Interval Estimation of Stress-Strength Reliability for a General Exponential Form Distribution with Different Unknown Parameters

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1 Internatonal Journal of Statstcs and Probablty; Vol. 6, No. 6; November 17 ISSN E-ISSN Publshed by Canadan Center of Scence and Educaton Interval Estmaton of Stress-Strength Relablty for a General Exponental Form Dstrbuton wth Dfferent Unknown Parameters Nahed A. Mokhls 1, Emad J. Ibrahm 1 & Dna M. Ghareb 1 1 Department of Mathematcs, An Shams Unversty, Caro, Egypt Correspondence: Emad J. Ibrahm, Department of Mathematcs, An Shams Unversty, Caro, Egypt. E-mal: emad j brahm@yahoo.co.uk Receved: July 4, 17 Accepted: August 8, 17 Onlne Publshed: September 1, 17 do:1.5539/jsp.v6n6p6 Abstract URL: Ths paper deals wth nterval estmaton of the stress-strength relablty, when the stress and strength varables follow a general exponental form dstrbuton. The dstrbuton parameters of both the stress and the strength are assumed to be unknown. Interval estmaton for relablty s dscussed, usng dfferent approaches. The results obtaned are applcable to many well known dstrbutons. For llustraton of the general results obtaned a smulaton study s performed wth applcaton on Webull dstrbuton. Numercal comparson of the nterval estmators s carred out based on average length, probablty coverage, and tal errors. Keywords: Fsher nformaton matrx, Generalzed varable method, Bootstrap method, Gbbs samplng, Metropols- Hastngs algorthm Math Subject Classfcaton: 6F15; 6F4; 6F5; 6N. 1. Introducton The stress-strength relablty can be expressed as R = P X 1 <X, where the random varables X 1 and X represent a random stress and a random strength, respectvely. The frst papers wth P X 1 <X n ther ttle were ntroduced by Brnbaum, 1956, and Brnbaum and McCarty, Because of the mportance of what we talk about there are a lot of research, whch took ths topc n dfferent ways and dfferent dstrbutons. One of these ways the pont estmaton of R, see, Saracoglu et al., 9, and Panah and Asad, 1. In many stuatons knowng an nterval estmator s better than just knowng a pont estmator because the nterval estmator covers the unknown parameter R wth a specfed probablty, confdence coeffcent. Many authors have studed the nterval estmaton of R. Among them, Kundu and Gupta, 6, Krshnamoorthy and Ln, 1, Asgharzadeh et al., 11, and Asgharzadeh et al., 13. Kotz et al., 3 have comprehensvely covered the problem of pont and nterval estmaton of R. Sngh et al., 15, and Asgharzadeh et al., 17 have also dscussed the problem of pont and nterval estmaton of R. Recently, Mokhls et al., 17 ntroduced pont and nterval estmaton of R = P X 1 <X by dfferent methods when X 1 and X follow a general exponental form or a general nverse exponental form wth survval functons gven by ether F X x; θ, c = exp [ θg 1 x;c ], or F X x; η, c = exp [ ηg x;c ], respectvely, where g 1 x;c s free from the unknown parameter θ, contnuous, monotone ncreasng, and dfferentable functon, wth g 1 x;c as x and g 1 x;c as x, whle g x;c s free from the unknown parameter η, contnuous, monotone decreasng, and dfferentable functon, wth g x;c as x and g x;c as x, and c s a known parameter. In ths paper, we dscuss the nterval estmaton of R =P X 1 <X, where X 1 and X follow a general exponental form dstrbuton. Dfferent methods of estmaton are dscussed. We obtan an approxmate confdence nterval of R va the maxmum lkelhood estmator of R. Usng the generalzed varable approach, a generalzed confdence nterval of R s presented. Two bootstrap confdence ntervals percentle and t-boot of R are also obtaned. Bayesan credble nterval of R s also consdered, usng Markov chan Monte Carlo method MCMC n two scenaros. In the frst scenaro we apply gamma prors, whle n the second one we apply gamma and unform prors. The dfferent nterval estmators are llustrated usng Webull dstrbuton, and a comparson s performed among the estmators obtaned, on the bass of average length, average coverage probablty, and tal errors. 6

