Microsoft Word - 82-91 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 82          Comparison Among Three Estimation Methods to Estimate Cascade Reliability Model (2+1) Based On Inverted Exponential Distribution Sairan Hamza Raheem University of Garmian, College of Computing and Information Technology, IT. Department sairan.hamza@garmian.edu.krd Abstract In this paper, we are mainly concerned with estimating cascade reliability model (2+1) based on inverted exponential distribution and comparing among the estimation methods that are used. The maximum likelihood estimator and uniformly minimum variance unbiased estimators are used to get the parameters of the strengths 𝑋 and the stress 𝑌 ;k=1,2,3 respectively then, by using the unbiased estimators, we propose Preliminary test single stage shrinkage (PTSSS) estimator when a prior knowledge is available for the scale parameter as initial value due past experiences . The Mean Squared Error [MSE] for the proposed estimator is derived to compare among the methods. Numerical results about conduct of the considered estimator are discussed including the study of mentioned expressions. The numerical results are exhibited and put it in tables. Keyword: Inverted exponential distribution, Maximum Likelihood method, Uniformly Minimum Variance Unbiased method, Single Stage Shrinkage Estimator, Mean Squared Error and Cascade Reliability Model (2+1). 1. Introduction An 𝑛 - cascade system is defined as a special type of standby system with 𝑛 components and it could be considered as a kind of stress-strength model. The reliability system of a cascade model can be described by a function of parameters of the identical and independent distributions with strength (X) and stress (Y) and the attenuation factor (K) in other words, the stresses on subsequent components are attenuated by a factor ‘k’, called attenuation factor that is generally assumed to be a constant for all the components or to be a parameter having different fixed values for different components, or it may be simply a random variable. This Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/33.4.2512 Article history: Received 13 January 2020, Accepted 12 February 2020, Published in October 2020   83  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 system was first proposed and studied by Pandit and Sriwastav (1975).Rekha and Chechu Raju (1999) presented a closed form solution of stress attenuated reliability function for n- cascade system with exponential stress and standby strengths following Rayleigh and exponential