Microsoft Word - 147-156     147 Ibn Al-Haitham Jour. for Pure & Appl.Sci. IHJPAS  https://doi.org/10.30526/32.1.1825 Vol. 32 (1) 2019 On Shrinkage Estimation for R (s, k) in Case of Exponentiated Pareto Distribution Eman Ahmed Abdulateef emanahmed88717@gmail.com Abbas Najim Salman abbasnajim66@yahoo.com  Department of Mathematics, College of Education for Pure Science Ibn Al – Haitham, University of Baghdad, Baghdad, Iraq. Article history: Received 2 May 2018, Accepted 26 September 2018, Publish January 2019 Abstract This paper concerns with deriving and estimating the reliability of the multicomponent system in stress-strength model R(s,k), when the stress and strength are identical independent distribution (iid), follows two parameters Exponentiated Pareto Distribution (EPD) with the unknown shape and known scale parameters. Shrinkage estimation method including Maximum likelihood estimator (MLE), has been considered. Comparisons among the proposed estimators were made depending on simulation based on mean squared error (MSE) criteria. Keywords: Exponentiated Pareto Distribution(EPD), Reliability of multi-component Stress – Strength models R(s,k), Maximum likelihood estimator (MLE), Shrinkage estimator and mean squared error (MSE). 1. Introduction The reliability of the multicomponent stress-strength model (s out of k) system, denoted by R(s,k)refers to the system functioning when at minimums (1≤s≤k) of components survive. In other words, this system works well if at least s out of k components resist the stress. Bhattacharyya and Johnson in (1974) was the first who studied and derived R(s,k) [1]. Noted that, when s=1 and s=k is respectively referring to parallel and series systems. The model mentioned used in many applications in physics and engineering and many authors had studied and estimated R(s,k)for example: Afify in (2010), showed that the Exponentiated Pareto distribution denoted by EP (α, λ) used quite successfully in studying many lifetime data and the EP (𝛼, λ) decreasing and upside-down bathtub shaped failure rates depending on shape parameter α[2]. Hassan& Basheikh in (2012), estimated 𝑅 , using Bayes and non-Bayes estimation methods when the strength and stress are non-identical and follows the Exponentiated Pareto distribution [3], Rao et al in (2016), estimated the reliability system in a multicomponent stress-strength when stress and strength follows Exponentiated Weibull distribution for different shape parameters [4], and in (2017) Abbas and Fatima, they estimated the reliability of the multicomponent system in stress–strength model for Exponentiated Weibull distribution, using; ML,     148 Ibn Al-Haitham Jour. for Pure & Appl.Sci. IHJPAS  https://doi.org/10.30526/32.1.1825 Vol. 32 (1) 2019 MOM and the conclude results approved that the Shrinkage estimator using Shrinkage weight function was the best[5]. In this paper we estimate R(s,k) based on