2 Internatonal Journal of Statstcs and Probablty Vol. 6, No. 6; 17 Ths paper s organzed as follows: In Secton, the stress-strength relablty, R, and ts maxmum lkelhood estmator are ntroduced. The approxmate confdence nterval of R s consdered, n Secton 3. In Secton 4, the generalzed confdence nterval of R s presented, whle n Secton 5, the percentle and t-bootstrap ntervals of R are obtaned. In Secton 6, the Bayesan credble ntervals of R are ntroduced. In Secton 7, an llustratve example of the obtaned nterval estmators of R s gven by usng the Webull dstrbuton. Then a numercal comparson of the ntervals obtaned s applyng the Webull dstrbuton, n Secton 8.. Maxmum Lkelhood Estmator of R Let X 1 and X be non-negatve ndependent and contnuous random varables, havng general exponental form dstrbutons, wth the survval functons SF and the probablty densty functons pdf gven by and F X x;b, c =exp [ φ b, c g x;c ], 1 f X x;b, c =φ b, c g x;c exp [ φ b, c g x;c ] ; = 1,. where φ b, c s a dfferentable functon of the unknown parameters b B, and c, and B, are the parametrc spaces of b, and c, respectvely. The functon g x;c s contnuous, monotone ncreasng, dfferentable functon such that, g x;c as x and g x;c as x, and g x;c s the frst dervatve of g x;c w.r.t x. Notce that, f F X x;b, c s defned on α, β, then g x;c as x α + and g x;c as x β. Then the stress-strength relablty s gven by R = P X 1 <X = φ 1 g z;c 1 exp φ g z;c dz, 3 where φ = φ b, c ; = 1,. In case of c 1 =c = c, the stress-strength relablty can be expressed as R = φ 1 Mokhls et al., 17. If X = X 1, X,..., X n ; = 1,, are two ndependent random samples from populatons wth dstrbutons gven by 1, then the lkelhood functon s gven as L x 1, x n n θ = exp n ln φ b, c + ln g x j ; c φ b, c gx j ; c, 4 =1 =1 where x j s the j th observaton n the sample X ;j = 1,..., n, = 1,, and θ = c 1, c, b 1, b. The log-lkelhood functon s ln L x 1, x n n θ = n ln φ b, c + ln g x j ; c φ b, c gx j ; c. 5 =1 Partally dfferentatng ln L x 1, x θ wth respect to θ, and equatng to, we get ln L b = =1 n n gx j ; c φ =1 =1 =1 φ 1 +φ as n φ b =, 6 and ln L c = n φ φ c + n c ln g x j ; c φ c n gx j ; c φ n c gx j ; c = ; = 1,. 7 From 6 and knowng that φ b, the MLEs, ˆφ of φ, are gven by ˆφ = n n gx ; = 1,. 8 j; ĉ The MLEs ĉ of c can be obtaned, by substtutng ˆφ nto 7 and solvng numercally. Once we obtan ĉ, the MLEs, ˆφ of φ are completely determned by substtutng ĉ n 8. Usng the nvarance property, the MLEs, ˆb of b can be deduced from ˆφ ˆb, ĉ ; = 1,. The correspondng MLE, ˆR of R can be obtaned by replacng the parameters n 3 wth ther MLEs. 61

