Microsoft Word - 158-166


   

158 
  

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

  

  

 

 

Applying Shrinkage Estimation Technique of P(Y<Max X1, X2,…, Xk) in 
Case of Generalized Exponential Distribution  

 
 
 
 
 

 
 
 

 

 
 

Abstract  

     This paper concerned with estimation reliability (𝑅  for K components parallel system of 
the stress-strength model with non-identical components which is subjected to a common 
stress, when the stress and strength follow the Generalized Exponential Distribution (GED) 
with unknown shape parameter α and the known scale parameter θ (θ=1) to be common. 
Different shrinkage estimation methods will be considered to estimate 𝑅  depending on 
maximum likelihood estimator and prior estimates based on simulation using mean squared 
error (MSE) criteria. The study approved that the shrinkage estimation using shrinkage weight 
function was the best. 
 

Keywords: Generalized Exponential Distribution (GED), Stress–Strength (S-S) models, 
Maximum likelihood estimator (MLE), Shrinkage estimator (Sh) and mean squares error 
(MSE). 
 

1- Introduction 
       At a specific time, the reliability is defined as a probability that the intended functions 
under specified operational conditions and environments the item will be performed. The 
reliability function is a function of the lifetime which is monotonically decreasing function; it 
is a quantitative measure of the quality of the item. The item will eventually cease to perform 
its intended function if the service life of the item is allowed to proceed unlimitedly. The 
reliability over time exhibit decreasing in all mechanical systems and their materials degrade 
as they age because their components are not ideal [1]. 
       The widest approach is the well-known application of reliability estimation is the model 
of stress–strength. This model is used in many applications of system collapse and 
engineering such as strength failure and physics. Some systems of engineering, which may 
have more than one component there may fail together or separately.      

Ibn Al Haitham Journal for Pure and Applied Science 

Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index 

Mohammad Khairallah Yaseen  Adel Abdulkadhim Hussein  Abbas Najim Salman 

Doi: 10.30526/33.3.2480 

Department of Mathematics, 
College of Education for Pure
Sciences Ibn Al – Haitham, 

University of  Baghdad 

Department of Studies and 
Planning University of  

Baghdad,  

Article history: Received 7 September 2019, Accepted 4 December 2019, Published in 
July 2020. 

adilabed57@gmail.com 

Department of Mathematics, 
College of Education for Pure 
Sciences Ibn Al – Haitham,  

University of  Baghdad 



   

159 
  

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

In the statistical approach, the model of stress-strength, most of the considerations depend on 
the assumption that the component strengths are independently and identically distributed 
(iid). The assumption of the components of a system are of the different structure identical 
strength distributions may not be quite realistic" in many practical situations [2,3]. 
      The issue of finding the reliability in a model of stress-strength has been discussed 
extensively in the literature when X and Y have some specified distribution. Here are a few 
references to contributions towards these models 
      "Hangal in 1996 obtained the estimation of system reliability with k components either 
series or parallel of a stress-strength model" [4]. He assumed that the Kth components strength 
and the common stress were independent and they follow the Exponential distribution with 
two parameters. Ali H.M. in 2013 estimated the reliability of the stress-strength system with 
non-identical parallel component subject to a common stress using Lomax distribution [5]. 
Sezer and Kinaci in 2013 estimated the stress-strength parameter (R) for a parallel system 
with two components based on masked data by using Exponential distribution; [6]. In 2014, 
Karam and Ali considered estimation of system Reliability for two components parallel 
stress-strength system model with non-identical parallel component which was subjected to 
common stress using Lomax distribution [7]. Recently, Karam in 2016 studied the reliability 
function for a component which had two components parallel which was subjected to 
common stress via Gompertz distribution [8]. 
     "The two-parameters Generalized Exponential Distribution (GED) has been introduced 
and studied quite extensively by Gupta & Kundu (1999, 2001, 2002, 2003, 2004), Raqab 
(2002), Raqab and Ahsanullah (2001), Zheng (2002) and Kundu, Gupta and Manglick (2004).  
The two-parameter GED was an alternative to the well-known two-parameter Gamma, two-
parameter Weibull" [9]. 
The estimation of reliability in the statistical literature is a very common problem. 
         The aim of this paper is to make comparisons between different proposal shrinkage 
estimator for the reliability (𝑅  of K parallel components system in stress-strength model  
(Rk=P(Y<Max. x1,x2,…xk)) when they are follows the Generalized Exponential Distribution 
(GED).  
    The probability density function for a r. v. X follows the Generalized Exponential 
Distribution (GED) is [9]. 

