IHJPAS. 36(1)2023 272 This work is licensed under a Creative Commons Attribution 4.0 International License Estimation of a Parallel Stress-strength Model Based on the Inverse Kumaraswamy Distribution Abstract The reliability of the stress-strength model attracted many statisticians for several years owing to its applicability in different and diverse parts such as engineering, quality control, and economics. In this paper, the system reliability estimation in the stress-strength model containing Kth parallel components will be offered by four types of shrinkage methods (Constant Shrinkage Estimation Method, Shrinkage Function Estimator, Modified Thompson Type Shrinkage Estimator, Squared Shrinkage Estimator). The Monte Carlo simulation study is compared among proposed estimators using the mean squared error. The result analyses of the shrinkage estimation methods showed that the shrinkage functions estimator was the best since it has a minor mean squared error than the other methods followed by the additional shrinkage estimator. The stress and strength belong to the In verse Kumaraswamy distribution. Keywords: Invers Kumaraswamy distribution, Stress ـ Strength reliability, Shrinkage estimator, Mean Squared Error. doi.org/10.30526/36.1.2972 Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Article history: Received 3 Augest 2022, Accepted 29 November 2022, Published in January 2023. Bayda Atiya Kalaf Department of Mathematics, College of Education for Pure Sciences, Ibn Al – Haitham, University of Baghdad Iraq, hbama75@yahoo.com Bsma Abdul Hameed Ministry of Education, Directorate of Education, Rusafa second, Baghdad, Iraq bbsmh896@gmail.com Abbas N. Salman Department of Mathematics, College of Education for Pure Sciences, Ibn Al – Haitham, University of Baghdad Iraq, abbasnajim66@yahoo.com Erum Rehman GEEDS-Group of energy, economy and system dynamics. University of Valladolid, Spain , Department of economics, Shandong University of Finance and Economics, Jinan, China Erum.rehman@uva.es https://creativecommons.org/licenses/by/4.0/ mailto:hbama75@yahoo.com mailto:bbsmh896@gmail.com mailto:abbasnajim66@yahoo.com mailto:Erum.rehman@uva.es IHJPAS. 36(1)2023 273 1. Introduction The word ‘Reliability’ refers to the ability of a system to execute its stated purpose adequately for a specified time under the operational conditions encountered [1]. The stress -strength model is used to compute reliability. It is found to be helpful in situations where the reliability of a component or system is defined by the probability that a random variable of strength is more significant than a random variable (stress). At the same time, it makes intuitive sense that a component is deemed to have failed when its strength is lower than the applied stress [2-3]. Several