad So NY. [4]. Padey, M., Borha, Md. U. (985. Estiatio of reliability i ulticopoet stressstregth odel followig Burr distributio, Proceedigs of the First Asia cogress o Quality ad Reliability, New Delhi, Idia, [5]. Padey, M., Uddi, Md. B. (99. Estiatio of reliability i ulti-copoet stressstregth odel followig a Burr distributio, Microelectrics Reliability, 3 (, -5. [6]. Rao, C. R. (973. Liear Statistical Iferece ad its Applicatio Wiley Easter Liited, Idia. [7]. Rao, G.S. (0. Estiatio of reliability i ulticopoet stress-stregth odel based o geeralized expoetial distributio, Colobia Joural of Statistic 35(, [8]. Rao, G.S., Kata, R.R.L. (00. Estiatio of reliability i ulticopoet stressstregth odel: log-logistic distributio, Electroic Joural of Applied Statistical Aalysi 3(, [9]. Raqab, M.Z., Kudu, D. (005. Copariso of differet estiators of py ( < X for a scaled Burr type X distributio, Cou. Statist. - Siula. & Cop., 34 (, [0]. Raqab, M.Z., Madi, M.T., Kudu, D. (008. Estiatio of P( Y < X for the 3-paraeter geeralized expoetial distributio, Cou. Statist - Theor. Meth., 37 (8,

9 []. Surle J.G., Padgett, W.J. (998. Iferece for P( Y < X i the Burr type X odel, Joural of Applied Statistical Sciece 7, []. Wu, J.W., Yu, H.Y. (005. Statistical iferece about the shape paraeter of the Burr type XII distributio uder the failure-cesored saplig pla, Applied Matheatics ad Coputatio, 63, [3]. Zier, W.J., Keat J.B., Wag, F.K.(998. The Burr XII distributio i reliability aalysi Joural of Quality Techology, 30, Table. Average bias of the siulated estiates of R (, ( (, (3.0,.5 (.5,.5 (.0,.5 (.5,.5 (.5,.0 (.5,.5 (.5,3.0 (0, (5, (,3 (0, (5, (30, (0, (5, (,4 (0, (5, (30, Table. Average MSE of the siulated estiates of R (, ( (, (3.0,.5 (.5,.5 (.0,.5 (.5,.5 (.5,.0 (.5,.5 (.5,3.0 (0, (,3 (5, (0,

10 (,4 (5, (30, (0, (5, (0, (5, (30, Table 3. Average cofidece legth of the siulated 95% cofidece itervals of R (, ( (, (3.0,.5 (.5,.5 (.0,.5 (.5,.5 (.5,.0 (.5,.5 (.5,3.0 (0, (5, (,3 (0, (5, (30, (0, (5, (,4 (0, (5, (30, Table 4. Average coverage probability of the siulated 95% cofidece itervals of R (, ( (, (3.0,.5 (.5,.5 (.0,.5 (.5,.5 (.5,.0 (.5,.5 (.5,3.0 (0, (5, (,3 (0, (5, (30, (0, (5, (,4 (0, (5, (30,