  𝑓 𝑥; α, θ  𝑒  1 𝑒 ,        x>0,  α,θ >0 

Whereas α refer to shape parameter and θ refer to scale parameter and the c.d.f. is:  

  𝐹 𝑥; α, θ 1 𝑒  
In this work, we employee the case when θ will be known (θ 1), this implies to: 
  𝑓 𝑥; 𝛼  𝛼𝑒 1 𝑒                x>0,  α>0                                                               (1) 
And the cumulative distribution function will be: 
F x; 𝛼 1 𝑒                                 x>0,  α>0                                       (2) 
      Assume  x1,x2,…xk arises strength have (GED) the distribution in (1) with parameter 𝛼 , 
i=1,2,…k all the components are subjected to the stress Y that has the same distribution with  
parameter 𝛼 ,  and assumes to be independent and not identical. 
     Now, the reliability of the reliability K component parallel system of the stress-strength 
model is defined as below[6]. 
Rk=P(Y<max x1,x2,…xk) 
Suppose  Z=max(x1,x2,…xk) 



   

160 
  

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

R 𝐹 𝑦 𝑓 𝑦 𝑑𝑦  
Fz(z)=P(Z<z) 
        =P(x1<z)P(x2 <z)…P(xk  <z) 
        = 1 𝑒  1 𝑒 … 1 𝑒  

        = 1 𝑒 ∑  

𝐹 𝑦  =1- 1 𝑒 ∑  

R 1 1 𝑒 ∑  𝛼  𝑒 1 𝑒 dy  

 R
∑

∑
                                                                                                                     (3) 

2. Estimation Methods of Rk: 
2-1 Maximum Likelihood Estimator (MLE): 
      Let k be a component of a system is put on the life–testing experiment. Suppose 
𝑥 , 𝑥 ,…, 𝑥 , 𝑥 , 𝑥 ,…,𝑥 ,…, 𝑥 , 𝑥 ,…,𝑥 , and 𝑦 ; t=1,2,…,m. then the 
likelihood function of the mentioned system will be[7]. 

𝑙 L 𝛼 , 1; 𝑥 , 𝑦 ∏ ∏ 𝑓 𝑥 ∏ 𝑔 𝑦  = ∏ 𝛼 ∏ ∏ 𝑒
∑ ∑  

 

   ∏ ∏ 1 𝑒 𝛼 ∏ 𝑒 ∑ ∏ 1 𝑒   

 
Take the logarithm of both sides, we get:  
 
Ln(l)=𝑛 ∑ 𝑙𝑛𝛼 ∑ ∑ 𝑥  

𝛼 1 ∑ ∑ 𝑙𝑛 1 𝑒 𝑚 𝑙𝑛𝛼 ∑ 𝑦 𝛼 1 ∑ ln 1 𝑒   

Derive the last equation with respect to the unknown shape parameters 𝛼  (i=1,2,…,k+1) and 
equate the result to zero, we get: 
 

∑ 𝑙𝑛 1 𝑒 0,      i=1,2,…,k  

∑ 𝑙𝑛 1 𝑒 0  

 
 Thus, the maximum likelihood estimator of the parameter 𝛼  (i=1,2,…,k+1) is as follows: 
𝛼

∑
 ;i=1,2,…,k                                                                         (4) 

𝛼
∑

                                                                                     (5) 

    To obtain the reliability estimation for K component parallel system of (S-S) model (𝑅  
using Maximum Likelihood method, we must substitute 𝛼  and 𝛼 in equation (3) 
then we get the following: 

R
∑

∑
                                                                                              (6) 

                                                                   
2 Shrinkage Estimation Method (Sh):-2  

      Thompson in 1968 suggested the problem of shrink a usual estimator 𝛼 unbiased of the 
parameter 𝛼 to prior information 𝛼  using shrinkage weight factor ∅ 𝛼), such that 0 
∅ 𝛼 1. Thompson said that "We are estimating 𝛼 and we believe 𝛼  is closed to 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 denoted by 𝛼  defined as below: see in [10-14]. 