researchers studied various lifetime distributions of the parallel system reliability in stress- strength model. [4] studied estimation, the system reliability consists of two parallel components, and the two strengths follow the Bivariate Pareto distribution subject to common stress, which follows the Pareto distribution. Also, [5] considered estimating the system of reliability in the S-S model in series and the parallel of K components when the stress and strengths follow exponential distributions. In 2013, [6] estimated S- S reliability consisting of two parallel components based on Exponential distribution. In addition, [7] offered estimations of system Reliability for one component, two parallel components, and s-out-of-k components in S-S system models with non-identical component strengths which are subjected to a common stress, based on Exponentiated Exponential distribution with common scale parameter. Also, a simulation study was used to compare the Maximum likelihood, Percentile, and Least squares estimators. The device combination of k independent components with the strengths X1, X2,…, X𝑘 and each component in the system is subject to common stress 𝑌.it is termed parallel in the analogy with electric circuits when the system operates successfully whenever at least one of the components survives. Consequently, this model is called the parallel stress-strength model [8]. This paper aims to estimate the reliability of the system containing Kth parallel components that have strengths X1, X2,…, X𝑘 subject to a common stress 𝑌 when the strengths and stress follow the Inverse Kumaraswamy distribution. Hence, the organization of this paper: Section 2 displays Inverse Kumaraswamy distribution. Section 3 includes the formula of the system reliability in the S-S model. Section 4 contains shrinkage estimation methods, while simulation studies are in Section 5. The efficiency of the user method appeared in Section 6. Lastly, in Section 7, a conclusion is delivered. 2. Inverse Kumaraswamy Distribution The Kumaraswamy distribution was defined by Ponndi Kumaraswamy in 1980 [9]. The Kumaraswamy distribution is like the beta distribution in many ways [10]. Then, [11] introduced the Inverse KumD using the 𝑋 = 1−𝑇 𝑇 transformation ; 𝑇~KumD (α, β). The probability density function (pdf) and cumulative distribution function (CDF) of a r.v. 𝑋, X~ IKum (α, β) can be written, respectively as [12], [13]: 𝑓(𝑥, 𝛼, 𝛽) = 𝛼𝛽(1 + 𝑥)−(𝛼+1)(1 − (1 + 𝑥)−𝛼 )𝛽−1 , 𝑥 > 0 ; 𝛼, 𝛽 > 0 (1) 𝐹(𝑥; 𝛼, 𝛽) = (1 − (1 + 𝑥)−𝛼 )𝛽 , 𝑥 > 0 , 𝛼, 𝛽 > 0 (2) IHJPAS. 36(1)2023 274 where β and α are shape parameters. 