Microsoft Word - 363-377 IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|363    Different Estimation Methods for System Reliability Multi-Components model: Exponentiated Weibull Distribution Abbas Najim Salman abbasnajim66@yahoo.com Dept. of Mathematical / College of Education (Ibn Al – Haitham)/ Bagdad University Fatima Hadi Sail sailfatima652@gmail.com Ministry of Education / Directorate of Education Al-Karkh II Abstract In this paper, estimation of system reliability of the multi-components in stress- strength model R(s,k) is considered, when the stress and strength are independent random variables and follows the Exponentiated Weibull Distribution (EWD) with known first shape parameter θ and, the second shape parameter α is unknown using different estimation methods. Comparisons among the proposed estimators through Monte Carlo simulation technique were made depend on mean squared error (MSE) criteria. Keywords: Exponentiated Weibull Distribution (EWD), Reliability of multi-component Stress – Strength models R(s,k), Maximum likelihood estimator (MLE), Moment estimator (MOM), shrinkage method (Sh), Least Squares Estimator (LS), Rank Set Sampling (RSS). and mean squared error (MSE). IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|364    1. Introduction The stress-strength (S-S) reliability of the system which contains one component is denoted by R=P(Y0, 𝛼 , θ>0 (1) 𝑓 𝑦; 𝛼 , 𝜃 𝛼 𝜃𝑦 𝑒 1 𝑒 y>0, 𝛼 , θ>0 (2) while the cumulative distribution function (CDF.) is: 𝐹 𝑥, 𝛼 1 𝑒 𝑥 0, 𝛼 , 𝜃 0 (3) 𝐹 𝑦, 𝛼 1 𝑒 𝑦 0, 𝛼 , 𝜃 0 (4) In order to find a system reliability consisting of kth identical components (when at least s out of k function) for the strength X1, X2, …, Xk which are random variables with EWD (𝛼 , 𝜃 , subjected to a stress Y which is a random variable follows EWD (𝛼 , 𝜃 depend on Bhattacharyya and Johnson (1974), the reliability of a multi-component stress-strength model R(s,k) ;[5] can be calculated as R(s,k) =P(at least s of the X1, X2,…, Xk exceed Y) =∑ 1 𝐹 𝑦 𝐹 𝑦 𝑑𝐺 𝑦 =∑ 1 1 𝑒 1 𝑒 𝛼 𝜃𝑦 𝑒 1 𝑒 𝑑𝑦 =∑ 1 1 𝑒 𝛼 𝜃𝑦 𝑒 1 𝑒 𝑑𝑦 And by some of the simplification, we get R(s,k)= ∑ ! ! ∏ 𝑘 𝑗 ; where k, i and j are integers (5) IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|366    2. Estimation methods of R(s,k): 2.1 Maximum Likelihood Estimator (MLE): Suppose that k of components are put on life-testing experiment, in this case, we consider that x1,x2,…,xn are a random sample of size n follows EWD (𝛼 , 𝜃 , and y1,y2,…,ym be a random sample of size m follows EWD (𝛼 , 𝜃 ,. Then the likelihood function of the mentioned system will be: l≡ L(𝛼 , 𝛼 , 𝜃; 𝑥, 𝑦)=∏ 𝑓 𝑥 ∏ 𝑔 𝑦 ) =∏ 𝛼 𝜃𝑥 𝑒 1 𝑒 ∏ 𝛼 𝜃𝑦 𝑒 1 𝑒 =𝛼 𝜃 ∏ 𝑥 𝑒 ∑ ∏ 1 𝑒 𝛼 𝜃 ∏ 𝑦 𝑒 ∑ ∏ 1 𝑒 Take logarithm for both sides, we get: Ln 𝑙 𝑛𝑙𝑛𝛼 𝑛𝑙𝑛𝜃 𝜃 1 ∑ 𝑙𝑛𝑥 ∑ 𝑥 𝛼 1 ∑ 1 𝑒 𝑚𝑙𝑛𝛼 𝑚𝑙𝑛𝜃 𝜃 1 ∑ 𝑙𝑛𝑦 ∑ 𝑦 𝛼 1 ∑ 1 𝑒 Derive the above equation