   

161 
  

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

𝛼 ∅ 𝛼 𝛼 1 ∅ 𝛼 𝛼                                                                               (7) 
    Now, we use the unbiased estimator 𝛼  as a usual estimator and 𝛼  estimator as initial 
estimate (prior estimation) of α in equation (7). 
    Where ∅ 𝛼  denotes the shrinkage weight factor as we mentioned above such that as 0
∅ 𝛼 1, which may be a function of 𝛼 , a function of sample size (ni, m) or  may be a 
constant or can be found by minimizing the mean square error of 𝛼 . 
Thus, the shrinkage estimator for the shape parameter 𝛼 of GED will be as follows: 
 𝛼 ∅ 𝛼 𝛼 1 ∅ 𝛼 𝛼                                                                           (8) 
 Where, 

𝛼 𝛼
∑

 , i=1,2,...,k 

Implies,  

E(𝛼 ) = 𝛼  and Var(𝛼 ) = 
 

 , 

Also,  𝛼 𝛼
∑

   

As well as, E(𝛼 ) = 𝛼  and Var(𝛼 ) =  . 

 
2-2-1 Shrinkage Estimator using Shrinkage Weight Function (sh1):   
      The shrinkage weight factor here is considered as a function of ni and m respectively as 
the following: 
∅ 𝛼 𝑒  , and  ∅ 𝛼 𝑒 ; i=1,2,…,k  
     Then the shrinkage estimator of 𝛼 which is defined in equation (8) using above shrinkage 
weight function will be as below, knowing that ni and m are defined in subsection (2-1). 
𝛼 ∅ 𝛼 𝛼 1 ∅ 𝛼 𝛼        for i=1,2,…,k+1                                     (9) 
 
   To obtain the reliability estimation for K component parallel system of (S-S) model 𝑅  
using shrinkage weight function, we must substitute equation (9) in equation (3) then we get 
the following: 
 

R
∑

∑
                                                                                              (10) 

2-2-2 Constant Shrinkage Weight Factor (sh2):  
    We suggest in this subsection constant shrinkage weight factor 𝜑 𝛼 0.3;  i=1,2,…,k+1. 
Therefore, the shrinkage estimator using specific constant weight factor will be as follows: 
𝛼 𝜑 𝛼 𝛼 1 𝜑 𝛼 𝛼        for i=1,2,…,k+1                               (11) 

   To find the reliability estimation for K component parallel system of (S-S) model using the 
constant shrinkage weight factor we must substitute equation (11) in equation (3) then we get 
the following: 

R
∑

∑
                                                                                                        (12) 

2-2-3 Modified Thompson Type Shrinkage Weight Function (th): 
      In this subsection, we introduce modifying the shrinkage weight factor considered by 
Thompson in 1968 as below.  

𝛾 𝛼
 

0.01 ;     for i =1,2,…,k+1.  

Therefore, the shrinkage estimator of using above modified shrinkage weight factor will be: 



   

162 
  

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

 𝛼 𝛾 𝛼 𝛼 1 𝛾 𝛼 𝛼       ;    for i =1,2,…,k+1                                 (13) 
To find the reliability estimation for K component parallel system of (S-S) model using 
modified Thompson type shrinkage we must substitute equation (13) in equation (3) then we 
get the following: 

R
∑

∑
                                   (14) 

