COMPARISON OF ESTIMATORS FOR STRESS-STRENGTH RELIABILITY IN THE GOMPERTZ CASE

Transcription

1 Haettepe Journal of Mathematis and Statistis olume , COMPARISON OF ESTIMATORS FOR STRESS-STRENGTH RELIABILITY IN THE GOMPERTZ CASE Buğra Saraçoğlu, Mehmet Fedai Kaya and A. M. Abd-Elfattah Reeived 23:8 :28 : Aepted 11 :9 :29 Abstrat In this paper, the Bayesian and non Bayesian estimation problem of reliability, R PY < X, will be onsidered when X and Y are two independent but not identially random variables belonging to the Gompertz distribution. The three estimation methods applied were the maximum likelihood, uniformly minimum variane unbiased, and Bayes estimators. A numerial illustration is used to ompare the three different estimators. Keywords: Stress-strength reliability, Gompertz distribution, Minimum variane unbiased estimation, Maximum likelihood estimation, Bayes estimation, Mean square error. 2 AMS Classifiation: 62 N 5, 62 F 15, 62 F Introdution The Gompertz distribution plays an important role in modeling survival times, human mortality and atuarial tables. It has many appliations, partiularly in medial and atuarial studies. Also, the Gompertz distribution is used as a survival model in reliability. It has an inreasing hazard rate for the life of the systems. This distribution does not seem to have reeived enough attention, possibly beause of its ompliated form. Reently, this distribution has been studied by some authors. For example, see Wu et al. [16], Jaheen [7], Wu et al. [17, 18], Ismail [6] and Al-Khedhair & El-Gohary [8]. Department of Statistis, Faulty of Siene, Seluk University, 4231, Konya, Turkey. B. Saraçoğlu bugrasara@seluk.edu.tr M. F. Kaya fkaya@seluk.edu.tr Corresponding author. This study forms part of the first author s Dotor of Philosophy Thesis entitled Estimation of System Reliability for some Distributions in Stress-Strength Models submitted to the Graduate Shool of Seluk University, 27. Department of Mathematial Statistis, Institute of Statistial Studies and Researh, Cairo University, Cairo, Egypt. a afattah@issr.u.edu.eg

2 34 B. Saraçoğlu, M.F. Kaya, A.M. Abd-Elfattah The problem of estimating R PY < X arises in the ontext of mehanial reliability of a system with strength X and stress Y, the reliability, R, is hosen as a measure of system reliability. In a stress strength model, the system fails if and only if, at any time, the applied stress is greater than its strength. This model, first onsidered by Birnbaum [2], is ommonly used in many engineering appliations, suh as ivil, mehanial and aerospae. Reently, a number of papers have dealt with the stress strength reliability problem. Several distributions have been used in the literature as failure models. For referenes see the book by Kotz et al. [9] or the artiles by Churh and Harris [4], Chao [3], Awad and Gharraf [1], Constantine and Karson [5], Surles and Padgett [15], Mokhlis [12], Raqab and Kundu [13] and Kundu and Gupta [1]. In this artile, we study various estimation proedures for the reliability, R, when X and Y are two independent but not identially Gompertz random variables. This paper is organized as follows: In Setion 2, the maximum likelihood estimator MLE for R with some of its properties is derived for the underlying Gompertz distribution. Setion 3 provides the uniformly minimum variane unbiased estimator UMUE for R. In setion 4, the Bayes estimator for R is onsidered based on the mean squared error loss funtion. In setion 5, a simulation study is performed to ompare the different estimators of R. 2. Maximum likelihood estimation Let X be the strength of a system and Y be the stress ating on it. Then X and Y will be random variables from Gompertz with parameters 1, and 2,, respetively. That is, the probability density funtions pdf and the umulative distribution funtions of X and Y are, respetively, and f X x exp 1xexp { 1 1 [exp 1x ] }, x >, 1 >, >, F x 1 exp { 1 1 [exp 1x ] }, f Y y exp 2yexp { 1 2 [exp 2y ] }, y >, 2 >, >, F y 1 exp { 1 2 [exp 2y ] }, where 1 and 2 are known parameters, and, are unknown parameters. Then R is 2.5 R P Y < X P Y < x f X x dx 1 k / 2 k 1/ k 2/ 1 1 exp {/ 1 + / 2} k! k [ 1 i / 1 2/ 1 ] k+i+1 Γ 2/ 1k / 1k + i + 1 i! If 1 2, then R has the following form: 2.6 R P Y < X P Y < xf X x dx [ { 1 exp λ2 ex λ2 +. }] i { } e x exp ex dx