with respect to the unknown shape parameters 𝛼 (i=1,2) and equating the result to zero, we get: 𝑑Ln 𝑙 𝑑𝛼 𝑛 𝛼 𝑙𝑛 1 𝑒 0 𝑑Ln 𝑙 𝑑𝛼 𝑚 𝛼 𝑙𝑛 1 𝑒 0 Thus, the maximum likelihood estimator of the parameter 𝛼 (i=1,2) will be as follows: 𝛼 ∑ (6) 𝛼 ∑ (7) By substituting 𝛼 (i=1,2) in equation (5) we get the reliability estimation for R(s,k) model via Maximum Likelihood method as below: 𝑅 , ∑ ! ! ∏ 𝑘 𝑗 (8) IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|367    2.2 Moment Method (MOM) R(s,k): Let x1, x2, …, xn be a random sample of size n for strength X follows EW (𝛼 , 𝜃 , and y1 , y2, …, ym be a random sample of size m for stress Y follows EW (𝛼 , 𝜃 . Let 𝑋 and 𝑌 are the means of samples of strength and stress respectively, then the population moments of X, Y are given by; see[20]. 𝐸 𝑋 = 𝛼 ∑ 1 𝑗 1 𝛤 1 , 𝑖𝑓𝛼 ∈ 𝑁 𝛼 ∑ ! 1 𝑗 1 𝛤 1 , 𝑖𝑓 𝛼 ∉ 𝑁 for r=1,2,3…. Where 𝛼𝑃 =α (α-1) (α-2)….. (α-j+1) and N is the set of natural number Therefore, the population means of X and Y are respectively as below: E(X)= 𝛼 ∑ 1 𝑗 1 𝛤 1 , 𝑖𝑓𝛼 ∈ 𝑁 𝛼 ∑ ! 1 𝑗 1 𝛤 1 , 𝑖𝑓 𝛼 ∉ 𝑁 … (9) And, E Y 𝛼 ∑ 1 𝑗 1 𝛤 1 , 𝑖𝑓𝛼 ∈ 𝑁 𝛼 ∑ ! 1 𝑗 1 𝛤 1 , 𝑖𝑓 𝛼 ∉ 𝑁 (10) Equating the samples mean with the corresponding populations mean for both X and Y as follows:- 𝑋 ∑ 𝛼 ∑ 1 𝑗 1 𝛤 1 𝑌 ∑ =𝛼 ∑ 1 𝑗 1 𝛤 1 By simplification, we obtain the estimation of unknown shape parameters 𝛼 (i=1,2) using moment method as follows: 𝛼 ∑ (11) 𝛼 ∑ (12) IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|368    Substitution 𝛼 (i=1,2) in equation (5), we conclude the reliability estimation for R(s,k) model via moment method as below: 𝑅 , ∑ ! ! ∏ 𝑘 𝑗 (13) 2.3 Shrinkage Estimation Method [1, 4, 5, 25 & 28]. Thompson in 1968 has suggested the problem of shrink a usual estimator 𝛼 of the parameter 𝛼 to prior information 𝛼 using shrinkage weight factor ∅ 𝛼), such that 0 ∅ 𝛼 1. Thompson says that "We are estimating 𝛼 and we believe 𝛼 is closed to the true value of 𝛼 or we fear that 𝛼 may be near the true value of 𝛼, that is mean something bad happens if 𝛼 𝛼 and we do not use 𝛼 ". Thus, the form of shrinkage estimator of α say 𝛼 will be: 𝛼 ∅ 𝛼 𝛼 1 ∅ 𝛼 𝛼 (14) In this work, we apply the unbiased estimator 𝛼 as a usual estimator and the moment estimator as a prior estimation of α in equation (14) above. Where ∅ 𝛼 denote the shrinkage weight factor as we mentioned above such that 0 ∅ 𝛼 1, which may be a function of 𝛼 ; can be found by minimizing the mean square error of 𝛼 .Thus, the shrinkage estimator for the shape parameter 𝛼 of EWD will be as follows: 𝛼 ∅ 𝛼 𝛼 1 ∅ 𝛼 𝛼 (15) Note that, 𝛼 𝛼 ∑ , Hence, E(𝛼 )= 𝛼 and Var(𝛼 )= And, 𝛼 𝑚 1 𝑚 𝛼 𝑚 1 ∑ 𝑙𝑛 1 𝑒 Implies, E(𝛼 )= 𝛼 and Var(𝛼 )= . 