                                                                                
3. Simulation Study 
     The performance and comparisons of the considered estimators of reliability are studied 
for numerical results and obtained using sample of different size =(10, 30, 50 and 100), based 
on 1000 repetition via MSE criteria. Monte Carlo simulation was used for this purpose as it 
steps below: -    
step1: the random sample was generated following the uniform continuous distribution which 
is defined on the interval (0,1) as ui1,ui2,…,uini ;i=1,2,…,k. 
step2: we will generate the random sample follows the continuous uniform distribution 
defined on the interval (0, 1) as w1, w2,…, wm. 
step3: the above uniform random samples transformed to GED using the cumulative 
distribution function (c. d. f.) as follow:  
F x   1 𝑒   
Uij  1 𝑒  

x ln 1 𝑈;        i=1,2…,k, j=1,2,...,n  

And, by the same method, we get 

y  ln 1 𝑊,      t=1,2,...,m 

in method of maximum likelihood using equation (6). kStep4: we compute the estimator of R  

Step5: we compute the estimators of Rk in three shrinkage methods using equations (10), (12) 
and (14). 
Step6: on (L=1000) repetition, we calculate the mean square error (MSE) as it comes: 

MSE ∑ 𝑅 𝑅   

of the proposed estimators. kindicates the real value of Reliability R 𝑅Whereas   
 
 

.= 0.64705882 , α1=1.5, α2=1.9, α 3=2.1, α 4=3 kThe estimation value when R Table 1. 
  

 n1, n2, n3, m 𝑅  𝑅  𝑅  𝑅  
10, 10, 10, 10 0.64974127 0.64705883 0.64710777 0.64718198 
10, 100, 50, 30 0.65146655 0.64705883 0.64732080 0.64728223 
30, 10, 50, 100 0.65669301 0.64705886 0.64721804 0.64741435 
30, 50, 100, 10 0.63666133 0.64705881 0.64715662 0.64652903 
50, 100, 30, 30 0.64701349 0.64705882 0.64719754 0.64711634 
50, 10, 30, 50 0.65380516 0.64705889 0.64706895 0.64721992 
100, 10, 50, 30 0.64988329 0.64705885 0.64703872 0.64713598 
100, 50, 30, 10 0.63613730 0.64705888 0.64723084 0.64677433 

  

  

  

 



   

163 
  

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

.= 0.64705882 , α1=1.5, α2=1.9, α 3=2.1, α 4=3 kR  value whenThe MSE  Table 2. 
 

n1, n2, n3, m 𝑅  𝑅  𝑅  𝑅  Best 
10, 10, 10, 10 0.0068345086009 0.0000000000166 0.0000785103279 0.0000379840661 𝑅  
10, 100, 50, 30 0.0023974820438 0.0000000000011 0.0000248009970 0.0000116520014 𝑅  
30, 10, 50, 100 0.0015326690134 0.0000000000016 0.0000152722700 0.0000076319598 𝑅  
30, 50, 100, 10 0.0061024701636 0.0000000000143 0.0000687911107 0.0000361160422 𝑅  
50, 100, 30, 30 0.0020754099172 0.0000000000000 0.0000212700068 0.0000101099334 𝑅  
50, 10, 30, 50 0.0021413961985 0.0000000000015 0.0000222919914 0.0000107131426 𝑅  
100, 10, 50, 30 0.0026738779103 0.0000000000018 0.0000287629381 0.0000144133293 𝑅  
100, 50, 30, 10 0.0056015578911 0.0000000000113 0.0000584342389 0.0000268982293 𝑅  

.= 0.6785714 , α1=1.1, α2=1.2, α 3=1.5, α 4=1.8 kThe estimation value when R Table 3.  
  

 n1, n2, n3, m 𝑅  𝑅  𝑅  𝑅  
10, 10, 10, 10 0.67866259 0.67857132 0.67836930 0.67852404 
10, 100, 50, 30 0.71429325 0.67856884 0.68137355 0.68053712 
30, 10, 50, 100 0.68400211 0.67857136 0.67839122 0.67867854 
30, 50, 100, 10 0.85657436 0.67857813 0.69362884 0.6904753 
50, 100, 30, 30 0.59101645 0.67857142 0.67160744 0.67225224 
50, 10, 30, 50 0.46316784 0.67856860 0.66490300 0.66659916 
100, 10, 50, 30 0.68193435 0.67857142 0.67868409 0.67870308 
100, 50, 30, 10 0.94405220 0.67858039 0.69525541 0.69315286 