distributions. Sundar (2012)has done a case study of cascade reliability with Rayleigh distribution. Devi (2016) studied Cascade System Reliability with Stress and Strength Follow Lindley Distribution. Dong and Cui studied System Reliability under Cascading Failure Models. Doloi and Gogoi (2017) presented a cascade reliability model for exponential and Lindley distributions. Reddy(2016) studied Cascade and System Reliability for Exponential Distributions. In (2019) Kanaparthi et el studied Cascade and System Reliability for the New Rayleigh-Pareto Distribution The aim of this study is to compare among three estimation methods (MLE ,UMVUE and PTSSs) in estimate the cascade reliability model(2+1) based on inverted exponential distribution (IE) with unknown scale parameter. For this purpose, we use Thamson shrinkage technique through the equation below: 𝛼 ∅ 𝛼 𝛼 𝛼 𝛼 (1) Where 𝛼 is a prior knowledge(estimation) about the parameter 𝛼 and 0 ∅ 𝛼 1 is a shrinkage weight factor to assign the degree of belief in 𝛼 ; also 𝛼 is the classical estimator of 𝛼 (MLE or UMVUE).  Several authors have studied the estimator defined in (1) for a special distribution for different parameters and suitable regions (R) as well as for estimate the parameters of linear regression model. See [13-16]. 2. Statistical Model In life distribution, if a random variable X following an exponential distribution then the variable 𝑌 has an inverted exponential distribution (IE) . The Invers Exponential Distribution IE 𝛼 has the following probability density function p.d.f and cumulative distribution function c.d.f. for 𝑥 0 : 𝑓 𝑥; 𝛼 𝑒𝑥𝑝 ; 𝛼 0 (2) 𝐹 𝑥; 𝛼 𝑒𝑥𝑝 ; 𝛼 0 (3) Killer and Kamath (1982) defined and studied IE for the first time. Lin et al (1989) descripted IE distribution as a lifetime model. It can be seen that IE distribution was widely used in the analytic study specially in engineering, biology and medicine fields Oguntunde et el(2017). Bayes estimators of the parameter and reliability function of the IE distribution were obtained by Singh et al(2013). Oguntunde et al. (2014) proposed exponentiated GIE (EGIE) distribution. Singh et al (2015) estimated the stress strength reliability parameter of IE distribution.  