Exponentiated Pareto distribution EP(α, λ) with unknown shape parameter α and known scale parameter λ using several shrinkage estimation methods depends on (MLE) methods and make a comparison of the considered estimation methods through Monte Carlo simulation via mean squared error (MSE) criteria. It is well known, the EP (α, λ) is a special case of Exponentiated Lomax distribution (ELD), when (𝜖 1 where the CDF of ELD has the form below [6]. 𝐹 𝑋 1 1 𝜖𝑥 𝑥 0; 𝛼, 𝜆𝑎𝑛𝑑 𝜖 0 (1) Let X be a random variable follows two parameters Exponentiated Pareto distribution EPD (α, λ). The probability density function (p. d. f.) of X will be. 𝑓 𝑥; 𝛼, 𝜆 𝛼𝜆 1 𝑥 1 1 𝑥 ; For x 0; 𝛼, 𝜆 0 0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (2) Here, 𝛼 refers to shape parameter and 𝜆 refers to scale parameter [2]. Implies, the cumulative distribution function (CDF) of X as below: 𝐹 𝑥, 𝛼, 𝜆 1 1 𝑥 𝑥 0; 𝛼, 𝜆 0 (3) Assume 𝑋 , 𝑋 , … , 𝑋 are strength random variable follows EP(α,λ) and subject to common stress random variable Y which is distributed as EP(β,λ) . Then, the reliability system for a multicomponent in stress-strength model R(s,k) will be [1]: R(s,k) =P(at least s of the X1, X2,…, Xkexceed Y) =∑ 1 𝐹 𝑦 𝐹 𝑦 𝑓 𝑦 𝑑𝑦 (4) =∑ 1 1 1 𝑦 1 1 𝑦 𝛽𝜆 1 𝑦 1 1 𝑦 𝑑𝑦 R (s, k) = ∑ ! ! ∏ 𝑘 𝑗 ; k, i, j are integers (5) Now we estimation methods of R(s,k) by the following: 2. Maximum Likelihood Estimator (MLE) Suppose the strength random sample be of size n say 𝑥 , 𝑥 , … , 𝑥 follow EPD (𝛼, 𝜆 and 𝑦 , 𝑦 , … … , 𝑦 be the stress random sample of size m follow EPD (𝛽, 𝜆 The Maximum Likelihood function for the observed sample is given as: 𝑙 =L (𝛼, 𝛽; 𝑥, 𝑦) =∏ 𝑓 𝑥 ∏ 𝑓 𝑦 ) (6) From equation (2) and the equation (6) become 𝑙=𝛼 𝛽 𝜆 𝑛 𝑚 ∏ 1 𝑥𝑖 𝜆 1 ∏ 1 1 𝑥𝑖 𝜆 𝛼 1 ∏ 1 𝑦𝑗 𝜆 1 ∏ 1 1 𝑦𝑗 𝜆 𝛽 1 Take Logarithm to both sides, we get:     149 Ibn Al-Haitham Jour. for Pure & Appl.Sci. IHJPAS  https://doi.org/10.30526/32.1.1825 Vol. 32 (1) 2019 ln 𝑙 𝑛𝑙𝑛𝛼 𝑚𝑙𝑛𝛽 𝑛 𝑚 𝑙𝑛𝜆 𝜆 1 ∑ 𝑙𝑛 1 𝑥 𝛼 1 ∑ ln 1 1 𝑥 𝜆 1 ∑ 𝑙𝑛 1 𝑦 𝛽 1 ∑ 𝑙𝑛 1 1 𝑦 (7) 𝑑Ln 𝑙 𝑑𝛼 𝑛 𝛼 𝑙𝑛 1 1 𝑥 0 𝑑Ln 𝑙 𝑑𝛽 𝑚 𝛽 𝑙𝑛 1 1 𝑦 0 Thus, the Maximum Likelihood estimator for the unknown shape parameters 𝛼 and β will be respectively as follows [2]: α ∑ (8) 𝛽 ∑ (9) Note that, α and𝛽 are biased estimators for α and β respectively since E (α 𝛼 , and E (β 𝛽 Hence, α α and 𝛽 𝛽 are respectively unbiased estimators for α and β.Therefore, E (α = 𝛼 and Var (α ) = (10) E (𝛽 = 𝛽 and Var (𝛽 ) = (11) By substitute Equations (8) and (9) in equation (5), we obtain the maximum likelihood estimator for 𝑅 , as below: 𝑅 , ∑ ! ! ∏ 𝑘 𝑗 ; k, i, j are integers (12) 3. Shrinkage Estimation Method (Sh) As Thompson suggested in (1968), the shrinkage estimator of α denoted by (α ) is defined as below: α ∅ α α 1 ∅ α α (13) He shrinks a usual estimator 𝛼 to prior information𝛼 using shrinkage weight factor ∅ α and he believed 𝛼 is closed to α [5], [7]. We apply the unbiased estimator 𝛼 as a usual estimator and 𝛼 𝛼 as a prior information