3. Reliability System of Parallel S-S Model We assume a system having 𝐾𝑡ℎ components with strengths 𝑋1, 𝑋2, … , 𝑋𝑘 independently distributed Inverse Kumaraswamy random variables with unknown shape parameter 𝛽𝑖 , 𝑖= 1,2, … , 𝑘 and known other shape parameter 𝛼 subjected to stress random variable 𝑌 such that 𝑌~𝐼𝐾𝑢𝑚𝐷(𝛼, 𝛽𝑘+1) with unknown shape parameter 𝛽𝑘+1 and known 𝛼. The formula of Parallel S-S model can be derived as fellows [14-16]: 𝑅 = 𝑃(𝑌 < 𝑚𝑎𝑥 (𝑋1𝑋2, … , 𝑋𝑘 )) Let 𝑍= 𝑚𝑎𝑥 𝑋1𝑋2, … , 𝑋𝑘 Therefore, 𝑅 = ∫ �̅�𝑧 (𝑦)𝑓(𝑦) 𝑑𝑦 ∞ 0 (3) 𝐹𝑧 (𝑍) = 𝑃(𝑍 < 𝑧) = 𝑃(𝑥1 < 𝑧)𝑃(𝑥2 < 𝑧) … 𝑃(𝑥𝑘 < 𝑧) = (1 − (1 + 𝑧)−𝛼 )𝛽1 (1 − (1 + 𝑧)−𝛼 )𝛽2 … (1 − (1 + 𝑧)−𝛼 )𝛽𝑘 = (1 − (1 + 𝑧)−𝛼 )∑ 𝛽𝑖 𝑘 𝑖=1 This implies that a random variable 𝑍 follows IKumD with the parameters 𝛼 and ∑ 𝛽𝑖 𝑘 𝑖=1 . Then, �̅�𝑧 (𝑦) = (1 − (1 − (1 + 𝑦) −𝛼 )∑ 𝛽𝑖 𝑘 𝑖=1 ) (4) In equation (3), substitute equation (4) and equation (1), we get: 𝑅 = ∫ (1 − (1 − (1 + 𝑦)−𝛼 )∑ 𝛽𝑖 𝑘 𝑖=1 ) 𝛼𝛽𝑘+1(1 − 𝑦) −(𝛼+1)(1 − (1 + 𝑦)−𝛼 )𝛽𝑘+1−1 𝑑𝑦 ∞ 0 𝑅 = ∫ 𝛼 ∞ 0 𝛽𝑘+1(1 − 𝑦) −(𝛼+1)(1 − (1 + 𝑦)−𝛼 )𝛽𝑘+1−1 𝑑𝑦 − ∫ 𝛼𝛽𝑘+1(1 − 𝑦) −(𝛼+1)(1 − (1 + ∞ 0 𝑦)−𝛼 )∑ 𝛽𝑖 𝑘+1 𝑖=1 −1𝑑𝑦 𝑅 = 1 − 𝛽𝑘+1 ∑ 𝛽𝑖 𝑘+1 𝑖=1 𝑅 = ∑ 𝛽𝑖 𝑘 𝑖=1 ∑ 𝛽𝑖 𝑘+1 𝑖=1 (5) IHJPAS. 36(1)2023 275 4. Estimation Methods for System Reliability 4.1 Shrinkage Estimation Method (Sh) In 1968, [17] proposed to shrink the usual estimator β̂ (ex. MLE or Unbiased estimator) of the parameter β to prior information 𝛽0 using shrinkage weight factor ψ(β̂), such that 0≤ ψ(β̂) ≤1. Thompson says that: "We are estimating β and we believe β0 is closed to the true value of β and something bad happens when β0 ≈ β and we do not use β0". Thus, Thompson gave the form of shrinkage estimator of β say β̂Sh as below: β̂Sh = ψ(β̂)β̂𝑢𝑏 + (1 − ψ(β̂))β0 (6) Unbiased estimator β̂𝑢𝑏 was applied as the usual estimator of β, and β0 is a very closed value of β as prior information due to previous studies or experience and ψ (β̂) denote the shrinkage weight factor as we mentioned above such that 0 ≤ ψ(β̂) ≤ 1, which may be a function of β̂𝑢𝑏: a function of sample size or may be constant. Also, it is possible to find ψ(β̂) through minimizing the mean square error of β̂𝑠ℎ (ad hoc basis) [18-20]. Note that β̂𝑖𝑢𝑏 of the strengths 𝑋1, 𝑋2, … , 𝑋𝑘 can be found depending on observation 𝑋𝑖𝑗 , 𝑖= 1,2, … , 𝑘 and 𝑗 = 1,2, … , 𝑛𝑖 as below: �̂�𝑖𝑢𝑏 = 𝑛𝑖−1 − ∑ ln(1−(1+𝑥𝑖𝑗) −𝛼 ) 𝑛𝑖 𝑗=1 (7) Likewise �̂�𝑘+1𝑢𝑏 of the stress 𝑌 can be found depends on observation 𝑌𝑟 , r= 1,2, … , 𝑛 as below: �̂�𝑘+1𝑢𝑏 = 𝑚−1 − ∑ ln(1−(1+𝑦𝑟) −𝛼)𝑚𝑟=1 (8) 4.1.1 Constant Shrinkage Estimation Method (Sh1) As we mention that 𝑋𝑖(𝑖= 1,2, … , 𝑘) follows 𝐼𝐾𝑢𝑚𝐷 (𝛼, 𝛽𝑖 ) and 𝑌~ 𝐼𝐾𝑢𝑚𝐷 (𝛼, 𝛽𝑘+1). A constant shrinkage weight factor Ψ(�̂�𝑖 ) = h = 0.01: 𝑖= 1,2, … , 𝑘 + 1 will be suggested in this subsection. Then, the constant shrinkage estimator for 𝛽𝑖, 𝑖= 1,2, … , 𝑘 + 1 will be: �̂�𝑖𝑆ℎ1 =h�̂�𝑖𝑢𝑏 + (1 − h)𝛽𝑖0 for 𝑖= 1,2, … , 𝑘 + 1 (9) Put the equation (9) in equation (5), and then the constant shrinkage estimator R̂𝑆ℎ1 for system reliability 𝑅 in S-S model consist 𝐾𝑡ℎ parallel components become: R̂𝑆ℎ1 = ∑ �̂�𝑖𝑆ℎ1 𝑘 𝑖=1 ∑ �̂�𝑖𝑆ℎ1 𝑘+1 𝑖=1 (10) 4.1.2 Shrinkage Function Estimator We suggest the shrinkage function estimator (𝑆ℎ2) for estimating the parameters 𝛽𝑖(𝑖= 1,2, … , 𝑘 + 1), and the reliability system R based on the shrinkage weight function depending on the sample size 𝑛𝑖 and 𝑚 respecting, as below: IHJPAS. 