2.4 Modified Thompson type shrinkage weight function (Th) In this subsection, we introduce and modify the shrinkage weight factor consider by Thompson in 1968 as below. IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|369    𝛾 𝛼 0.01 for i=1,2 (16) Therefore, the shrinkage estimator of 𝛼 (i=1,2) by using above modified shrinkage weight factor will be: 𝛼 𝛾 𝛼 𝛼 1 𝛾 𝛼 𝛼 for i=1,2 (17) Substitute equation (17) in equation (5), we conclude the reliability estimation of R(s,k) based on modified Thompson type shrinkage weight factor as follows: 𝑅 , ∑ ! ! ∏ 𝑘 𝑗 (18) 2.5 Least Squares Estimator Method (LS) [2 & 13] In this subsection, we discuss the estimator of Least Squares method, this method used for many mathematical and engineering application [2]. The main idea for this method by minimizes the sum of squared error between the values and the expected value. To find the parameters αi (i=1,2) via mentioned least squares method, let: S=∑ 𝐹 𝑥 𝐸 𝐹 𝑥 i=1,2,3……n (19) Where, 𝐹 𝑥 refer to the CDF of two parameters EWD which is defined in equation (3) as follows: 𝐹 𝑥 = 1 𝑒 And, 𝐸 𝐹 𝑥 =𝑃 Such as; 𝑃 = , i=1,2,…,n 𝐹 𝑥 𝐸 𝐹 𝑥 1 𝑒 = (20) Now, taking the natural logarithm of both sides in equation (20) as below: 𝛼 Ln 1 𝑒 =Ln𝑃 𝛼 Ln 1 𝑒 -Ln𝑃 =0 (21) Now, by putting equation (21) in equation (19) we obtain: IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|370    𝑆 ∑ 𝛼 Ln 1 𝑒 Ln𝑃 (22) And be finding the partial derivatives for the equation (22) with respect to α1. 2 ∑ 𝛼 Ln 1 𝑒 Ln𝑃 Ln 1 𝑒 (23) Then, equal the equation (23) to zero get as: ∑ 𝛼 Ln 1 𝑒 – Ln𝑃 Ln 1 𝑒 0 ∑ 𝛼 Ln 1 𝑒 Ln 1 𝑒 ∑ Ln𝑃 Ln 1 𝑒 𝛼 = ∑ ∑ (24) By same way, one can estimate the parameter 𝛼 for Y which is represent stress random variable and follows EWD (α2,θ) with size m as follow: 𝛼 = ∑ ∑ (25) ; 𝑃 = , j=1,2,…,m. Put 𝛼 and 𝛼 in equation (5), will be obtain the reliability estimation of R(s,k) based on LS as follows: 𝑹 𝒔,𝒌 𝑳𝑺 𝜶𝟐𝑳𝑺 𝜶𝟏𝑳𝑺 ∑ 𝒌! 𝒌 𝒊 ! ∏ 𝒌 𝜶𝟐𝑳𝑺 𝜶𝟏𝑳𝑺 𝒋𝒊𝒋 𝟎 𝟏𝒌 𝒊 𝒔 i, j, k are integer (26) Ranked Set Sampling Method (RSS) [8, 12, 26 & 27] “Ranked set sampling was introduced and applied to the problem of estimating mean pasture yields by Mclntyre in 1952 this function was to improve the efficiency of the sample mean as an estimator of the population mean in situations in which the characteristic of interest was difficult or expensive to measure, but using ranked, it become cheaper” [27 & 12] “The concept of rank set sampling is a recent development that enables one to provide more structure to the collected sample items” [26] Let x1, x2, …, xn be a random sample EWD, let x(1), x(2),…, x(n) be order statistics increasing order. The PDF of x(q) is: IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|371    𝑓 𝑥 ! ! ! 𝐹 𝑥 1 𝐹 𝑥 𝑓 𝑥 (27) by substituting the PDF from equation (1) and the CDF from equation (3) in equation (27) will get: 𝑓 𝑥 𝑛! 𝑞 1 ! 𝑛 𝑞 ! 