 

.= 0.6785714, α1=1.1, α2=1.2, α 3=1.5, α4=1.8 kR  value whenMSE  The Table 4. 
  

n1, n2, n3, m 𝑅  𝑅  𝑅  𝑅  Best 

10, 10, 10, 10 0.006413473894 0.000000000016 0.000076589118 0.000039866197 𝑅  
10, 100, 50, 30 0.0021457805724 0.0000000000066 0.0000125077396 0.0000079031698 𝑅  
30, 10, 50, 100 0.0012709588070 0.0000000000011 0.0000127008484 0.0000058904503 𝑅  
30, 50, 100, 10 0.031954410612 0.0000000000450 0.0002301025638 0.0001437716440 𝑅  
50, 100, 30, 30 0.009892213008 0.000000000000 0.0000670207090 0.0000491702945 𝑅  
50, 10, 30, 50 0.0479186501084 0.0000000000079 0.0001973440002 0.0001483221229 𝑅  
100, 10, 50, 30 0.002481844844 0.0000000000012 0.0000261782216 0.0000129859998 𝑅  
100, 50, 30, 10 0.0704976787510 0.000000000080 0.0002800972547 0.0002136971265 𝑅  

 

.α1=1, α2=2, α 3=3, α 4=2.7 0.68965517,=  kestimation value when RThe . Table  5 
  

 n1, n2, n3, m 𝑅  𝑅  𝑅  𝑅  
10, 10, 10, 10 0.80040930 0.68965568 0.69089509 0.69099792 
10, 100, 50, 30 0.6884277 0.68965515 0.68950451 0.68959020 
30, 10, 50, 100 0.69469071 0.68965516 0.68945879 0.68974836 
30, 50, 100, 10 0.67838443 0.68965516 0.68974088 0.68913416 
50, 100, 30, 30 0.68716399 0.68965517 0.68953574 0.68950827 
50, 10, 30, 50 0.69398643 0.68965514 0.68954182 0.68976537 
100, 10, 50, 30 0.68908654 0.68965516 0.68934005 0.68950500 
100, 50, 30, 10 0.67662788 0.68965513 0.68951009 0.68918936 

  

  

 



   

164 
  

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

.= 0.68965517, α1=1, α2=2, α 3=3, α 4=2.7 kR  value whenThe MSE  Table 6. 
  

n1, n2, n3, m 𝑅  𝑅  𝑅  𝑅  Best 
10, 10, 10, 10 0.01229792159789 0.00000000000026 0.0000015438839 0.0000018105906 𝑅  
10, 100, 50, 30 0.0020793896045 0.0000000000003 0.0000212660377 0.0000100571815 𝑅  
30, 10, 50, 100 0.0013116339341 0.0000000000011 0.0000135119636 0.0000065066202 𝑅  
30, 50, 100, 10 0.0055564329633 0.0000000000126 0.0000609082708 0.0000319882618 𝑅  
50, 100, 30, 30 0.002072936381 0.0000000000000 0.000022190642 0.000011232268 𝑅  
50, 10, 30, 50 0.0020661841322 0.0000000000013 0.0000219388121 0.0000106684559 𝑅  
100, 10, 50, 30 0.0022984507903 0.0000000000012 0.0000240920439 0.0000111287847 𝑅  
100, 50, 30, 10 0.0056989431761 0.0000000000118 0.0000604307903 0.0000308152485 𝑅  

  

.α1=2, α2=3, α 3=3, α 4=2 0.8000000,=  kThe estimation value when R Table 7.                    
  