3 Comparison of Estimators 341 Let X 1, X 2,..., X n and Y 1, Y 2,..., Y m be two independent random samples taken from the Gompertz distribution with parameters, and,, respetively, and let be known. To obtain the MLE for R, firstly we must obtain the MLEs for and. Using the likelihood funtion for the two random samples, the MLEs for and are, respetively, λ 1 n/ n e x i and λ 2 m/ m e y j. Hene R 1, the MLE of R, is written as follows; i1 2.7 R1 +. The distribution of R 1 an be found as follows; Γ n + m n f R1 r 1 Γ n Γ m 2.8 j1 n r n r 1 } n+m {1 r 1 1 n, with < r 1 < 1. For s >, the s th moment of R 1 is given by n Γ n + s Γn + m n E Rs 1 Γ n + m + s Γ n mλ F 2,1 n + s, n + m, n + m + s,1 n, where F p,q n,d, r is the generalized hypergeometri funtion. This funtion is also known as Barnes s extended hypergeometri funtion. The definition of F p,qn,d, r is as follows; 2.1 F p,qn,d, r k r k p Γn i + k Γ 1 n i i1 Γ k + 1 q, Γ d i + k Γ 1 d i where n [n 1, n 2,..., n p], p is the number of operands of n, d [d 1, d 2,..., d q] and q is the number of operands of d. The above generalized hypergeometri funtion is quikly evaluated and readily available in standard software programmes suh as Maple. By substituting s 1 in Eq. 2.9, the expeted value of R 1 an be found as follows; 2.11 E R 1 n n + m n i1 n F 2,1 n + 1, n + m, n + m + 1, 1 n. Using Eq. 2.9, the mean squared error of the R 1 is obtained as follows 2 MSE R1 E R1 R n nn + 1 n n + mn + m F 2,1 n + 2, n + m, n + m + 2,1 n 2n n λ2 n n + m + see Saraçoğlu and Kaya, [14]. 2 λ2 + + F 2,1 n + 1, n + m, n + m + 1, 1 n,

4 342 B. Saraçoğlu, M.F. Kaya, A.M. Abd-Elfattah 3. Uniformly minimum variane unbiased estimation Let θ, Θ, be an arbitrary value of θ. Denote by gt θ and g θ T X 1 x 1,..., X n x k, Y 1 y 1,..., Y k y k the pdf of TX, Y and the onditional density of T for given X j x j, Y j y j, j 1,..., k, respetively, at θ θ. Then the UMUE of fx 1,..., x k ; y 1,..., y k θ is of the form 3.1 ˆfx 1,..., x k ; y 1,..., y k k j1 fx j, y g θ T X 1 x 1,..., X k x k, Y 1 y 1,..., Y k y k j. gt θ The UMUE of R is in the form 3.2 R2 Iy < x fx,ydx dy, Lumelski and Sapoznikov [11]. Let X 1, X 2,..., X n and Y 1, Y 2,..., Y m be the two independent random samples from the Gompertz distribution, with parameters, and,, respetively, and let is known. To find the UMUE ˆR 2 for R, firstly we must find ˆfx 1,..., x k ; y 1,..., y k, whih is the UMUE for fx 1,..., x k ; y 1,..., y k. Now W 1 n i1 ex i and 1 m i1 ey i have Gamma distributions with parameters n, and m,, respetively. If is known, the Gompertz family is an exponential family. Therefore W and are suffiient and omplete. The probability density funtions of W and are respetively f Ww λn 1 Γn wn 1 e w, w >, > f v λm 2 Γm v e v, v >, >. Sine X and Y are independent, Eq. 3.1 an be written as follows, 3.5 ˆfx1,..., x k ; y 1,..., y k ˆfx 1,..., x k ˆfy 1,..., y k. Then 3.6 with 3.7 ˆfx 1,..., x k ; y 1,..., y k k j1 fx j gw X1 x1,..., X k x k gw k j1 fy j gv Y1 y1,..., Y k y k gv gw X 1 x 1,..., X k x k λn k 1 e x Γn k i n k 1 w i1 e x i exp { λ } 1 w i1 e x i I w i1