1 𝑒 1 1 𝑒 𝛼 𝜃𝑥 𝑒 1 𝑒 … (28) 𝑓 𝑥 Q𝛼 𝜃𝑥 𝑒 1 𝑒 1 1 𝑒 Such that; Q ! ! ! the likelihood function of order sample x(1),x(2),…,x(n) is: L 𝑥 , 𝑥 , … , 𝑥 ; 𝛼 , 𝜃 𝑄 𝛼 𝜃 ∏ 𝑥 ∏ 𝑒 ∏ 1 𝑒 ∏ 1 1 𝑒 …(29) Where, take (Ln) for both side of the equation (29) as below: Ln 𝑙 nLnQ nLn𝛼 nLn𝜃 𝜃 1 ∑ 𝐿𝑛𝑥 – ∑ 𝑥 𝑞𝛼 1 ∑ 𝐿𝑛 1 𝑒 𝑛 𝑞 ∑ 𝐿𝑛 1 1 𝑒 Take the partial derivative with respect to the parameter 𝛼 get the following: ∑ 𝑞 𝐿𝑛 1 𝑒 ∑ 𝑛 𝑞 (30) Now, equal the equation (30) to zero, we get: ∑ 𝑞 𝐿𝑛 1 𝑒 ∑ 𝑛 𝑞 0 ∑ 𝑛 𝑞 ∑ 𝑞 𝐿𝑛 1 𝑒 Then, IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|372    𝛼 ∑ ∑ (31) And by same way assume the stress y random sample follows EWD with two parameters (α2and θ) with size m to find α as below: 𝛼 ∑ ∑ (32) Substitution the estimation parameter 𝛼 (i=1,2) in equation (5) will be obtain the reliability estimation of R(s,k) based on RSS as follows: 𝑹 𝒔,𝒌 𝑹𝑺𝑺 𝜶𝟐𝑹𝑺𝑺 𝜶𝟏𝑹𝑺𝑺 ∑ 𝒌! 𝒌 𝒊 ! ∏ 𝒌 𝜶𝟐𝑹𝑺𝑺 𝜶𝟏𝑹𝑺𝑺 𝒋𝒊𝒋 𝟎 𝟏𝒌 𝒊 𝒔 (33) Simulation study In this section we numerical results were studied to compare the performance of the different estimators of reliability which is obtained in section 2, using different sample size =(25,50,75 and 100), based on 1000 replication via MSE criteria. For this purpose, Monte Carlo simulation was used the following steps:-[14] Step1: We generate the random sample which follows the continuance uniform distribution defined on the interval (0,1) as u1,u2,…, un. Step2: We generate the random sample which follows the continuance uniform distribution defined on the interval (0, 1) as w1, w2,…, wm. Step3: Transform the uniform random samples in step1 to random samples follows EWD, applying the theorem that using the inverse cumulative probability distribution function (CDF) as below shown: F x 1 𝑒 𝑈 1 𝑒 𝑥 ln 1 𝑈 And, by the same method, we get: 𝑦 ln 1 𝑊 Step4: We recall the R(s,k) from equation (5). IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|373    Step5: We compute the Maximum Likelihood Estimator of the R(s,k) using equation (8). Step6: We compute the Moment Method of R(s,k) using equation (13). Step7: We compute the Modified Thompson types Shrinkage Estimator of R(s,k) using equation (18). Step8: We compute the Least Squares method of R(s,k) using equation (26). Step9: We compute the Ranked set sampling of R(s,k) using equation (33). Step10: based on (L=1000) Replication, we calculate the MSE as follows: 𝐌𝐒𝐄 𝟏 𝑳 ∑ 𝑹 𝒔,𝒌 𝒊 𝐑 𝐬,𝐤 𝟐𝑳 𝒊 𝟏 Where 𝑅 , refers the proposed estimators of real value of reliability R , . Note that in this paper, we consider (s,k)=(2,3), (2,4) and (3,4), and all the results are put it in the tables (1-3) below: Table (1): Shown estimation value of R(s,k), and MSE when (s,k)=(2,3), and α1=2, α2=5, 𝜽=3; R(s,k)= 0.24242 𝑹 𝒔,𝒌 MSE (n,m) 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝑻𝒉 𝑹 𝒔,𝒌 𝑳𝑺 𝑹 𝒔,𝒌 𝑹𝑺𝑺 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝑻𝒉 𝑹 𝒔,𝒌 𝑳𝑺 𝑹 𝒔,𝒌 𝑹𝑺𝑺 Best (25,25) 0.24791 0.24283 0.49692 0.23294 0.00488 0.00024 0.08271 0.00029 Th (25,50) 0.43344 