 n1, n2, n3, m 𝑅  𝑅  𝑅  𝑅  
10, 10, 10, 10 0.79732879 0.79999992 0.79981682 0.79994799 
10, 100, 50, 30 0.80021734 0.79999996 0.79999223 0.80000082 
30, 10, 50, 100 0.80422188 0.79999999 0.79992073 0.80014540 
30, 50, 100, 10 0.78689067 0.79999990 0.79980236 0.79949901 
50, 100, 30, 30 0.79698163 0.79999999 0.79987571 0.79986822 
50, 10, 30, 50 0.80392120 0.80000001 0.80000297 0.80016313 
100, 10, 50, 30 0.80006170 0.79999996 0.79987705 0.79999486 
100, 50, 30, 10 0.79144712 0.80000007 0.80024377 0.79978967 

 

.= 0.8000000 , α1=2, α2=3, α 3=3, α 4=2 kR  value whenThe MSE  Table 8. 
 

n1, n2, n3, m 𝑅  𝑅  𝑅  𝑅  Best 

10, 10, 10, 10 0.0036930321488 0.0000000000088 0.0000416870241 0.0000218510918 𝑅  
10, 100, 50, 30 0.0011807809520 0.0000000000003 0.0000119623355 0.0000055447844 𝑅  
30, 10, 50, 100 0.0007760955807 0.0000000000009 0.0000082061304 0.0000039207273 𝑅  
30, 50, 100, 10 0.0035662706436 0.0000000000068 0.0000348598683 0.0000183278607 𝑅  
50, 100, 30, 30 0.0011429353051 0.0000000000000 0.0000115756637 0.0000058533095 𝑅  
50, 10, 30, 50 0.0010247431814 0.0000000000009 0.0000110864286 0.0000055558376 𝑅  
100, 10, 50, 30 0.0014165597010 0.0000000000009 0.0000148389019 0.0000071877688 𝑅  
100, 50, 30, 10 0.0031290741177 0.0000000000059 0.0000309491744 0.0000151569884 𝑅  

 

.α1=2, α2=3, α 3=3, α 4=2 0.6296296,= kThe estimation value when R Table 9. 
  

 n1, n2, n3, m 𝑅  𝑅  𝑅  𝑅  
10, 10, 10, 10 0.63076360 0.62962956 0.62951463 0.62964048 
10, 100, 50, 30 0.63172841 0.62962958 0.62951579 0.62962435 
30, 10, 50, 100 0.63790618 0.62962956 0.62958384 0.62986333 
30, 50, 100, 10   0.620838632 0.62962972 0.62995512 0.62935807 
50, 100, 30, 30 0.62658135 0.62962962 0.62943595 0.62939530 
50, 10, 30, 50 0.63765206 0.62962964 0.62974108 0.62990918 
100, 10, 50, 30 0.63290386 0.62962961 0.62962576 0.62977961 
100, 50, 30, 10 0.62207442 0.62962977 0.63004524 0.62929717 

 

 



   

165 
  

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

.= 0.6296296 , α1=2, α2=3, α 3=3, α 4=2 kR  value when. The MSE Table 10 
  

n1, n2, n3, m 𝑅  𝑅  𝑅  𝑅  Best 
10, 10, 10, 10 0.0069000376819 0.0000000000169 0.0000799910406 0.0000388496851 𝑅  
10, 100, 50, 30 0.0026648513273 0.0000000000014 0.0000282562972 0.0000140228606 𝑅  
30, 10, 50, 100 0.0019060448282 0.0000000000020 0.0000199887466 0.0000100559705 𝑅  
30, 50, 100, 10 0.0057617706197 0.0000000000126 0.0000630168023 0.0000304912831 𝑅  
50, 100, 30, 30 0.0025093087038 0.0000000000000 0.0000262623068 0.0000129639110 𝑅  
50, 10, 30, 50 0.0022561911791 0.0000000000023 0.0000244149758 0.0000123993849 𝑅  
100, 10, 50, 30 0.0031588200355 0.0000000000023 0.0000338946302 0.0000168170931 𝑅  
100, 50, 30, 10 0.0059428138792 0.0000000000136 0.0000667599379 0.0000335650838 𝑅  

 
Numerical results:                                                                                                
      In this work, when we notice for any ni =(10,30,50 and 100) and all m=(10,30,50 and 100) 
the minimum mean square error (MSE) for the reliability estimator of K components parallel 
system of (S-S) model with the Generalized Exponential Distribution held using the shrinkage 
estimator as function of ni and m based on weight factor of shrinkage. 
This implies that shrinkage weight factor as function of ni and m is the best and it is follow by 

using modified Thompson type shrinkage estimator R . For all ni and all m, the third order 
best estimator is the shrinkage estimator based on constant shrinkage weight function R  

after R  and ( R  . 
 