36(1)2023 276 𝛹 (�̂�𝑖𝑢𝑏 )=𝑒 −𝑛𝑖 for i= 1, 2,…, 𝑘 and 𝛹 (�̂�𝑘+1𝑢𝑏 )=𝑒 −𝑚 Thus, based on equation (6), the shrinkage function estimators of 𝛽𝑖 (𝑖= 1,2, … , 𝑘 + 1) using the above shrinkage weight functions will be: �̂�𝑖𝑆ℎ2 = 𝛹 (�̂�𝑖𝑢𝑏 )�̂�𝑖𝑢𝑏 +(1 − Ψ (�̂�𝑖𝑢𝑏 )) 𝛽𝑖0 i=1, 2, …, 𝑘 (11) �̂�𝑘+1𝑆ℎ2 =𝛹(�̂�𝑘+1𝑢𝑏 )�̂�𝑘+1𝑢𝑏 +(1 − Ψ (�̂�𝑘+1𝑢𝑏 )) 𝛽𝑘+10 (12) Substitute �̂�𝑖𝑆ℎ2 i= 1, 2,…, 𝑘 + 1 in equation (5), we get the shrinkage function estimator of reliability system R based on shrinkage weight function, as follows: R̂𝑆ℎ2 = ∑ �̂�𝑖𝑆ℎ2 𝑘 𝑖=1 ∑ �̂�𝑖𝑆ℎ2 𝑘+1 𝑖=1 (13) 4.1.3 Modified Thompson Type Shrinkage Estimator (Th) The shrinkage weight factor considered by Thompson in 1968 will be modified and used in shrinkage estimator to estimate the reliability system for the proposed of S-S models. The modified shrinkage weight factor (Th) has the form, as below: 𝜑(�̂�𝑖𝑢𝑏 ) = (�̂�𝑖𝑢𝑏 −�̂�𝑖0 ) 2 (�̂�𝑖𝑢𝑏 −�̂�𝑖0 ) 2 +𝑣𝑎𝑟(�̂�𝑖𝑢𝑏 ) (0.001), for i=1,2, …,k+1 (14) We can estimate the parameters 𝛽𝑖 (𝑖= 1,2, … , 𝑘 + 1) by the modified Thompson type shrinkage estimator, as follows: �̂�𝑖𝑇ℎ = 𝜑(�̂�𝑖𝑢𝑏 )�̂�𝑖𝑢𝑏 + (1 − 𝜑(�̂�𝑖𝑢𝑏 )) 𝛽𝑖0 , for i = 1, 2, … , 𝑘 + 1 (15) When substituting the equation (15) in equation (5), the reliability system for estimating of R using the modified Thompson type shrinkage estimator become: R̂ 𝑇ℎ = ∑ �̂�𝑖𝑇ℎ 𝑘 𝑖=1 ∑ �̂�𝑖𝑇ℎ 𝑘+1 𝑖=1 (16) 4.1.4 Squared Shrinkage Estimator (Sq) In this subsection, we propose the squared shrinkage estimator (Sq) for �̂�𝑖 for the parameters 𝛽𝑖(𝑖= 1,2, … , 𝑘 + 1), as below: �̂�𝑖𝑆𝑞 = 𝛾(�̂�𝑖𝑢𝑏 )�̂�𝑖𝑢𝑏 + (1 − 𝛾(�̂�𝑖𝑢𝑏 )) 𝛽𝑖0 , i = 1, 2,…, 𝑘 + 1 (17) Substituting (17) in (5), we conclude the squared shrinkage estimator of the reliability system 𝑅 which consist 𝐾𝑡ℎ Parallel component, as below: R̂𝑆𝑞 = ∑ �̂�𝑖𝑆𝑞 𝑘 𝑖=1 ∑ �̂�𝑖𝑆𝑞 𝑘+1 𝑖=1 (18) 5. Simulation Study To verify the performance of the proposed estimation methods introduced to estimate the reliability system of 𝐾𝑡ℎ components (𝑅), Mote Carlo simulation was used. The proposed IHJPAS. 36(1)2023 277 estimation methods in S-S models have been implemented using various samples (20, 30, 50, and 100). In addition, the statistical outcomes for every sample are based on Mean Squared Errors criteria with 1000 replicates. Therefore, the following subsections explain the steps of Monte Carlo simulation. Step1: Generate random samples following the continuance uniform distribution defined on the interval (0,1) for the strength 𝑋𝑖 , stress 