Fatima and Ahmad (2018) used a Bayesian Approximation Techniques of Inverse Exponential Distribution with Applications in Engineering. For cascade model (2+1), assume that 𝑋 , 𝑋 and 𝑋 are the strengths with two fundamental components and one standby component is i.i.d with p.d.f follows IED 𝛼 with scale parameter 𝛼 ; 𝑖 1,2,3, when activated faces the stresses random variables 𝑌 , 𝑌 and 𝑌 are imposed on the strengths components and followed IED 𝛽 with scale parameter 𝛽 ; 𝑗 1,2,3. In cascade system, after every failure the stress gets modified by attenuation factor (k) such that: 𝑌 𝑘𝑌 , 𝑌 𝑘𝑌 𝑘 𝑌 , … , 𝑌 𝑘 𝑌 ; 𝑘 1   84  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 And we suppose a factor (m) to modified the strength such that: 𝑋 𝑚𝑋 , 𝑋 𝑚𝑋 𝑚 𝑋 , … , 𝑋 𝑚 𝑋 ; m 0 The real reliability function for the (2+1) cascade model is given by: 𝑅 𝑝 𝑋 𝑌 , 𝑋 𝑌 𝑝 𝑋 𝑌 , 𝑋 𝑌 , 𝑋 𝑌 𝑝 𝑋 𝑌 , 𝑋 𝑌 , 𝑋 𝑌 (4) 𝑅 𝑅 𝑅 𝑅 (5) 𝑅 𝑝 𝑋 𝑌 , 𝑋 𝑌 (6) 𝐹 𝑦 𝑔 𝑦 𝑑𝑦 𝐹 𝑦 𝑔 𝑦 𝑑𝑦 1 𝑒𝑥𝑝 𝛼 𝑦 𝛽 𝑦 𝑒𝑥𝑝 𝛽 𝑦 𝑑𝑦 1 𝑒𝑥𝑝 𝛼 𝑦 𝛽 𝑦 𝑒𝑥𝑝 𝛽 𝑦 𝑑𝑦 So we get 𝑅 (7) 𝑅 𝑝 𝑋 𝑌 , 𝑋 𝑌 , 𝑋 𝑌 (8) 𝑝 𝑋1 𝑌1, 𝑚𝑋1 𝑘𝑌1 𝑝 𝑋 𝑌 𝑝 𝑋 𝑌 where 𝑝 𝑋 𝑌 , 𝑚𝑋 𝑘𝑌 𝐹 𝑦 𝐹 𝑦 𝑔 𝑦 𝑑𝑦 (9) 𝑒𝑥𝑝 1 𝑒𝑥𝑝 𝑒𝑥𝑝 Then 𝑝 𝑋 𝑌 , 𝑚𝑋 𝑘𝑌 (10) And 𝑝 𝑋 𝑌 𝐹 𝑦 𝑔 𝑦 𝑑𝑦 (11) So 𝑅 (12) By the same way finding R R 𝑝 𝑋 𝑌 , 𝑋 𝑌 , 𝑋 𝑌 (13) (14) Substitution (6), (11) and (13) in (4) 𝑅 𝛼 𝛼 𝛽 𝛼 𝛼 𝛽 𝑘 𝑚 𝛼 𝛽 𝑘 𝑚 𝛼 𝛽 𝛼 𝑘 𝑚 𝛼 𝛽 𝛼 𝛼 𝛽 (15)   85  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 3. Maximum Likelihood Estimation (MLE) The Maximum likelihood method is an important and commonly, since it contained properties for good estimate. Suppose𝑋 ; 𝑖 1,2,3 strength random sample follows IED 𝛼 with the sample size 𝑛.The likelihood function is given by; 𝑙 𝐿(𝑥 , 𝑥 , … , 𝑥 ; α) ∏ 𝑓 𝑥 ∏ 𝑒𝑥𝑝 𝛼 ∏ 𝑒𝑥𝑝 ∑ (16) Taking logarithm of (16) and then differentiating the result partially with respect to 𝛼: ∑ (17) Equalizing (17) to zero to get the estimated scale parameter of IED 𝛼 ∑ (18) Hence, for strength random samples, 𝑋 ~𝐼𝐸 𝛼 ; 𝑖 1,2, … , 𝑛 , 𝑋 ~𝐼𝐸 𝛼 ; 𝑖 1,2, … , 𝑛 and 𝑋 ~𝐼𝐸 𝛼 ; 𝑖 1,2, … , 𝑛 with samples size 𝑛 , 𝑛 and 𝑛 respectively MLE for unknown parameters 𝛼 , 𝛼 and 𝛼 is : 𝛼 ; 𝑇 ∑ ; 𝛬 1,2,3 (19) By the same way for the stress random variables𝑌 ~𝐼𝐸 𝛽 ; 𝑗 1,2, … , 𝑚 , 𝑌 ~𝐼𝐸 𝛽 ; 𝑗 1,2, … , 𝑚 and 𝑌 ~𝐼𝐸 𝛽 ; 𝑗 1,2, … , 𝑚 with samples size 𝑚 , 𝑚 and 𝑚 respectively, the MLE estimator for unknown parameters 𝛽 , 𝛽 and 𝛽 will be as follows: 𝛽 ; 𝐻 ∑ , 𝛬 1,2,3 (20) Substitution (19) and (20) in (15) the MLE for Cascade reliability, invariability will be as: 𝑅 (21) 4. Uniformly Minimum Variance Unbiased Estimators The UMVUE method depends on minimizing the mean square error among unbiased estimators. The unbiased estimator 𝛼 of 𝛼 is called (UMVUE) if and only if Var (𝛼 )≤ Var(𝛼 ) for any 𝑥 ∈ 𝑋 and any other unbiased estimator of α, see Devor and Berk (2012). We could find the UMVU of the scale parameters 𝛼 ,𝑎𝑛𝑑 𝛽 ; 𝛬 =1,2,3 of the stress 𝑋 ; 𝑖 1,2, … , 𝑛 and strength 𝑌 𝑗 1,2, … , 𝑚 ;of cascade reliability model with IED that belongs to exponential family densities function as it shown below : 𝑄 𝑥; 𝜎 𝑎 𝜎 𝑏 𝑥 exp ∑ 𝜌 𝜎 𝑟 𝑥 , where 𝑎 𝜎 , 𝑏 𝑥 𝑜 , and 𝜎𝜎 , 𝜎 , … , 𝜎 with 𝛾 𝜎 𝛿 and each of 𝜎, 𝛾 𝑎𝑛𝑑𝛿 are constant .   