3 Internatonal Journal of Statstcs and Probablty Vol. 6, No. 6; Approxmate Confdence Interval of R ACI As we know from the asymptotc maxmum lkelhood propertes, the approxmate 1 α1% confdence nterval for R s ˆR±z 1 α/ ˆσ ˆR, where z 1 α/ s the 1 α/th quantle of the standard normal dstrbuton and ˆσ s the estmator ˆR of varance of ˆR, ˆσ ˆR =At B 1 A θ=ˆθ, where ˆθ s the MLE of θ, B 1 s the nverse of the Fsher nformaton matrx B of θ, A t s the transpose of matrx A, see, Rao, 1965, where [ ] [ ] ln L R B = E, A = ;, j = 1,, 3, 4, θ θ j θ ln L c = n φ φ n + c ln L = ln L = c b b c ln L b = n φ φ, b n φ c φ c + ln g x j ; c φ n gx j ; c c n c φ b, ln L = ln L = ln L = ; j and, j = 1,. c c j b b j b c j gx j ; c φ c n c gx j ; c, We can see that the explct expresson of σ ˆR depends on the forms of φ, g x j ;c and g xj ;c ; j = 1,..., n, = 1,. 4. Generalzed Confdence Interval of R GCI We obtan the generalzed confdence nterval for R by applyng the generalzed varable approach. The generalzed pvotal quantty GPQ for R, G R = RG φ1, G φ, G c1, G c s obtaned by replacng the parameters n 3 wth G φ =φ G b, G c, G b, and G c, where G φ, G b, and G c denote the GPQs for φ, b, and c ; = 1,, respectvely. The GPQ s a functon of observed statstcs and random varables whose dstrbuton s free of unknown parameters. The 1 α1% generalzed confdence nterval of R can be obtaned as G Rα/, G R1 α/, where GRα/ and G R1 α/ are the α/th and 1 α/th quantles of R, respectvely. 5. Bootstrap Confdence Intervals of R boot The bootstrap confdence nterval of R can be estmated usng ether percentle bootstrap or t-bootstrap. The bootstrap samples wll be generated frstly, usng the followng bootstrap samplng algorthm, see, Efron, Algorthm Generate X ; = 1,, from 1, and compute the MLEs ĉ 1, ĉ, ˆφ 1, ˆφ, ˆR of c 1, c,φ 1, φ, R.. Resample two ndependent random samples X ; = 1,, from X ; = 1,, respectvely; compute the MLEs ĉ 1, ĉ, ˆφ 1, ˆφ, ˆR of c 1, c,φ 1, φ, R. 3. Repeat Step, N tmes to obtan a set of bootstrap samples of R, say { ˆR j ;j = 1,..., N }. Order ˆR j, n an ncreasng order. 4. Construct two dfferent bootstrap ntervals of R: a Percentle bootstrap P-boot The 1 α1% percentle bootstrap confdence nterval of R s ˆR α/, ˆR are the α/th and 1 α/th quantles of R, respectvely. b T-bootstrap T-boot 1 α/, where ˆR α/ and ˆR 1 α/ The 1 α1% t-bootstrap confdence nterval of R s gven by ˆR ˆt 1 α/ S, ˆR ˆt α/ S, where S be the sample standard devaton of { ˆR j ;j = 1,..., N } { ˆR } and ˆt α be the αth quantle of j ˆR ;j = 1,..., N. S 6

4 Internatonal Journal of Statstcs and Probablty Vol. 6, No. 6; Bayesan Credble Interval of R BCI We suggest two dfferent scenaros to estmate the Bayesan credble nterval of R by applyng MCMC. For the frst scenaro, we assume gamma prors for φ 1, φ, c 1, and c, whle n the second scenaro, we consder ndependent gamma prors for φ 1, φ and unform prors for c 1, c as the avalable prors nformaton s weak Gamma Prors G-BCI Let the pror densty of φ ; = 1, be gamma gven by f φ φ = βα φ α 1 e β φ ; φ, α, β >, 9 Γα and also assume that φ 1 and φ are ndependent. Moreover, assume that c ; = 1, have gamma prors wth probablty densty functons f c c = λδ c δ 1 e λ c ; c, δ, λ >, 1 Γδ and c 1 and c are ndependent. From 4, 9, and 1, the jont posteror densty functon of φ 1, φ, c 1, and c can be obtaned as π L x 1, x θ fφ1 φ 1 f φ φ f c1 c 1 f c c φ 1, φ, c 1, c x 1, x = L x 1, x. θ fφ1 φ 1 f φ φ f c1 c 1 f c c dφ 1 dφ dc 1 dc Snce, the jont posteror densty functon cannot be obtaned analytcally, we apply MCMC method to estmate the Bayesan credble nterval of R. There are generally two algorthms, the Gbbs samplng and the Metropols-Hastngs algorthm. If the condtonal dstrbuton for each parameter s a known dstrbuton, the Gbbs samplng can be used. If the condtonal dstrbuton doesn t look lke any known dstrbuton, n ths case the Metropols-Hastngs algorthm can be useful. To perform the MCMC method, we frst fnd the margnal posteror dstrbutons of φ and c. The margnal posteror dstrbuton of φ s β + n π 1 φ c, x 1, x gx j; c n +α = Γ n + α φ n +α 1 n exp φ β + gx j ; c ; = 1,. 11 The margnal posteror dstrbuton of c s n n π 1 c x 1, x = K 1 exp δ 1 ln c λ c + ln g x j ; c n + α ln β + gx j ; c, 1 where K = n n exp δ 1 ln c λ c + ln g x j ; c n + α ln β + gx j ; c dc ; = 1,. However, we shall use the unon of the Metropols-Hastngs wth Gbbs samplng, see, Asgharzadeh et al., 13. The procedure s shown by the followng algorthm. Algorthm. 1. Choose the startng values c 1 and c.. For j = 1 to N tmes. 3. Generate φ j ; = 1,, from Generate c j ; = 1,, from 1. Usng the Metropols-Hastngs algorthm wth the normal proposal dstrbuton π Nc j 1, 1; = 1,. a Generate ξ from the proposal dstrbuton π ; = 1,, respectvely. 63