𝑌 as 𝑢𝑖1, 𝑢𝑖2, … , 𝑢𝑖𝑛𝑖 ; 𝑖= 1,2, … , 𝑘. and 𝑣1, 𝑣2, … , 𝑣𝑚, respectively. Step2: convert the uniform random samples using the cumulative distribution function for random samples of 𝑋𝑖 ~ IKumD (𝛼, 𝛽𝑖) for 𝑖= 1,2, … 𝑘, as follow: F(𝑥𝑖𝑡 ) = (1 − (1 + 𝑥𝑖𝑡 ) −𝛼 ) 𝛽1 𝑢𝑖𝑡 = (1 − (1 + 𝑥𝑖𝑡 ) −𝛼 ) 𝛽1 𝑥𝑖𝑡 = [1 − (𝑢𝑖𝑡 ) 1 𝛽1 ] − 1 𝛼 − 1 ; 𝑖= 1,2, … , 𝑘. Using the same method for 𝑌~ IKumD (𝛼, 𝛽𝑘+1): 𝑦𝑗 = [1 − (𝑣𝑗 ) 1 𝛽2 ] − 1 𝛼 − 1 Step3: Recall R in equation (5) Step4: Compute Shrinkage estimators of reliability using equations (10), (13), (16) and (18) Step5: Based on L=1000 replication, MSE will be calculated as follows; MSE = 1 𝐿 ∑ (�̂�𝑖 − 𝑅) 2𝐿 𝑖=1 6. Numerical Results In this model the reliability system 𝑅 = 𝑃(𝑚𝑎𝑥 ( 𝑋1, 𝑋2, … , 𝑋𝑘)) in the S-S model when k=3 and 𝛼 = 5 will be estimated. Simulation results based on the four parameters (𝛽1, 𝛽2,𝛽3, 𝛽4), clarified in tables 1 to 8 to show the order rank of the proposed estimators as follows; �̂�Sh2,�̂�𝑇ℎ , �̂�Sq 𝑎𝑛𝑑 �̂�Sh1 depending on the mean square error. Shrinkage estimator (�̂�𝑆ℎ2) was the best of others estimators followed by the modified Thompson type shrinkage weight factor ( �̂�𝑇ℎ ). The Squared Shrinkage Estimator (�̂�𝑆𝑞 ) has the third rank, and finally was constant shrinkage estimator �̂�𝑆ℎ1. The following Tables (1-8) will be presented the simulation results. IHJPAS. 36(1)2023 278 Table 1: Estimation value of 𝑅 = 0.802259, when 𝛼 = 5, 𝛽1 = 4, 𝛽2 = 5.2 , 𝛽3 = 5 𝑎𝑛𝑑 𝛽4 = 3.5 (𝑛 , 𝑚1, 𝑚2, 𝑚3) �̂�𝑆ℎ1 �̂�𝑆ℎ2 �̂�𝑇ℎ �̂�𝑆𝑞 (20,20,20,20) 0.802262 0.8022716997468 0.8022598 0.802268 (20,50,100,50) 0.802264 0.8022716998468 0.8022595 0.802272 (20,30,20,50) 0.802295 0.8022716998151 0.8022600 0.802277 (20,100,100,30) 0.802246 0.8022716998033 0.8022599 0.802266 (30,50,100,100) 0.802278 0.8022716998191 0.8022602 0.802282 (30,30,50,30) 0.802249 0.8022716998191 0.8022595 0.802270 (30,20,50,50) 0.802279 0.8022716997659 0.8022597 0.802282 (30,100,20,30) 0.802238 0.8022716997839 0.8022579 0.802243 (50,50,50,50) 0.802252 0.8022716998191 0.8022589 0.802262 (50,20,30,100) 0.802283 0.8022716997827 0.8022587 0.802289 (50,100,100,50) 0.802245 0.8022716998191 0.8022593 0.802263 (50,30,100,20) 0.802204 0.8022716998490 0.8022586 0.802238 (100,20,30,30) 0.802263 0.8022716997906 0.8022602 0.802280 (100,30,50,20) 0.802205 0.8022716997592 0.8022584 0.802252 (100,100,20,50) 0.802272 0.8022716997937 0.8022602 0.802271 (100,50,20,20) 0.802218 0.8022716997321 0.8022588 0.802245 Table 2. MSE value of 𝑅 = 0.802259, when 𝛼 = 5, 𝛽1 = 4, 𝛽2 = 5.2, 𝛽3 = 5 𝑎𝑛𝑑 𝛽4 = 3.5 (𝑛,𝑚1, 𝑚2, 𝑚3) �̂�𝑆ℎ1 �̂�𝑆ℎ2 �̂�𝑇ℎ �̂�𝑆𝑞 (20,20,20,20) 2.0732E-07 1.39541E-010 1.4043E-09 6.0409E-08 (20,50,100,50) 8.1335E-07 1.39543E-09 5.8360E-09 2.4349E-07 (20,30,20,50) 1.0351E-07 1.39542E-010 7.3684E-010 9.7220E-08 (20,100,100,30) 1.1580E-07 1.39542E-010 8.0165E-010 