86  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 Let 𝑎 𝜎 = 𝛼 ; 𝑏 𝑥 1/𝑥 ; 𝜌 𝜎 𝛼 ; 𝑟 𝑥 Thus, 𝑇 is a complete sufficient statistic for (𝛼 ) for 𝛬 =1, 2,3. To find the distribution of 𝑇 ∑ , suppose 𝑍 consequently, 𝑋 𝓅 𝑧 𝑓 𝑋 (22) Substitute (2) in (22) 𝓅 𝑧 𝛼 exp 𝛼 𝑧 (23) Thus 𝑍 ~𝐸𝑥𝑝 𝛼 ,and 𝑇 ~𝛤 𝑛 , 𝛼 𝑈 𝑡 𝑡 . exp 𝛼 𝑡 ; 𝑡 0 , 𝛼 0 , 𝑛 0 (24) then 𝐸 So the unbiased estimator of (𝛼 ) is ( ), therefore according to Lehmann-Scheffe theorem the (UMVUE) of (𝛼 ) is 𝛼 Hence 𝛼 ; 𝑇 ∑ , 𝑘1,2,3 (25) By the same way, we can obtain (UMVUE) of (𝛽 ) as below: 𝛽 ; 𝐻 ∑ , 𝛬 1,2,3 (26) Substituting (25), (26) in (15) to obtain UMVU estimator for cascade reliability model of IED as the following : 𝑅 (27) 5. Preliminary Test Single Shrinkage Estimator (PTSSSE). In this section, we use (1) to estimate the cascade reliability parameters 𝛼 , 𝛽 ; 𝛬 1,2,3 of 𝑋 ~𝐼𝐸𝐷 and 𝑌 ~𝐼𝐸𝐷 respectively. First in order to find PTSSSE of 𝛼 choose ∅ 𝛼 as follows: ∅ 𝛼 𝜗 𝛼 𝑖𝑓𝛼 ∈ 𝑅 𝜗 𝛼 𝑖𝑓𝛼 ∉ 𝑅 (28) Where 𝑅 is a preliminary test region of acceptances of size (𝜆 ) for testing hypotheses 𝐻 : 𝛼 𝛼 vs. 𝐻 : 𝛼 𝛼 using the test statistic 𝜏 ; 𝛼 𝛼 thus, by rewriting equation(1) as below: 𝛼 𝜗 𝛼 𝛼 𝛼 𝛼 , 𝑖𝑓𝛼 ∈ 𝑅 𝜗 𝛼 𝛼 𝛼 𝛼 , 𝑖𝑓𝛼 ∉ 𝑅 (29) Where 𝜗 𝛼 ; 𝑟 1,2 represents the shrinkage weight factor which may be a function of 𝛼 or may be constants with the condition 0 𝜗 𝛼 1 . Using the form (29), we proposed   87  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 two preliminary test single stage shrinkage estimators 𝛼 when a prior knowledge 𝛼 is available for 𝛼 𝛼 𝛼 , 𝑖𝑓𝛼 ∈ 𝑅 𝛼 , 𝑖𝑓𝛼 ∉ 𝑅 (30) Then 𝜗 𝛼 0 and 𝜗 𝛼 1 in (28) And 𝛼 𝛼 , 𝑖𝑓𝛼 ∈ 𝑅 𝑐 𝛼 𝛼 , 𝑖𝑓𝛼 ∉ 𝑅 (31) Thus 𝜗 𝛼 0 and 𝜗 𝛼 𝑐 in (28) while: 𝑅 𝑎 , 𝑎 ; 𝑎 𝑋 ,⁄ And 𝑎 𝑋 ,⁄ where 𝑋 ,⁄ , 𝑋 ,⁄ are respectively the lower and upper 100 𝜆 2⁄ precedential point of chi-square distribution with (2𝑛 ) degree of freedom. Similarly for parameters 𝛽 ; 𝛬 1,2,3 of stress random variables 𝑌 ~𝐼𝐸𝐷 choose ∅ 𝛽 as below: ∅ 𝛽 𝛩 𝛽 𝑖𝑓𝛽 ∈ 𝑅 𝛩 𝛽 𝑖𝑓𝛽 ∉ 𝑅 (32) Where 𝑅 is a preliminary test region of acceptances of size (𝜆 ) for testing hypotheses Њ : 𝛽 𝛽 vs. Њ : 𝛽 𝛽 using the test statistic 𝜏 ; 𝛽 𝛽 thus, we get: 𝛽 𝛩 𝛽 𝛽 𝛽 𝛽 , 𝑖𝑓𝛽 ∈ 𝑅 𝛩 𝛽 𝛽 𝛽 𝛽 , 𝑖𝑓𝛽 ∉ 𝑅 (33) Such that 0 𝛩 𝛽 1, 𝑟 1,2 and 𝛩 𝛽 represents as shrinkage weight factor which may be constants or functions of 𝛽 . We proposed two preliminary test single stage shrinkage estimators 𝛽 to estimate the reliability parameters when a prior