5 Internatonal Journal of Statstcs and Probablty Vol. 6, No. 6; 17 b Defne Q =mn 1, π 1ξ x 1, x π c j 1 π 1 c j 1 x 1, x π ξ c Generate u from Unform, 1. Take c j 5. Compute the R j from End j loop. ; = 1,. = ξ ; u Q, c j 1 ; otherwse 7. Repeat the steps -6, N tmes, and sort R j ;j = 1,..., N, ascendng. ; = 1,, respectvely. 8. Construct the 1 α1% Bayesan credble nterval of R as R gα/, R g1 α/, where R gα/ and R g1 α/ are the α/th and 1 α/th quantles of R, respectvely. 6. Mxed Prors M-BCI Let the pror densty functon of φ be as 9; = 1,, and assume that, c has unform pror dstrbuton wth probablty densty functon f c = 1; c >, = 1,. From the lkelhood functon n 4 and the pror densty functons of φ 1, φ, c 1, and c, the jont densty functon can be obtaned as β α 1 L x1, x, φ 1, φ, c 1, c = 1 βα n n exp Γα 1 Γα n + α 1 ln φ + ln g x j ; c φ β + gx j ; c. 13 The jont posteror densty functons of φ 1, φ, c 1, and c based on 13 s gven as L x1, x, φ 1, φ, c 1, c L φ1, φ, c 1, c x 1, x = =1 =1 =1 L x1, x, φ 1, φ, c 1, c dφ1 dφ dc 1 dc. The margnal posteror dstrbuton of φ s gven by the same as n 11, whle the margnal posteror dstrbuton of c becomes n n π c x 1, x = T 1 exp ln g x j ; c n + α ln β + gx j ; c, where T = n n exp ln g x j ; c n + α ln β + gx j ; c dc ; = 1,. As we notce that, the margnal posteror dstrbutons of c 1 and c do not have a known form. Usng a technque smlar to that n Algorthm except for the posteror dstrbuton of c ; = 1,. The 1 α1% Bayesan credble nterval of R can be obtaned as R mα/, R m1 α/, where R mα/ and R m1 α/ are the α/th and 1 α/th quantles of R, respectvely. 7. Illustratve Example We see that, the Webull dstrbuton follows the general exponental form n 1, wth φ = 1 b c and g x;c = x c. If X ; = 1,, are two [ ndependent ] random samples from Webull dstrbutons wth the survval functon gven as F X x;b, c =exp 1 b c x c ; = 1,. Then from 3, the stress-strength relablty s gven as ] dz. Usng the MLEs, ĉ 1, ĉ, ˆφ 1, ˆφ, ˆR, the approxmate 1 α1% confdence n- R = c 1 b c 1 z c1 1 exp b c z c terval for R can be obtaned. Usng the Newton Raphson teratve method, the MLE ĉ, of c from 7 s gven by 1 c + 1 n ln x j n n xc j ln x j n xc j [ =1 1 of b can be deduced as ˆb = =, and the MLE, ˆφ, of φ from 8 can be expressed as ˆφ = 1 ˆbĉ n 1/ĉ xĉ j n ; = 1,. The MLE, ˆR of R s gven as ˆR= ĉ1 ˆbĉ1 1 = n n xĉ j zĉ1 1 exp, and also the MLE, ˆb, [ ] =1 1 dz. zĉ ˆbĉ 64