of α in this paper. Thus, the shrinkage estimator of the shape parameterα of EP (α,λ) will be as follows: 𝛼 ∅ 𝛼 𝛼 1 ∅ 𝛼 𝛼 (14) And the same way, the shrinkage estimator for the shape parameter β of EP(β,λ) will be as the following: 𝛽 ∅ 𝛽 𝛽 1 ∅ 𝛽 𝛽 (15)     150 Ibn Al-Haitham Jour. for Pure & Appl.Sci. IHJPAS  https://doi.org/10.30526/32.1.1825 Vol. 32 (1) 2019 3.1. The Shrinkage Weight Function (Sh1): In this subsection, the shrinkage weight factor will be considered as a function of sizes n and m respectively and taking the forms below: ∅ 𝛼 |sin 𝑛/𝑛| , and ∅ 𝛽 |sin 𝑚/𝑚| where, n and m refer to the sample size of X and Y. Therefore, the shrinkage estimator using shrinkage weight function of 𝛼 and β which is defined in equations (14) and (15), will be. 𝛼 |sin 𝑛/𝑛|𝛼 1 |sin 𝑛/𝑛| 𝛼 (16) 𝛽 |sin 𝑚/𝑚|𝛽 1 |sin 𝑚/𝑚| 𝛽 (17) Then, the estimation of𝑅 , which is defined in Equation (5) using shrinkage weight function will be:- 𝑅 , ∑ ! ! ∏ 𝑘 𝑗 (18) 3.2. Constant Shrinkage Weight Function (Sh2) We suggest in this subsection constant shrinkage weight factor ∅ 𝛼 0.1, and∅ 𝛽 0.1. Therefore, the shrinkage estimator using specific constant weight factor will be as follows: 𝛼 0.1 𝛼 0.9 𝛼 (19) 𝛽 0.1 𝛽 0.9 𝛽 (20) Substitute equation (19) and (20) in equation (5) to obtain the shrinkage estimation of R(s,k) using the above constant shrinkage weight factor as below: 𝑅 , ∑ ! ! ∏ 𝑘 𝑗 ; k, i, j are integers (21) 3.3. Modified Thompson Type Shrinkage Weight Function (Th) In this subsection, we modify the shrinkage weight factor of Thompson type estimator as below, [7] 𝛾 𝛼 *0.001 (22) 𝛾 𝛽 ∗ 0.001 (23) where Var (α ) = and Var (𝛽 ) = Therefore, the shrinkage estimator ofα and β using modified shrinkage weight factor are respectively as below: 𝛼 𝛾 𝛼 𝛼 1 𝛾 𝛼 𝛼 (24) 𝛽 𝛾 𝛽 𝛽 1 𝛾 𝛽 𝛽 (25)     151 Ibn Al-Haitham Jour. for Pure & Appl.Sci. IHJPAS  https://doi.org/10.30526/32.1.1825 Vol. 32 (1) 2019 Substitute equation (24) and (25) in equation (5), then the shrinkage estimation of R(s,k) based on modified Thompson type shrinkage weight factor will be : 𝑅 , ∑ ! ! ∏ 𝑘 𝑗 ; k, i, j are integers (26) 4. Simulation Study In this section, numerical results were studied to compare the performance of the suggested estimators for𝑅 , , using different sample size n and m = (15, 25, 50 and 100), based on 1000 replication via MSE criteria. For this purpose, Mote Carlo simulation was employed by generating the random sample from continuous uniform distribution defined on the interval (0,1) as u1,u2,…, un; v1, v2,…, vm. Transform uniform random samples to follow EPD (α, λ) using (c.d.f) as below, [8]: 𝐹 𝑥 1 1 𝑥 → 𝑈 1 1 𝑥 → 𝑥 1 𝑈 1 ; i=1,2,…,n. And by the same way, calculate 𝑉 to obtain the 𝑦 : 𝑦 1 𝑉 1 ; j=1,2,…,m. Compute the real value of R(s,k) in equation (5) and the value of estimation methods of all suggested methods 𝑅 , , 𝑅 , , 𝑅 , and 𝑅 , in Equations (12), (18), (21) and (26) respectively. Based on (L=1000) replication, we