knowledge 𝛽 is available for 𝛽 𝛽 𝛽 , 𝑖𝑓𝛽 ∈ 𝑅 𝛽 , 𝑖𝑓𝛽 ∉ 𝑅 (34) Then 𝛩 𝛽 0 and 𝛩 𝛽 1 in (28) And 𝛽 𝛽 , 𝑖𝑓𝛽 ∈ 𝑅 𝑐 𝛽 𝛽 , 𝑖𝑓𝛽 ∉ 𝑅 (35) Thus 𝛩 𝛽 0 and𝛩 𝛽 𝑐 in (28) Such that 𝑅 𝑠 , 𝑠 while 𝑠 𝑌 ,⁄ and 𝑠 𝑌 ,⁄ 𝑌 ,⁄ , 𝑌 ,⁄ are respectively the lower and upper 100 𝜆 2⁄ precedential point of chi-square distribution with (2𝑚 ) degree of freedom. Substituting (30),(31),(34) and (35) in (15) to obtain preliminary test single Shrinkage estimator for cascade reliability model(2+1) of IED as the following   88  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 𝑅 (36) 6. Simulation Study In this section, we present some results based on Monte Carlo simulations to compare the performance of different methods: MLE, UMVUE and PTSSSE of Cascade Reliability Model(2+1) using different sample (20, 40, and 60). For this purpose the following steps of Monte Carlo simulation are used based on Mean Squared Errors criteria with 1000 replicates: Step1: Generate random samples which follow the continuous uniform distribution defined on the interval (0,1), as 𝑢 , 𝑢 , … , 𝑢 and 𝑤 , 𝑤 , … , 𝑤 ,for all 𝑖 1,2, … , 𝑘 𝑎𝑛𝑑 𝑗 1,2, … , 𝑛 . respectively., Step2: Applying an inverse transformation approach to generate random variables follows IED as follows: 𝐹 𝑥 𝑒𝑥𝑝 : 𝑈𝑖 𝑒𝑥𝑝 : 𝑥 𝛼 / ln 𝑈𝑖 And, by the same method, we get: 𝑦 𝛼 / ln 𝑉𝑗 Step3: Recall the R from equation 15 . Step4: find the cascade reliability R of the MLE using equation 21 . Step5: find the uniformly minimum variance unbiased method of R using equation 27 . Step6: Compute (PTSSSE) of cascade reliability using equations (36). Step7: Calculate the MSE based on (L=1000) trials as follows: MSE ∑ 𝑅 𝑅 7. Results of Simulation In this section, the simulation results are used to determine the best outcome of the conceder estimation methods ( 𝑀𝐿𝐸, UMVUE, PTSSSE) of cascade reliability model (2+1) estimator based on one parameter IED. In the cascade reliability model (2+1) of estimate the system reliability 𝑅 𝑝 𝑋 𝑌 , 𝑋 𝑌 𝑝 𝑋 𝑌 , 𝑋 𝑌 , 𝑋 𝑌 𝑝 𝑋 𝑌 , 𝑋 𝑌 , 𝑋 𝑌 )).The following tables of mean square error show, at most the orders rank of the estimators as follows: 𝑅 , 𝑅 and 𝑅 respectively, that means 𝑅 is the best than the others estimators. The following tables (1-9) will present the simulation results. 