3.9204E-08 (30,50,100,100) 4.6577E-07 1.39542E-09 3.4071E-09 1.2336E-07 (30,30,50,30) 1.2119E-07 1.39543E-010 8.4560E-010 2.7734E-08 (30,20,50,50) 9.1350E-07 1.39541E-09 6.6416E-09 2.7654E-07 (30,100,20,30) 1.3840E-07 1.39542E-010 9.9736E-010 1.4193E-07 (50,50,50,50) 7.2987E-07 1.39542E-09 5.3090E-09 3.0470E-07 (50,20,30,100) 6.2701E-08 1.39542E-010 4.5194E-010 2.6724E-08 (50,100,100,50) 6.6753E-07 1.39542E-09 4.7857E-09 1.6291E-07 (50,30,100,20) 1.8216E-07 1.39543E-010 1.2248E-09 2.3098E-07 (100,20,30,30) 1.3797E-09 1.39542E-010 9.7206E-010 8.5443E-8 (100,30,50,20) 1.6688E-07 1.39541E-010 1.0929E-09 6.2777E-08 (100,100,20,50) 8.2541E-07 1.39541E-09 5.9453E-09 3.5773E-07 (100,50,20,20) 1.9497E-07 1.39541E-010 1.3397E-09 1.6416E-07 IHJPAS. 36(1)2023 279 Table 3. Estimation value of 𝑅 = 0.659863, when 𝛼 = 5, 𝛽1 = 3.5, 𝛽2 = 2.5 , 𝛽3 = 3.7 𝑎𝑛𝑑 𝛽4 = 5 (𝑛 ,𝑚1, 𝑚2, 𝑚3) �̂�𝑆ℎ1 �̂�𝑆ℎ2 �̂�𝑇ℎ �̂�𝑆𝑞 (20,20,20,20) 0.659867 0.6598394124288 0.6598644 0.659847 (20,50,100,50) 0.659883 0.6598394120692 0.6598645 0.659851 (20,30,20,50) 0.659922 0.6598394120750 0.6598648 0.659865 (20,100,100,30) 0.659856 0.6598394120572 0.6598647 0.659833 (30,50,100,100) 0.659915 0.6598394120849 0.6598666 0.659854 (30,30,50,30) 0.659830 0.6598394120849 0.6598621 0.659820 (30,20,50,50) 0.659889 0.6598394121347 0.6598640 0.659850 (30,100,20,30) 0.659866 0.6598394120528 0.6598641 0.659828 (50,50,50,50) 0.659875 0.6598394120849 0.6598651 0.659847 (50,20,30,100) 0.659923 0.6598394121063 0.6598653 0.659860 (50,100,100,50) 0.659849 0.6598394120849 0.6598640 0.659832 (50,30,100,20) 0.659812 0.6598394122460 0.6598646 0.659805 (100,20,30,30) 0.659838 0.6598394121219 0.6598625 0.659837 (100,30,50,20) 0.659817 0.6598394121855 0.6598653 0.659815 (100,100,20,50) 0.659885 0.6598394120706 0.6598647 0.659846 (100,50,20,20) 0.659822 0.6598394121940 0.6598641 0.659795 Table 4. MSE value of 𝑅 = 0.659863, when 𝛼 = 5, 𝛽1 = 3.5, 𝛽2 = 2.5, 𝛽3 = 3.7 𝑎𝑛𝑑 𝛽4 = 5 (𝑛,𝑚1, 𝑚2, 𝑚3) �̂�𝑆ℎ1 �̂�𝑆ℎ2 �̂�𝑇ℎ �̂�𝑆𝑞 (20,20,20,20) 4.0707E-07 6.01875E-010 2.8366E-09 1.9030E-07 (20,50,100,50) 1.5669E-07 6.01893E-010 1.1314E-09 3.3125E-08 (20,30,20,50) 2.0169E-07 6.01893E-010 1.4103E-09 5.1564E-08 (20,100,100,30) 2.4504E-07 6.01894E-010 1.7792E-09 1.2937E-07 (30,50,100,100) 9.3516E-08 6.01892E-010 6.7290E-010 4.1893E-08 (30,30,50,30) 2.4678E-07 6.01892E-010 1.8055E-09 1.1665E-07 (30,20,50,50) 1.6287E-07 6.01890E-010 1.1784E-09 6.3491E-08 (30,100,20,30) 2.8990E-07 6.01894E-010 2.1492E-09 1.0802E-07 (50,50,50,50) 1.4567E-07 6.01892E-010 1.0915E-09 33611E-08 (50,20,30,100) 1.2185E-07 6.01891E-010 8.5750E-010 3.6065E-08 (50,100,100,50) 1.2732E-07 6.01892E-010 9.5287E-010 3.8116E-08 (50,30,100,20) 3.3889E-07 6.01884E-010 2.4187E-09 2.1102E-08 (100,20,30,30) 2.6669E-07 6.01890E-010 1.9410E-09 2.7221E-07 (100,30,50,20) 3.3352E-07 6.01887E-010 2.3632E-09 1.2724E-07 (100,100,20,50) 1.5450E-07 6.01893E-010 1.1495E-09 4.3320E-08 (100,50,20,20) 3.8060E-07 6.01887E-010 2.7720E-09 7.0661E-07 IHJPAS. 