calculate the MSE for all proposed estimation methods of R(s,k) as follows: MSE 1 𝐿 𝑅 , R , where 𝑅 , refers to the proposed estimators of real value of R , . All the results are put on the tables listed below, noted that the odd label Tables (1- 7). contain the value of real reliability R , and the value of reliability estimator 𝑅 , for different methods when (s,k)=(2,3),(2,4) and (𝛼, 𝛽 =(2.5,4) and (4,2.5) and (𝛼 , 𝛽 ) = ( 𝛼 0.001, 𝛽 0.001). and the even label tables (2-8) contain the value of MSE for the reliability estimator𝑅 , for different methods when (s,k)=(2,3),(2,4) and (𝛼, 𝛽 =(2.5,4) and (4,2.5) and (𝛼 , 𝛽 )=( 𝛼 0.001, 𝛽 0.001).     152 Ibn Al-Haitham Jour. for Pure & Appl.Sci. IHJPAS  https://doi.org/10.30526/32.1.1825 Vol. 32 (1) 2019 Table 1. Shows 𝑅 , for (s,k)=(2,3), α=2.5,β=4 & λ=2 when 𝑅 , 0.36232,. (n,m) 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝒔𝒉𝟏 𝑹 𝒔,𝒌 𝒔𝒉𝟐 𝑹 𝒔,𝒌 𝑻𝒉 (15,15) 0.37128 0.36262 0.36297 0.36237 (15,25) 0.36447 0.36214 0.36193 0.36236 (15,50) 0.37216 0.36244 0.36245 0.36236 (15,100) 0.37221 0.36231 0.36224 0.36236 (25,15) 0.36163 0.36240 0.36253 0.36236 (25,25) 0.36432 0.36236 0.36235 0.36236 (25,50) 0.37004 0.36238 0.36272 0.36236 (25,100) 0.36997 0.36237 0.36253 0.36236 (50,15) 0.35514 0.36228 0.36217 0.36236 (50,25) 0.35773 0.36234 0.36203 0.36236 (50,50) 0.36239 0.36236 0.36228 0.36236 (50,100) 0.36249 0.36235 0.36213 0.36236 (100,15) 0.35464 0.36229 0.36224 0.36236 (100,25) 0.35810 0.36235 0.36228 0.36236 (100,50) 0.35886 0.36235 0.36208 0.36236 (100,100) 0.36441 0.36237 0.36252 0.36236 Table 2. Shows MSE of 𝑅 , for (s,k)=(2,3), α=2.5,β=4 & λ=2 when 𝑅 , 0.36232. (n,m) 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝒔𝒉𝟏 𝑹 𝒔,𝒌 𝒔𝒉𝟐 𝑹 𝒔,𝒌 𝑻𝒉 Best (15,15) 0.01029 0.23E-4 0.12E-3 0.76E-8 Th (15,25) 0.00843 0.12E-4 0.96E-4 0.60E-8 Th (15,50) 0.00714 0.13E-4 0.81E-4 0.56E-8 Th (15,100) 0.00605 0.11E-4 0.66E-4 0.46E-8 Th (25,15) 0.00752 0.10E-4 0.88E-4 0.59E-8 Th (25,25) 0.00619 0.19E-6 0.68E-4 0.47E-8 Th (25,50) 0.00489 0.15E-6 0.53E-4 0.44E-8 Th (25,100) 0.00406 0.12E-6 0.45E-4 0.38E-8 Th (50,15) 0.00666 0.12E-4 0.77E-4 0.53E-8 Th (50,25) 0.00465 0.14E-6 0.50E-4 0.39E-8 Th (50,50) 0.00326 0.97E-7 0.34E-4 0.34E-8 Th (50,100) 0.00215 0.61E-7 0.22E-4 0.27E-8 Th (100,15) 0.00622 0.12E-4 0.72E-4 0.52E-8 Th (100,25) 0.00423 0.13E-6 0.45E-4 0.39E-8 Th (100,50) 0.00244 0.70E-7 0.26E-4 0.30E-8 Th (100,100) 0.00177 0.49E-7 0.18E-4 0.27E-8 Th Note: 0.95E-5=0.0000095     153 Ibn Al-Haitham Jour. for Pure & Appl.Sci. IHJPAS  https://doi.org/10.30526/32.1.1825 Vol. 32 (1) 2019 Table 3. Shows 𝑅 , for (s,k)=(2,4) α=2.5,β=4 & λ=2when 𝑅 , 0.46584. (n,m) 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝒔𝒉𝟏 𝑹 𝒔,𝒌 𝒔𝒉𝟐 𝑹 𝒔,𝒌 𝑻𝒉 (15,15) 0.46058 0.46565 0.46535 0.46588 (15,25) 0.47149 0.46584 0.46604 0.46588 (15,50) 0.47133 0.46592 0.46576 0.46588 (15,100) 0.47616 0.46589 0.46605 0.46588 (25,15) 0.45646 0.46576 0.46541 0.46588 (25,25) 