4. Conclusion 
      Of the previous item, the proposal can be found that the shrinkage estimation method 

using shrinkage weight factor as function of ni and m (𝑅 ) which depends on prior 
estimate and unbiased estimator as a linear combination is the best estimator and performs 
good behavior than the others in the sense of  MSE when estimate reliability for system 
contains Kth non-identical strength parallel components which are subjected to a common 
stress in stress-strength model and when the stress and strength follow the Generalized 
Exponential Distribution (GED). 
 
References  

1. Azarkhail. M.; Modarres. M. The Evolution and History of Reliability Engineering: Rise 
of Mechanistic Reliability Modeling. International Journal of Performability 
Engineering.2012, 8, 1, 35-47. 

2. Hassan. A.S.; Basheikh, H.M. Reliability Estimation of Stress - Strength Model with 
Non-Identical Component Strengths: The Exponentiated Pareto Case. International 
Journal of Engineering Research and Applications.2012, 2, 3, 2774-2781. 

3. Hussian, M.A. Estimation of Stress -Strength Model for Generalized Inverted 
Exponential Distribution Using Ranked Set Sampling. International Journal of 
Advances in Engineering & Technology.2014, 6, 6, 2354-2362. 

4.  Hangal, D.D. On the estimation of system reliability in stress-strength models. 
Economic Quality Control Journal.1998, 12, 17-22. 

5. Ali, H.M. Comparison of some methods to estimate some of the stress–strength Lomax 
models. M.Sc. thesis, Al -Mustansiriya University, 2013.      



   

166 
  

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (3) 2020 

6. Sezer, D.; Kinaci, I. Estimation of a stress-strength parameter of a parallel system for 
Exponential Distribution Based on Masked Data. Journal of Selçuk University Natural 
and Applied Science.2013, 2, 3, 60-68. 

7. Karam. N.S.; Ali, H.M. Estimation of Two Component Parallel System Reliability 
Stress-Strength Using Lomax distribution. Magistra.2014, 26, 88, 78- 84. 

8. Karam, N.S. One, Two and Multi–Component Gompertz Stress –Strength Reliability 
Estimation. Mathematical Theory and Modeling.2016, 6, 3, 77-92. 

9. Kundu, D.; Gupta, R.D. Estimation of P(Y < X) for the Generalized Exponential 
Distribution. Metrika.2005, 613, 291-308. 

10. Al-Joboori, A.N.; Hamad, A.M. Mohammad.M.A. On shrunken Estimation of 
Generalized Exponential Distribution. Journal of Economics and Administrative 
Sciences.2011, 17, 1-8. 

11. Al-Joboori, A.N.; Ali, A.H.; Salman, M.D.; Mohammad, M.A. Single and Double Stage 
Shrinkage Estimators for the Normal Mean with the Variance Cases. International 
Journal of statistic.2014, 38, 2, 1127-1134. 

12. Al-Joboori, A.N.; Ameen, M.M. Estimate the Shape Parameter of Generalize Rayleigh 
Distribution Using Bayesian-Shrinkage Technique.  International Journal of Innovative 
Science, Engineering, and Technology.2015, 2, 675-683. 

13. Thompson, J.R. Some shrinkage Techniques for Estimating the mean. J. Amer. Statist. 
Assoc.1968, 63, 113-122. 

14. Al-Hemyari, Z.N.; Al-Joboori, A.N. On Thompson Type Estimators for the Mean of 
Normal Distribution. REVISTA INVESTIGACION OPERACIONAL J.2009, 30, 2, 109-
116.