36(1)2023 280 Table 5. Estimation value of 𝑅 = 0.791666, when 𝛼 = 5, 𝛽1 = 4, 𝛽2 = 3, 𝛽3 = 2.5 𝑎𝑛𝑑 𝛽4 = 2.5 (𝑛 ,𝑚1, 𝑚2, 𝑚3) �̂�𝑆ℎ1 �̂�𝑆ℎ2 �̂�𝑇ℎ �̂�𝑆𝑞 (20,20,20,20) 0.791656 0.7916805603479 0.7916659 0.791682 (20,50,100,50) 0.791684 0.7916805601685 0.7916667 0.791690 (20,30,20,50) 0.791716 0.7916805602060 0.7916675 0.791711 (20,100,100,30) 0.791676 0.7916805601476 0.7916682 0.791690 (30,50,100,100) 0.791697 0.7916805601867 0.7916676 0.791697 (30,30,50,30) 0.791672 0.7916805601867 0.7916674 0.791681 (30,20,50,50) 0.791706 0.7916805602331 0.7916679 0.791698 (30,100,20,30) 0.791651 0.7916805602568 0.7916654 0.791678 (50,50,50,50) 0.791673 0.7916805601867 0.7916672 0.791683 (50,20,30,100) 0.791714 0.7916805602663 0.7916679 0.791704 (50,100,100,50) 0.791649 0.7916805601867 0.7916659 0.791675 (50,30,100,20) 0.791606 0.7916805599520 0.7916649 0.791659 (100,20,30,30) 0.791655 0.7916805602083 0.7916663 0.791678 (100,30,50,20) 0.791613 0.7916805600171 0.7916656 0.791658 (100,100,20,50) 0.791650 0.7916805602617 0.7916654 0.791677 (100,50,20,20) 0.791600 0.7916805600296 0.7916642 0.791655 Table 6.MSE value of 𝑅 = 0.791666, when 𝛼 = 5, 𝛽1 = 4, 𝛽2 = 3, 𝛽3 = 2.5 𝑎𝑛𝑑 𝛽4 = 2.5 (𝑛,𝑚1, 𝑚2, 𝑚3) �̂�𝑆ℎ1 �̂�𝑆ℎ2 �̂�𝑇ℎ �̂�𝑆𝑞 (20,20,20,20) 2.3171E-07 1.93034E-010 1.4743E-09 1.0633E-07 (20,50,100,50) 9.3801E-07 1.93029E-09 6.2256E-09 54754E-07 (20,30,20,50) 1.1352E-07 1.93030E-010 7.2320E-010 1.7540E-07 (20,100,100,30) 1.3498E-07 1.93029E-010 8.7464E-010 4.6398E-08 (30,50,100,100) 5.7235E-07 1.93029E-09 3.8307E-09 1.3931E-07 (30,30,50,30) 1.3215E-07 1.93030E-010 8.5837E-010 4.5733E-08 (30,20,50,50) 1.1198E-06 1.93031E-09 7.4709E-09 4.2545E-07 (30,100,20,30) 1.4282E-07 1.93032E-010 9.2913E-010 3.9023E-08 (50,50,50,50) 7.6742E-07 1.93029E-09 5.0665E-09 1.8428E-07 (50,20,30,100) 6.3415E-08 1.93032E-010 4.0528E-010 2.0606E-08 (50,100,100,50) 7.2895E-07 1.93029E-09 4.8046E-09 2.1633E-07 (50,30,100,20) 1.8009E-07 1.93023E-010 1.0874E-09 6.0397E-07 (100,20,30,30) 1.2715E-07 1.93031E-010 7.9716E-010 3.1972E-08 (100,30,50,20) 1.8568E-07 1.93025E-010 1.1338E-09 1.0036E-07 (100,100,20,50) 7.9998E-07 1.93031E-09 5.2898E-09 3.4745E-07 (100,50,20,20) 2.0092E-07 1.93026E-010 1.2253E-09 7.0238E-08 IHJPAS. 36(1)2023 281 Table 7. Estimation value of 𝑅 = 0.764705, when 𝛼 = 5, 𝛽1 = 6, 𝛽2 = 7.5 , 𝛽3 = 6 𝑎𝑛𝑑 𝛽4 = 6 (𝑛 ,𝑚1, 𝑚2, 𝑚3) �̂�𝑆ℎ1 �̂�𝑆ℎ2 �̂�𝑇ℎ �̂�𝑆𝑞 (20,20,20,20) 0.764741 0.7647081895378 0.7647086 0.764707 (20,50,100,50) 0.764729 0.7647081895554 0.7647068 0.764714 (20,30,20,50) 0.764747 0.7647081895190 0.7647059 0.764728 (20,100,100,30) 0.764693 0.7647081895096 0.7647062 0.764695 (30,50,100,100) 0.764735 0.7647081895199 0.7647068 