6 Internatonal Journal of Statstcs and Probablty Vol. 6, No. 6; 17 Applyng the method n Secton 4, we can have G R by replacng φ, b and c wth G φ, G b and G c, respectvely, where Gc 1 G φ = G b, G c = ĉ 1 c ĉ ĉ = ĉ ĉ b, and G b = Gc ˆb ˆb = ˆb ; = 1,, and ĉ 1, ĉ, ˆb 1, ˆb denote the observed value ĉ 1 ˆb of the MLEs ĉ ĉ 1, ĉ, ˆb 1, ˆb. The MLEs 1, ĉ, ˆb 1, ˆb of c1, c, b 1, b are based on two ndependent samples from standard exponental dstrbutons see, Krshnamoorthy et al., 9, and Krshnamoorthy and Ln, 1. The quantty ĉ c s dstrbuted as ĉ and the quantty ĉ ln ˆb b s dstrbuted as ĉ ln ˆb ; = 1,, and the dstrbutons of ĉ c and ĉ ln ˆb b ; = 1,, do not depend on any unknown parameters. So they are pvotal quanttes and can be obtaned by generatng two ndependent samples from standard exponental dstrbuton, see, Thoman et al., Usng R- language, the followng algorthm s used to estmate the generalzed confdence nterval of R, Algorthm Generate two ndependent random samples X from Webullb, c ; = 1,, respectvely, compute the MLEs ĉ1, ĉ, ˆb 1, ˆb of c1, c, b 1, b.. Generate two ndependent random samples X from Exp1; = 1,, compute the MLEs ĉ 1, ĉ, ˆb 1, ˆb. 3. Compute the GPQs, G c, G b, G φ, and G R ; = 1,. 4. Repeat the steps -3, N tmes to obtan a set of samples of G R, say { G Rj ;j = 1,..., N }, and the ordered G Rj ;j = 1,..., N, wll be denoted as G 1 R j < < G N R j. 5. Construct the 1 α1% generalzed confdence nterval of R as G Rα/, G R1 α/. The bootstrap confdence ntervals and the Bayesan credble ntervals are obtaned, usng Algorthm 1 and, respectvely. 8. Smulaton Study In ths secton, we present a smulaton study to observe the behavor of the dfferent nterval estmators of P X 1 <X for dfferent sample szes and dfferent parameter values for the Webull dstrbuton. The comparson s based on average length, average coverage, left and rght tal errors when α =.5. We generate 1 samples of sample szes n 1, n = 1, 1 and 3, 3 from the Webull dstrbutons of X 1 and X. We select the parameter values that produce the values of R =.634,.716,.811,.966,.956, and.977. In Bayesan estmaton, the hyper-parameters of prors n the two scenaros have the same means but dfferent varances. 1. Let α 1, β 1 =, 1, α, β = 1, 1/, δ 1, λ 1 = 3, 3/, and δ, λ = 4,.. Let α 1, β 1 =, 1, and α, β = 1, 1/. 65

7 Internatonal Journal of Statstcs and Probablty Vol. 6, No. 6; 17 Fgure 1. Average length for Webull dstrbuton 66

8 Internatonal Journal of Statstcs and Probablty Vol. 6, No. 6; 17 Fgure. Average coverage probablty for Webull dstrbuton 67

9 Internatonal Journal of Statstcs and Probablty Vol. 6, No. 6; 17 Fgure 4: Rght tal error for Webull dstrbuton. Fgure 3. Left tal error for Webull dstrbuton 68