0.46517 0.46588 0.46589 0.46588 (25,50) 0.46751 0.46588 0.46577 0.46588 (25,100) 0.47039 0.46588 0.46589 0.46588 (50,15) 0.46224 0.46605 0.46635 0.46588 (50,25) 0.46428 0.46589 0.46609 0.46589 (50,50) 0.46608 0.46589 0.46593 0.46588 (50,100) 0.46628 0.46588 0.46576 0.46588 (100,15) 0.45569 0.46576 0.46576 0.46588 (100,25) 0.46121 0.46588 0.46596 0.46588 (100,50) 0.46567 0.46589 0.46603 0.46588 (100,100) 0.46616 0.46588 0.46592 0.46588 Table 4. Shows MSE of𝑅 , for(s,k)=(2,4), α=2.5,β=4 & λ=2 when𝑅 , 0.46584. (n,m) 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝒔𝒉𝟏 𝑹 𝒔,𝒌 𝒔𝒉𝟐 𝑹 𝒔,𝒌 𝑻𝒉 Best (15,15) 0.01047 0.00002 0.00012 0.69E-8 Th (15,25) 0.00880 0.00001 0.00009 0.65E-8 Th (15,50) 0.00711 0.00001 0.00008 0.56E-8 Th (15,100) 0.00658 0.00001 0.00007 0.53E-8 Th (25,15) 0.00904 0.00001 0.00010 0.63E-8 Th (25,25) 0.00656 0.21E-6 0.00007 0.51E-8 Th (25,50) 0.00516 0.16E-6 0.00006 0.45E-8 Th (25,100) 0.00448 0.14E-6 0.00005 0.42E-8 Th (50,15) 0.00713 0.00001 0.00008 0.58E-8 Th (50,25) 0.00469 0.14E-6 0.00005 0.43E-8 Th (50,50) 0.00322 0.79E-7 0.00003 0.35E-8 Th (50,100) 0.00245 0.70E-7 0.00003 0.30E-8 Th (100,15) 0.006889 0.00001 0.00008 0.56E-8 Th (100,25) 0.00407 0.12E-6 0.00004 0.39E-8 Th (100,50) 0.00251 0.73E-8 0.00003 0.33E-8 Th (100,100) 0.00173 0.47E-7 0.00002 0.28E-8 Th     154 Ibn Al-Haitham Jour. for Pure & Appl.Sci. IHJPAS  https://doi.org/10.30526/32.1.1825 Vol. 32 (1) 2019 Table 5. Shows 𝑅 , for (s,k)=(2,3), α=4,β=2.5 & λ=2 when 𝑅 , 0.63054. (n,m) 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝒔𝒉𝟏 𝑹 𝒔,𝒌 𝒔𝒉𝟐 𝑹 𝒔,𝒌 𝑻𝒉 (15,15) 0.62297 0.63042 0.63028 0.63050 (15,25) 0.62694 0.63041 0.630291 0.63050 (15,50) 0.63207 0.63043 0.63036 0.63050 (15,100) 0.63831 0.63059 0.63078 0.63050 (25,15) 0.61813 0.63038 0.63009 0.63050 (25,25) 0.62791 0.63052 0.63056 0.63050 (25,50) 0.63026 0.63050 0.63042 0.63050 (25,100) 0.63394 0.63051 0.63063 0.63050 (50,15) 0.62026 0.63055 0.63051 0.63050 (50,25) 0.62475 0.63050 0.63050 0.63050 (50,50) 0.62789 0.63049 0.63042 0.63050 (50,100) 0.62923 0.63049 0.63036 0.63050 (100,15) 0.61780 0.63046 0.63038 0.63050 (100,25) 0.62514 0.63051 0.63061 0.63050 (100,50) 0.62882 0.63051 0.63059 0.63050 (100,100) 0.62890 0.63049 0.63042 0.63050 Table 6. Shows MSE of 𝑅 , for (s,k)=(2,3), α=4,β=2.5 & λ=2 when 𝑅 , 0.63054 (n,m) 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝒔𝒉𝟏 𝑹 𝒔,𝒌 𝒔𝒉𝟐 𝑹 𝒔,𝒌 𝑻𝒉 Best (15,15) 0.00938 0.00002 0.00010 0.63E-8 Th (15,25) 0.00664 0.00001 0.00008 0.51E-8 Th (15,50) 0.00564 0.95E-5 0.00006 0.46E-8 Th (15,100) 0.00489 0.94E-5 0.00006 0.39E-8 Th (25,15) 0.00746 0.99E-5 0.00008 0.52E-8 Th (25,25) 0.00531 0.16E-6 0.00006 0.41E-8 Th (25,50) 0.00362 0.11E-6 0.00004 0.33E-8 Th (25,100) 0.00326 0.98E-7 0.00003 0.30E-8 Th (50,15) 0.00616 0.98E-5 0.00007 0.43E-8 Th (50,25) 0.00390 0.11E-6 0.00004 0.32E-8 Th (50,50) 0.00257 0.75E-7 0.00003 0.27E-8 Th (50,100) 0.00193 0.56E-7 0.00002 0.25E-8 Th (100,15) 0.00554 0.97E-4 0.00006 