𝐓𝐚𝐛𝐥𝐞 𝟏: 𝛼 1 , 𝛼 1, 𝛽 1.5 , 𝛽 1.5 , 𝑅 0.800000000 MLE PT UM (n1, n2,m1,m2) m k 0.800055821748313 0.800000399098201 0.800040031839184 (20, 20, 20, 20) 0.2 1.5 0.800045359839790 0.800000862941454 0.800040032049879 ( 60,60, 20, 40) 0.800053589528536 0.799999583412720 0.800040032014180 ( 40, 20, 60, 60) 0.5 1.8 0.800019751832772 0.799999000261255 0.800040031806876 (20,40,60, 20) 0.800033275056292 0.799999163925098 0.800040032055934 ( 40, 60, 20, 40) 0.7 1.3 0.800056080018052 0.800000092631102 0.800040032013649 (60, 20, 40, 60) 0.800050772085346 0.800000881956533 0.800040032025629 (40, 40, 40, 40) 0.9 1.1 0.8000403019787380.7999997959707440.800040032025629(60,60,60, 60)   89  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 𝐓𝐚𝐛𝐥𝐞𝟐: 𝑀𝑆𝐸 𝛼 1 , 𝛼 1, 𝛽 1.5 , 𝛽 1.5   𝐓𝐚𝐛𝐥𝐞 𝟑: 𝛼 1.5 , 𝛼 1.5 , 𝛽 2 , 𝛽 2 , 𝑅 0.7857142857  MLE PT UM (n1, n2,m1,m2) m k 0.785720768248288 0.785714149871148 0.785734705446280 (20, 20, 20, 20) 0.2 1.5 0.785737729080948 0.785713939601543 0.785734705585085 ( 60,60, 20, 40) 0.785750736020131 0.785715533029282 0.785734705520629 (40, 20, 60, 60) 0.5 1.8 0.785718895194405 0.785714520693756 0.785734705528084 (20,40,60, 20) 0.785725567713847 0.785712470091343 0.785734705594252 ( 40, 60, 20, 40) 0.7 1.3 0.785748611031530 0.785714534388167 0.785734705530414 (60, 20, 40, 60) 0.785728999913387 0.785713842247919 0.785734705546020 (40, 40, 40, 40) 0.9 1.1 0.785729005582190 0.785714339660318 0.785734705546020 (60,60,60, 60) 𝐓𝐚𝐛𝐥𝐞 4: 𝛼 1.5 , 𝛼 1.5 , 𝛽 2 , 𝛽 2  MLE PT UM (n1,n2,m1,m2) m k 0.000000291526563 0.000000001318706 0.000000000416965 (20, 20, 20, 20) 0.2 1.5 0.000026615438193 0.000000578485132 0.000000416971124 ( 60,60, 20, 40) 0.000020849454688 0.000000456480257 0.000000416968491 (40, 20, 60, 60) 0.5 1.8 0.000000123002043 0.000000001195640 0.000000000416969 (20,40,60, 20) 0.000000045033693 0.000000000680612 0.000000000416971 (40, 60, 20, 40) 0.7 1.3 0.000037108268999 0.000000466212429 0.000000416968890 (60, 20, 40, 60) 0.000054516476000 0.000000576798620 0.000000416969528 (40, 40, 40, 40) 0.9 1.1 0.000028051216264 0.000000383695325 0.000000416969528 (60,60,60, 60) 𝐓𝐚𝐛𝐥𝐞 𝟓 𝛼 3 , 𝛼 3 , 𝛽 1 , 𝛽 1, 𝑅 0.6250000000  MLE PT UM (n1,n2,m1,m2) m k 0.624932272202343 0.624999705238945 0.624937468791855 (20, 20, 20, 20 ) 0.2 1.5 0.624953403330766 0.624999441035243 0.624937468537429 ( 60,60, 20, 40) 0.624940516955036 0.625000056038205 0.624937468714946 ( 40, 20, 60, 60) 0.5 1.8 0.624913690218326 0.624999904721153 0.624937469058640 (20,40,60, 20) 0.624943162050123 0.624998720248892 0.624937468534079 ( 40, 60, 20, 40) 0.7 1.3 0.624939686353734 0.624998458153044 0.624937468736565 (60, 20, 40, 60) 0.624942577777575 0.625002127090182 0.624937468734366 (40, 40, 40, 40) 0.9 1.1 0.624942314238157 0.625000176762664 0.624937468734366 (60,60,60, 60)     MLE PT UM (n1, n2,m1,m2) m k 0.000000071936176 0.000000000868428 0.000000001602548 (20, 20, 20, 20) 0.2 1.5 