5 Comparison of Estimators 343 and 3.8 gv Y 1 y 1,..., Y k y k λm k 2 e y Γm k i m k 1 v i1 e y i exp { λ } 2 v i1 e y i I v. i1 Substituting Eq.2.1, Eq.2.3, Eq.3.7 and Eq.3.8 in Eq. 3.6, ˆfx 1,..., x k ; y 1,..., y k is obtained as 3.9 ˆfx 1,..., x k ; y 1,..., y k { } ΓnΓm exp x Γn kγm kw n 1 v i + y i i1 i1 e x i n k 1 e y i m k 1 w v i1 i1 e x i e y i I w I v. i1 i1 From Eq.3.2, the UMUE for R is ˆR 2 fx 1, y 1/w, vdx 1dy where G n m w n 1 v n 2 m 2 e x+y w ex v ey d y d x, G G { x, y : < x < From Eq.3.1, for w < v, lnw + 1, < y < } lnv + 1, y < x ˆR 2 n m w n 1 v lnw+1 x x y 1 n w n 1 v n 2 m 2 e x+y w ex v ey d y d x w u k v u 1 k m, k v k u k v u w u n 2 du

6 344 B. Saraçoğlu, M.F. Kaya, A.M. Abd-Elfattah fulfil Eq.3.12 to Eq.3.11, so the UMUE for R is obtained as follows: 3.13 ˆR k ΓnΓm w k, w < v. Γn + kγm k v k For v < w the UMUE for R is obtained as follows 3.14 ˆR 2 n m w n 1 v lnv+1 y lnw+1 xy n 2 m 2 e x+y w ex v ey d x d y n 1 1 k ΓnΓm v k, v < w. Γn kγm + k w k Thus, 1 i W d i, W < i 3.15 R2 n 1 j j, W j W where 3.16 j 1 j Γ n Γ m Γ n j Γ m + j and 3.17 d i 1 i Γ n Γ m Γn + i Γ m i. The MSE of R 2 is 3.18 MSE R2 2 E R2 R E R2 2 R 2. The seond moment of R 2 in 3.18 ould be written as 3.19 E R2 2 n 1 n 1 j j { j+j j j E W} P W W + P > W 2 d ie + i í i { W { W d id í E i > W} P > W i+í > W} P > W, where the probability density funtions of the random variables T W/ and S /W are as follows: n n+m Γ n + m tn f Tt 1 + t, t >, Γ mγn m n+m Γ n + m s λ f Ss 1 + λ2 s, s >. Γ mγn

7 Comparison of Estimators 345 Using 3.2 and 3.21, 3.22 j E W P W W j E W W 1 P W 1 1 s j f Ss ds Γ n + m Γ m Γn λ2 1 m Γ n + m Γ n Γ mm + j λ2 s m+j λ2 n+m s ds m F 2,1 n + m, m + j, m + j + 1, λ2, 3.23 i > W P > W W E W E 1 t i f Tt dt Γ n + m Γ m Γn i W W < 1 P < 1 Γ n + m Γ n Γ mn + i 1 n λ2 t n+i n+m t dt n F 2,1 n + m, n + i, n + i + 1, and 3.24 P > W P 1 W < 1 f Ttdt Γ n + m Γm Γn 1 n n Γn + m mγnγm n+m t n t dt { F 2,1 n, m + n, n, λ m n 2 + m n+1 } are obtained. The MSE of R 2 is given by:

8 346 B. Saraçoğlu, M.F. Kaya, A.M. Abd-Elfattah 3.25 MSE R2 m λ2 j j Γ n + m n 1 n 1 j j + Γ nγmm + j + j F 2,1 n + m, m + j + j, m + j + j + 1, λ2 n Γn + m {F 2,1 n, m + n, n, mγnγm λ n+m } 2 + n+ n d i Γ n + m 2 λ2 + i Γn Γ mn + i F 2,1 n + m, n + i, n + i + 1, 2 + i i n d id í Γ n + m Γ n Γ mn + i + í F 2,1 n + m,n + i + í, n + i + í + 1, where F p,q n,d, r is the generalized hypergeometri funtion in Eq.2.1, j and d i are defined in 3.16 and 3.17, respetively., 4. Bayes estimation In this setion we onsider the Bayes estimator R 3 for R with respet to the mean square error. Let X 1, X 2,..., X n and Y 1, Y 2,..., Y m be the two independent random samples from a Gompertz distribution with parameters, and,, respetively. Assuming is known, the likelihood funtions are 4.1 L 1x 1,..., x n λ n 1 exp and 4.2 L 2y 1,..., y n λ m 2 exp Let and be independent with priors n x i exp i1 i1 m y i exp λ2 n e x i i1 m e y i. i π βα 1 1 Γα 1 λα e β 1, α 1 >, β 1 >, >, π βα 2 2 Γα 1 λα e β 2, α 2 >, β 2 >, >,

9 Comparison of Estimators 347 respetively. The posterior distributions of and are as follows. 4.5 where π, X, Y f x, y, Π, f x, y, Π, d d β1 + k1n+α 1 β 2 + k 2 m+α 2 λ n+α λ m+α n + α 1 m + α 2 exp { β 1 + k 1 β 2 + k 2}, k 1 1 n 4.6 i1 ex i, k 2 1 m 4.7 i1 ey i. Let r / + and u +, u >, < r < 1. Then 4.8 where π r, u X, Y Sk 1, k 2, ru n+m+α 1+α 2 1 exp { u1 r β 1 + k 1 ru β 2 + k 2} 4.9 Sk 1, k 2, r β1 + k1n+α 1 β 2 + k 2 m+α 2 r m+α2 1 1 r n+α 1 1. n + α 1 m + α 2 The marginal posterior distribution of R is 4.1 π R r X, Y u u n+m+α 1+α 2 1 Sk 1, k 2, r exp { u1 r β 1 + k 1 ruβ 2 + k 2} du Sk 1, k 2, r n + m + α 1 + α 2 [1 r β 1 + k 1 + r β 2 + k 2] n+m+α 1+α 2. Hene the Bayes Estimator R 3 of R with respet to the mean squared error loss funtion is the posterior mean 4.11 where R 3 1 r r.π R r x, y dr 1 vδ 1 1 vδ 2 2 δ1 + δ2 r δ 2 1 r δ 1 1 δ 1 δ 2 r [1 rv 1 + rv 2] δ 1+δ 2 dr δ2 v 2 δ 2 v 1 δ 1 +δ 2 F 2,1 δ 1 + δ 2, δ 2 + 1, δ 1 + δ 2 + 1, v 2 v 1 v 1, v 2 v 1, δ1 v 1 δ 2 v 2 δ 1 +δ 2 F 2,1 δ 1 + δ 2, δ 1, δ 1 + δ 2 + 1, v 2 v 1 v 2, v 2 > v 1, 4.12 δ 1 n + α 1 δ 2 m + α 2, v 1 β 1 + k 1, v 2 β 2 + k Comparison of the estimators In this setion, the software pakage MathCAD 21 was used for the simulation study. The numerial illustration was arried out to obtain and study the properties of the MLE, UMUE and Bayes Estimators for R. An extensive numerial investigation was used to make a omparison among the three estimators, MLE, UMUE and Bayes estimators for R. The following steps were onsidered when obtaining these numerial results: Step 1. First, 1 uniform,1 random samples were generated. The usual transformation tehnique was used to get the orresponding Gompertz random samples. In

10 348 B. Saraçoğlu, M.F. Kaya, A.M. Abd-Elfattah this way, samples X 1,..., X n from the Gompertz distribution with sample sizes n 5, 2 and 5, and sale parameter 4 were obtained. Table 1. The MSE values of the MLE, UMUE and Bayes Estimator samp. param. reliability MLE UMUE Bayes n m R R1 MSE R 1 R2 MSE R 2 R3 MSE R Step 2. Similarly, 1 random samples Y 1,..., Y m were generated from the Gompertz distribution with sample sizes m 3, 5 and 1, and sale parameters 2, 4 and 6. The MLE for the parameters were obtained using n n/ e x i and λ m 2 m/ e y j. i1 Step 3. Using the results for the MLE of the two unknown parameters of the Gompertz distribution and using Eq. 2.7, the MLE R 1 of R was obtained. Then using Eq the MSE of UMUE was alulated. Step 4. The MathCAD program and Eq was used for omputing the UMUE R 2 of R. Then using Eq.3.25 the MSE of UMUE was omputed. Step 5. To obtain the Bayes estimator of R under the Gompertz distribution, the simulated random samples with different values of the parameters and sample sizes were j1