0.40E-8 Th (100,25) 0.00370 0.11E-6 0.00004 0.32E-8 Th (100,50) 0.00194 0.54E-7 0.00002 0.24E-8 Th (100,100) 0.00131 0.36E-7 0.00001 0.21E-8 Th     155 Ibn Al-Haitham Jour. for Pure & Appl.Sci. IHJPAS  https://doi.org/10.30526/32.1.1825 Vol. 32 (1) 2019 Table 7. Shows 𝑅 , for, (s,k)=(2,4), α=4,β=2.5 & λ=2 when 𝑅 , 0.71575. (n,m) 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝒔𝒉𝟏 𝑹 𝒔,𝒌 𝒔𝒉𝟐 𝑹 𝒔,𝒌 𝑻𝒉 (15,15) 0.71125 0.71581 0.71589 0.71572 (15,25) 0.71194 0.71574 0.71567 0.71572 (15,50) 0.71423 0.71557 0.71534 0.71571 (15,100) 0.71529 0.71562 0.71539 0.71571 (25,15) 0.70901 0.71582 0.71589 0.71572 (25,25) 0.71098 0.71572 0.71567 0.71572 (25,50) 0.71496 0.71572 0.71569 0.71572 (25,100) 0.71776 0.71573 0.71583 0.71572 (50,15) 0.70333 0.71558 0.71542 0.71572 (50,25) 0.71236 0.71573 0.71592 0.71572 (50,50) 0.71332 0.71572 0.71567 0.71572 (50,100) 0.71743 0.71573 0.71593 0.71572 (100,15) 0.70434 0.71572 0.71566 0.71572 (100,25) 0.70971 0.71572 0.71575 0.71572 (100,50) 0.71124 0.71571 0.71554 0.71571 (100,100) 0.71516 0.71572 0.71577 0.71572 Table 9. Shows MSE of𝑅 , for (s,k)=(2,4), α=4,β=2.5 & λ=2 when 𝑅 , 0.63054. (n,m) 𝑹 𝒔,𝒌 𝑴𝑳𝑬 𝑹 𝒔,𝒌 𝒔𝒉𝟏 𝑹 𝒔,𝒌 𝒔𝒉𝟐 𝑹 𝒔,𝒌 𝑻𝒉 Best (15,15) 0.00631 0.00001 0.00007 0.39E-8 Th (15,25) 0.00549 0.00001 0.00006 0.39E-8 Th (15,50) 0.00382 0.00001 0.00004 0.30E-8 Th (15,100) 0.00376 0.68E-5 0.00004 0.31E-8 Th (25,15) 0.00528 0.71E-5 0.00006 0.35E-8 Th (25,25) 0.00409 0.12E-6 0.00004 0.30E-8 Th (25,50) 0.00277 0.86E-7 0.00003 0.25E-8 Th (25,100) 0.00239 0.73E-7 0.00003 0.23E-8 Th (50,15) 0.00509 0.90E-5 0.00006 0.39E-8 Th (50,25) 0.00306 0.90E-7 0.00003 0.24E-8 Th (50,50) 0.00185 0.52E-7 0.00002 0.20E-8 Th (50,100) 0.00140 0.40E-7 0.00001 0.17E-8 Th (100,15) 0.00425 0.75E-5 0.00004 0.30E-8 Th (100,25) 0.00251 0.71E-7 0.00003 0.22E-8 Th (100,50) 0.00147 0.41E-7 0.00001 0.18E-8 Th (100,100) 0.00105 0.28E-7 0.00001 0.15E-8 Th 5. Discussion Numerical Simulation Results From the tables above, for all n=(15,25,50,100) and m=(15,25,50,100) we conclude that, the shrinkage estimator using Modified Thompson type shrinkage weight factor to estimate the reliability 𝑅 , is the bestsince the (MSE)of RTh(s,k) was less than in the other methods.     156 Ibn Al-Haitham Jour. for Pure & Appl.Sci. IHJPAS  https://doi.org/10.30526/32.1.1825 Vol. 32 (1) 2019 6. Conclusion Form the numerical results, we conclude that the shrinkage estimation method performance good behavior especially when using modified Thompson type shrinkage weight factor. References 1. Bhattacharyya, G.K.; Johnson, R.A., Estimation of Reliability in a Multi-Component Stress-Strength Model, Journal of the American Statistical Association. 1974, 69, 348, 966-970. 2. Afify, W.M. On estimation of the Exponentiated Pareto distribution under different sample schemes, Applied Mathematical Sciences. 2010, 4, 8, 393 - 402. 3. Rao,G.S.; Aslam, M.; Arif, O.H. 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