0.764721 (30,30,50,30) 0.764707 0.7647081895199 0.7647064 0.764701 (30,20,50,50) 0.764743 0.7647081894968 0.7647068 0.764722 (30,100,20,30) 0.764704 0.7647081895328 0.7647062 0.764708 (50,50,50,50) 0.764711 0.7647081895199 0.7647063 0.764704 (50,20,30,100) 0.764747 0.7647081894660 0.7647058 0.764727 (50,100,100,50) 0.764690 0.7647081895199 0.7647056 0.764701 (50,30,100,20) 0.764639 0.7647081895568 0.7647039 0.764676 (100,20,30,30) 0.764701 0.7647081894775 0.7647055 0.764718 (100,30,50,20) 0.764664 0.7647081894843 0.7647063 0.764684 (100,100,20,50) 0.764713 0.7647081895303 0.7647062 0.764705 (100,50,20,20) 0.764673 0.7647081894824 0.7647065 0.764698 Table 8. MSE value of 𝑅 = 0.764705, when 𝛼 = 5, 𝛽1 = 6, 𝛽2 = 7.5, 𝛽3 = 6 𝑎𝑛𝑑 𝛽4 = 6 (𝑛,𝑚1, 𝑚2, 𝑚3) �̂�𝑆ℎ1 �̂�𝑆ℎ2 �̂�𝑇ℎ �̂�𝑆𝑞 (20,20,20,20) 2.6414E-07 5.323E-012 1.9644E-09 9.9664E-08 (20,50,100,50) 9.9863E-07 5.3231E-011 7.7581E-09 2.0165E-07 (20,30,20,50) 1.2929E-07 5.323E-012 9.9643E-010 5.8344E-08 (20,100,100,30) 1.5517E-07 5.323E-012 1.2063E-09 1.0039E-07 (30,50,100,100) 5.8353E-08 5.323E-012 4.5743E-010 1.0984E-08 (30,30,50,30) 1.5375E-07 5.323E-012 1.1708E-09 4.8042E-08 (30,20,50,50) 1.1503E-07 5.323E-012 8.9958E-010 3.0406E-08 (30,100,20,30) 1.5690E-07 5.323E-012 1.1992E-09 7.4901E-08 (50,50,50,50) 9.2584E-07 5.3230E-011 7.3307E-09 3.3730E-07 (50,20,30,100) 7.9172E-08 5.323E-012 6.0768E-010 2.7153E-08 (50,100,100,50) 9.0335E-07 5.3230E-011 7.1707E-09 2.3430E-07 (50,30,100,20) 2.1534E-07 5.323E-012 1.5579E-09 1.8696E-07 (100,20,30,30) 1.6145E-07 5.323E-012 1.2280E-09 5.0486E-08 (100,30,50,20) 2.3598E-07 5.323E-012 1.7385E-09 1.8249E-07 (100,100,20,50) 1.0578E-06 5.3230E-011 83475E-09 5.8070E-07 (100,50,20,20) 2.1892E-07 5.323E-012 1.6044E-09 8.6640E-08 IHJPAS. 36(1)2023 282 7. Conclusion The estimation of S-S reliability for two parameters Invers Kumaraswamy distribution was introduced in this paper using different shrinkage methods and estimation methods, namely: Constant Shrinkage Estimation Method, Modified Thompson Type Shrinkage Estimator, Shrinkage Function Estimator, and Squared Shrinkage Estimator. The simulation was exhibited. Based on the results, the performance of the Shrinkage Function Estimator was appropriate behavior, and it is an efficient estimator than the others in the sense of MSE based on four parameters (𝛽1, 𝛽2 , 𝛽3, 𝛽4). While Modified Thompson Type Shrinkage Estimator had the second rank and followed by Squared Shrinkage Estimator and Constant Shrinkage Estimation Method, respectively. References 1.Hameed, B. A.; Salman, A. N.; Kalaf, B. A. On Estimation of P (Y< X) in Case Inverse Kumaraswamy Distribution. Ibn AL-Haitham Journal For Pure and Applied Sciences, 2020, 33(1), 108-118 2.Raheem, S. H., Kalaf, B. A.; Salman, A. N. 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