10 Internatonal Journal of Statstcs and Probablty Vol. 6, No. 6; 17 Fgure 4. Rght tal error for Webull dstrbuton In Fgure 1, the average length of ACI s the smallest at all the values of R. All lengths of the ntervals estmator of R decrease when n ncreases. Lengths of ACI, GCI, and boot decrease when R ncreases. In Fgure we observed that, the average coverage probablty of ACI and T-boot are the worst, and the GCI and BCI are roundly approxmated 1 α 1%, the boot s affected by n and R. In Fgure 3 we see the followng; the ACI has the smallest left tal error. All left tal errors of the ntervals decrease when n ncreases. All left tal errors of the ntervals except BCI decrease when R ncreases. As we see n Fgure 4, the rght tal error of boot s the largest and all rght tal errors of the ntervals are affected by n and R. Acknowledgment The authors would lke to thank the Edtor-n-Chef, two anonymous referees for ther valuable comments and nsghtful suggestons on ths paper. References Asgharzadeh, A., Kazem, M., & Kundu, D. 17. Estmaton of PX > Y for Webull Dstrbuton Based on Hybrd Censored Samples,Internatonal Journal of Systems Assurance Engneerng and Management, 8,

11 Internatonal Journal of Statstcs and Probablty Vol. 6, No. 6; 17 Asgharzadeh, A., Valollah, R., & Raqab, M. Z. 11. Stress-Strength Relablty of Webull Dstrbuton Based on Progressvely Censored Samples, Statstcs and Operatons Research Transactons, 35, Asgharzadeh, A., Valollah, R. & Raqab, M. Z. 13. Estmaton of the Stress-Strength Relablty for the Generalzed Logstc Dstrbuton, Statstcal Methodology, 15, Brnbaum, Z., & McCarty, R A Dstrbuton-Free Upper Confdence Bound for PY < X Based on Independent Samples of X and Y, Annals of Mathematcal Statstcs, 9, Brnbaum, Z On a Use of Mann-Whtney Statstcs, Statstcs and Probablty, Unversty of Calforna Press. Efron, B An Introducton to the Bootstrap, Chapman & Hall. Kotz, S., Lumelsk, Y., & Pensky, M. 3. The Stress-Strength Model and ts Generalzatons: Theory and Applcatons, Sngapore: World Scentfc. Krshnamoorthy, K., & Ln, Y. 1. Confdence Lmts for Stress-Strength Relablty Involvng Webull Models, Statstcal Plannng and Inference, 14, Krshnamoorthy, K., Ln, Y., & Xa, Y. 9. Confdence Lmts and Predcton Lmts for a Webull Dstrbuton Based on the Generalzed Varable Approach, Statstcal Plannng and Inference, 139, Kundu, D., & Gupta, R. D. 6. Estmaton of PY < X for Webull Dstrbuton, IEEE Transactons on Relablty, 55, Mokhls, N., Ibrahm, E., & Ghareb, D. 17. Stress-Strength Relablty wth General Form Dstrbutons, Communcatons n Statstcs-Theory and Methods, 46, Panah, H., & Asad, S. 1. Estmaton of R = PY < X for Two-Parameter Burr Type XII Dstrbuton, World Academy of Scence, Engneerng and Technology Internatonal Journal of Mathematcal, Computatonal, Physcal, Electrcal and Computer Engneerng, 1, 1. Rao, C Lnear Statstcal Inference and ts Applcatons, New York: John Wley and Sons. Saracoglu, B., Kaya, M. F., & Abd-Elfattah, A. M. 9. Comparson of Estmators for Stress-Strength Relablty n the Gompertz Case, Hacettepe Journal of Mathematcs and Statstcs, 38, Sngh, S. K., Sngh, U., Yadav, A. S., & Vshwkarma, P. K. 15. On the Estmaton of Stress-Strength Relablty Parameter of Inverted Exponental Dstrbuton, Internatonal Journal of Scentfc World, 3, Thoman, D. R., Ban, L. J., & Antle, C. E Inferences on the Parameters of the Webull Dstrbuton, Technometrcs, 11, Copyrghts Copyrght for ths artcle s retaned by the authors, wth frst publcaton rghts granted to the journal. Ths s an open-access artcle dstrbuted under the terms and condtons of the Creatve Commons Attrbuton lcense 7