0.000042789507494 0.000000456950252 0.000001602565018 ( 60,60, 20, 40) 0.000042894580602 0.000000397091475 0.000001602562159 ( 40, 20, 60, 60) 0.5 1.8 0.000000345959007 0.000000000960463 0.000000001602546 (20,40,60, 20) 0.000032505999070 0.000000433528185 0.000001602565503 ( 40, 60, 20, 40) 0.7 1.3 0.000044937037854 0.000000382471540 0.000001602562117 (60, 20, 40, 60) 0.000032823139814 0.000000389473349 0.000001602563075 (40, 40, 40, 40) 0.9 1.1 0.000014943488773 0.000000306936893 0.000001602563075 (60,60,60, 60)   90  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 𝐓𝐚𝐛𝐥𝐞 𝟔: 𝛼 3 , 𝛼 3 , 𝛽 1 , 𝛽 1  8. Conclusion the cascade reliability model (2+1) 𝑅 𝑝 𝑋 𝑌 , 𝑋 𝑌 𝑝 𝑋 𝑌 , 𝑋 𝑌 , 𝑋 𝑌 𝑝 𝑋 𝑌 , 𝑋 𝑌 , 𝑋 𝑌 based on inverted exponential distribution were used in this paper to verify the performance of different estimators which are ; Maximum likelihood estimation, Unbiased estimation method and Preliminary test single stage shrinkage (PTSSS) estimator using different samples 20, 40, and 60 . The Monte Carlo Simulation was exhibited to analyses and comparison between these methods. the results show that the performance of (PTSSS) estimator was appropriate behavior and it is efficient estimator than the others in the sense of 𝑀𝑆𝐸 at most. While 𝑅 had the second rank and followed by 𝑅 . References 1. Pandit, S. N. N.; Sriwastav, G. L. Studies in Cascade Reliability I. IEEE Trans. on Reliability.1975, 24,1, 53-56. 2. Rekha, A.; Raju, V.C. C. Cascade System Reliability with Rayleigh Distribution. Botswana Journal of Technology.1999, 8, 14-19. 3. 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MLE PT UM n1,n2,m1,m2) m k 0.000000239325138 0.000000003234843 0.000000003910152 (20, 20, 20, 20) 0.2 1.5 0.000000085454692 0.000000001870957 0.000000003910184 ( 60,60, 20, 40) 0.000000045087534 0.000000001122373 0.000000003910162 (40, 20, 60, 60) 0.5 1.8 0.000000181399978 0.000000002522772 0.000000003910119 (20,40,60, 20) 0.000000189712498 0.000000001857193 0.000000003910184 (40, 60, 20, 40) 0.7 1.3 0.000000030728521 0.000000001040114 0.000000003910159 (60, 20, 40, 60) 0.000000056503599 0.000000001408268 0.000000003910159 (40, 40, 40, 40) 0.9 1.1 0.000000038498170 0.000000000964106 0.000000003910159 (60,60,60,60   91  Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 9. Salman, A.S.; Ali, A.H; Salman, M.D. Preliminary Test Single Stage Shrinkage Estimator for the Scale Parameter of Gamma Distribution. American Journal of Mathematics and Statistics. 2014, 4, 3, 131-136. 10. Salman, A.S. , Hadi, R.A. 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