0.24312 0.17813 0.43157 0.04141 0.00019 0.01516 0.03600 Th (25,75) 0.55136 0.24343 0.01305 0.55219 0.09928 0.00018 0.05345 0.09613 Th (25,100) 0.62907 0.24342 0.03564 0.63177 0.15264 0.00018 0.04533 0.15174 Th (50,25) 0.10944 0.24014 0.58209 0.09797 0.01892 0.00016 0.12174 0.02090 Th (50,50) 0.24405 0.24254 0.13274 0.23333 0.00253 0.00012 0.01865 0.00018 Th (50,75) 0.35270 0.24296 0.14738 0.34225 0.01472 0.00010 0.01531 0.01007 Th (50,100) 0.43686 0.24371 0.08694 0.42684 0.04038 0.00010 0.02523 0.03411 Th (75,25) 0.06155 0.23740 0.38130 0.05372 0.03315 0.00016 0.02095 0.03561 Th (75,50) 0.15610 0.24118 0.36197 0.14613 0.00864 0.00009 0.02494 0.00931 Th (75,75) 0.24192 0.24214 0.15093 0.23350 0.00155 0.00007 0.00976 0.00013 Th (75,100) 0.31836 0.24290 0.11883 0.30854 0.00755 0.00007 0.01784 0.00444 Th (100,25) 0.03960 0.23560 0.72816 0.03387 0.04134 0.00019 0.24874 0.04349 Th (100,50) 0.11040 0.24081 0.57515 0.09964 0.01807 0.00008 0.11484 0.02040 Th (100,75) 0.17674 0.24118 0.17903 0.16956 0.00535 0.00007 0.00824 0.00534 Th (100,100) 0.24444 0.24270 0.12698 0.23309 0.00119 0.00005 0.01604 0.00013 Th IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|374    Table (2): Shown estimation value of R(s,k), and MSE when (s,k)=(2,4), and α1=2, α2=5, 𝜽=3; R(s,k)= 0.33566 𝑹 𝒔,𝒌 MSE (n,m) 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝑻𝒉 𝑹 𝒔,𝒌 𝑳𝑺 𝑹 𝒔,𝒌 𝑹𝑺𝑺 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝑻𝒉 𝑹 𝒔,𝒌 𝑳𝑺 𝑹 𝒔,𝒌 𝑹𝑺𝑺 Best (25,25) 0.33902 0.33616 0.40165 0.32468 0.00612 0.00030 0.01323 0.00039 Th  (25,50) 0.53639 0.33595 0.14739 0.53480 0.04521 0.00027 0.03988 0.03988 Th  (25,75) 0.65045 0.33767 0.07510 0.64660 0.10217 0.00023 0.06972 0.09682 Th (25,100) 0.71426 0.33733 0.02873 0.71654 0.14586 0.00025 0.09450 0.14518 Th (50,25) 0.16928 0.33330 0.46971 0.15389 0.02996 0.00021 0.03347 0.03312 Th  (50,50) 0.33761 0.33591 0.34261 0.32479 0.00293 0.00014 0.01547 0.00023 Th (50,75) 0.45225 0.33613 0.19639 0.44531 0.01634 0.00013 0.02012 0.01213 Th  (50,100) 0.53925 0.33703 0.16807 0.53043 0.04380 0.00013 0.03956 0.03803 Th  (75,25) 0.10250 0.33124 0.86866 0.08935 0.05540 0.00021 0.28612 0.06070 Th  (75,50) 0.23217 0.33458 0.43460 0.21818 0.01276 0.00013 0.01351 0.01387 Th  (75,75) 0.33637 0.33573 0.37699 0.32495 0.00206 0.00010 0.00330 0.00019 Th  (75,100) 0.42069 0.33658 0.30343 0.40879 0.00911 0.00009 0.00514 0.00542 Th  (100,25) 0.06675 0.32741 0.37284 0.05869 0.07278 0.00025 0.08851 0.07672 Th  (100,50) 0.16750 0.33292 0.60934 0.15680 0.02943 0.00012 0.07658 0.03203 Th  (100,75) 0.25687 0.33418 0.54797 0.24809 0.00760 0.00009 0.05578 0.00771 Th  (100,100) 0.33762 0.33584 0.43285 0.32471 0.00166 0.00008 0.01129 0.00018 Th  IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|375    Table (3): Shown estimation value of R(s,k), and MSE when (s,k)=(3,4), and