11 Comparison of Estimators 349 used with Eq. 4.12, the values of the prior parameters being β 1 2, α 1 3, β 2 2 and α 2 3. Then the MSE for the bayes estimator was omputed using simulation. Table 1 shows the simulation results for the MSEs of the MLE, UMUE and Bayes estimators for R, for different values of the parameters and sample sizes. From Table 1, we note that the mean square error of Bayes is smaller than the mean square error for maximum likelihood, and the minimum variane unbiased, exept for small sample size and with R Also, maximum likelihood has mean square error less than UMUE. It an be noted that the MSE of the three estimators derease with inreasing sample size n and fixed sample size m, also MSE inreases with inrease sample size m and fixed n. Referenes [1] Awad, A. M. and Gharraf, M.K. Estimation of PY < X in the Burr ase: A omparative study, Commun. Statist. Simul. Comp. 152, , [2] Birnbaum, Z. W., On a use of the Mann-Whitney statisti Proeedings of the Third Berkeley Symposium Math. Statisti Probability I, 1956, [3] Chao, A. On omparing estimators of PY < X in the exponential ase, IEEE Trans. Reliab. 314, , [4] Churh, J. D. and Harris, B. The estimation of reliability from stress strength relationships, Tehnometris 12, 49 54, 197. [5] Constantine, K. and Karson, M. Estimators of PY < X in the gamma ase, Commun. Statist. Simul. Comp. 15, , [6] Ismail, A. A. On the optimal design of step-stress partially aelerated life tests for the Gompertz distribution with type-i ensoring, [7] Jaheen, Z. F. A Bayesian analysis of reord statistis from the Gompertz model, Appl. Math. Comp , 37 32, 23. [8] Khedhairi, A. and Gohary, A. E. A new lass of bivariate Gompertz distributions and its mixture, Int. J. Math. Anal. 2 5, , 28. [9] Kotz, S., Lumelskii, Y. and Pensky, M. The Stress-Strength Model and its Generalizations: Theory and Appliations World Sientifi Publishing, Singapore, 23. [1] Kundu, D. and Gupta R.D. Estimation of PY < X for the generalized exponential distribution, Metrika 613, , 25. [11] Lumelski, Ya. P. and Sapoznikov, P. N. Unbiased estimators of probabilities densities, Theor. eroyat. Primen. 14, , [12] Mokhlis, N.A. Reliability of a stress-strength model with Burr Type III distributions, Commun. Statist.Theory Meth. 347, , 25. [13] Raqab M. Z. and Kundu, D. Comparison of different estimators of PY < X for a saled Burr type X distribution, Commun. Statist. Simul. Comp. 342, , 25. [14] Saraçoğlu, B. and Kaya, M. F. Maximum likelihood estimation and onfidene intervals of system reliability for Gompertz distribution in stress-strength models, Selçuk J. Appl. Math. 82, 25 36, 27. [15] Surles, J. G. and Padgett, W. J. Inferene for reliability and stress-strength for a saled Burr-type X distribution, Lifetime Data Analy. 7, 187 2, 21. [16] Wu, S. J., Chang, C. T. and Tsai, T. R. Point and interval estimations for the Gompertz distribution under progressive type-ii ensoring, Metron - International J. Stat. LXI 3, , 23. [17] Wu, J.W., Hung, W.L. and Tsai, C.H. Estimation of parameters of the Gompertz distribution using the least squares method, Appl. Math. Comp. 1581, , 24. [18] Wu, C. C., Wu, S. F. and Chan, H.Y. MLE and the estimated expeted test time for the two-parameter Gompertz distribution under progressive ensoring with binomial removals, Appl. Math. Comp. 1812, , 26.