α1=2, α2=5, 𝜽=3; R(s,k)= 0.14918 𝑹 𝒔,𝒌 MSE (n,m) 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝑻𝒉 𝑹 𝒔,𝒌 𝑳𝑺 𝑹 𝒔,𝒌 𝑹𝑺𝑺 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝑻𝒉 𝑹 𝒔,𝒌 𝑳𝑺 𝑹 𝒔,𝒌 𝑹𝑺𝑺 Bes t (25,25) 0.1582 9 0.1495 4 0.0540 2 0.1408 0 0.00367 0.0001 7 0.0139 8 0.0002 0 Th (25,50) 0.3371 9 0.1501 3 0.0474 5 0.3271 4 0.04068 0.0001 4 0.0105 6 0.0319 0 Th  (25,75) 0.4590 6 0.1504 9 0.1145 1 0.4567 7 0.10095 0.0001 3 0.0262 0 0.0948 4 Th  (25,100) 0.5477 3 0.1507 9 0.0221 2 0.5457 3 0.16312 0.0001 3 0.0163 3 0.1574 4 Th  (50,25) 0.0499 7 0.1471 0 0.4219 2 0.0421 1 0.01034 0.0001 1 0.0833 1 0.0114 7 Th  (50,50) 0.1528 9 0.1493 4 0.1268 3 0.1413 3 0.00185 0.0000 8 0.0039 1 0.0001 3 Th  (50,75) 0.2508 5 0.1497 2 0.0462 8 0.2396 3 0.01302 0.0000 8 0.0113 8 0.0082 8 Th  (50,100) 0.3325 6 0.1500 0 0.1722 7 0.3236 0 0.03638 0.0000 7 0.0107 2 0.0305 3 Th (75,25) 0.0229 0 0.1457 2 0.3805 0 0.0177 6 0.01607 0.0001 1 0.0807 0 0.0172 7 Th  (75,50) 0.0833 0 0.1485 3 0.2898 8 0.0734 3 0.00499 0.0000 6 0.0260 3 0.0057 5 Th  (75,75) 0.1528 7 0.1494 8 0.0986 3 0.1411 4 0.00125 0.0000 5 0.0034 3 0.0001 0 Th  (75,100) 0.2194 8 0.1497 7 0.0292 0 0.2074 7 0.00647 0.0000 5 0.0155 1 0.0034 5 Th (100,25) 0.0121 0 0.1438 6 0.4206 8 0.0091 1 0.01883 0.0001 2 0.0841 6 0.0196 2 Th  (100,50) 0.0491 3 0.1475 1 0.3445 1 0.0431 9 0.01025 0.0000 6 0.0419 0 0.0112 4 Th  (100,75) 0.0997 1 0.1489 5 0.4359 3 0.0901 9 0.00300 0.0000 4 0.0955 7 0.0034 9 Th  (100,10 0) 0.1501 1 0.1490 5 0.3288 5 0.1415 2 0.00085 0.0000 4 0.0370 9 0.0000 8 Th IHSCICONF 2017 Special Issue Ibn Al-Haitham Journal for Pure and Applied science https://doi.org/ 10.30526/2017.IHSCICONF.1809 For more information about the Conference please visit the websites: http://www.ihsciconf.org/conf/   www.ihsciconf.org   Mathematics|376    3. Discussion Numerical Simulation Results: For all n=(25,50,75, and100) and for all m=(25,50,75, and100), in this work, the minimum mean square error (MSE) for reliability estimation of R(s,k) model for the exponentiated Weibull distribution is held using Modified Thompson type shrinkage estimator 𝑹 𝒔,𝒌 𝑻𝒉. This implies that, the shrinkage for reliability estimation (𝑹 𝒔,𝒌 𝑻𝒉) is the best and follows by using. For any n, some of the proposed estimator (MLE, LS, and RSS) with m the methods are vibration. Finally, for any n and m the second order best estimator is Ranked Set Sampling (𝑹 𝒔,𝒌 𝑹𝑺𝑺) or Maximum Likelihood Estimator (𝑹 𝒔,𝒌 𝑴𝑳𝑬). 4. Conclusion: From the numerical results, one can find the proposal using Modified Thompson type shrinkage estimator 𝑹 𝒔,𝒌 𝑻𝒉) which depends on unbiased estimator and prior estimate (moment method) as a linear combination, performance good behavior and it is the best estimator than the others in the sense of MSE. 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