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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 
 

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Employ Shrinkage Estimation Technique for the Reliability System in Stress-Strength 
Models: special case of Exponentiated Family Distribution   

Eman A.A.                                                                     Abbas N .S. 
Department of Mathematics, College of Education for Pure Sciences (Ibn Al – Haitham), University 

o f  Baghdad 
emanahmed88717@gmail.com                                   abbasnajim66@yahoo.com 

 

 

 

 

Abstract  

       A reliability system of the multi-component stress-strength model R(s,k) will be considered 
in  the present paper ,when the stress and strength are independent and non-identically 
distribution have the Exponentiated Family Distribution(FED) with the unknown  shape 
parameter α and known scale parameter λ  equal to two and parameter θ equal to three. 
Different estimation methods of R(s,k) were introduced corresponding to Maximum likelihood 
and Shrinkage estimators. Comparisons among the suggested estimators were prepared 
depending on simulation established on mean squared error (MSE) criteria. 

Keywords: Exponentiated Family Distribution, Reliability of multicomponent Stress–Strength 
models, Maximum likelihood estimator, Shrinkage estimator and mean squared error. 

1. Introduction  
    A multicomponent system consuming kth strength independently and non-identically random 
variables and everyone subjected to random stress was presented by Bhattacharyya and 
Johnson (1974)[1].The system reliability model, (s out of k ) was denoted by R(s,k) when at 

least s ( 1≤ s≤ k) of components survive. In (2012), Pandit and Kantu studied the reliability 
estimation in multicomponent stress-strength(s-s) with the assumption that strengths and 
stresses followed the Pareto distribution [2] .Rao & Naidu in (2013) studied the 
multicomponent system reliability in (s-s) model for Exponentiated half Logistic 
distribution[3]. In (2014), Rao estimated the reliability system of the multicomponent (s-s) 
model for Generalized Rayleigh distribution through simulation [4].  
   In (2015), Kizilaslan and Nader, studied the multicomponent system assumed that the stress 
and strength of identical independent followed Weibull distribution using ML and Bayes 
methods [5].  In (2016) Rao et al, they estimated the multicomponent reliability in model when 
the stress and strength followed Exponentiated Weibull distribution [6]. In (2017), Abbas and 
Fatima estimated the system reliability of the multicomponent (s-s) model via Exponentiated 

Ibn Al Haitham Journal for Pure and Applied Science 

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

Doi: 10.30526/33.4.2508 

Article history: Received 13 November 2019, Accepted 15 December 2019, Published in October 2020 



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 

Weibull distribution, consuming; ML and MOM estimators and they accomplished results 
agreed that the shrinkage estimator was the best [7]. And in (2019), Abbas and Eman  derived 
and estimated the formula of reliability for multicomponent in (s-s) model in case of 
Exponentiated Pareto distribution and they concluded that the shrinkage estimator was the best 
[8].  
  The aim of this paper is to estimate the reliability of multicomponent system in stress-strength 
model R(s,k) based on Family of Exponentiated distribution with unknown  shape parameter α 
and known the parameters λ=2 and θ=3 and a=0 via different estimation methods like MLE, as 
well as some shrinkage methods and make comparisons among the proposed estimator 
methods using simulation, depending on mean squared error criteria by using special case of 
Family Exponentiated distribution . 

       The probability density function (p. d. f.) of a r.v. X following Exponentiated Family 
Distribution EFD has the form below [9]: 

𝑓 𝑥; 𝛼, 𝜆 𝛼𝜆�̀� 𝑥; 𝑎, 𝜃 𝑒 ; , 1 𝑒 ; , ;   𝑥 0, 𝑎 0, 𝛼, 𝜆 0           (1)   

Here g x; a, 𝜃  is the function of x and may also be of the parameter 𝜃 (may be vector valued) 

and a is constant. Furthermore, g(𝑥; 𝑎, 𝜃) refer to real valued, monotonically increasing 

function of 𝑋 with g(a; a, 𝜃)=0, g(∞; a, 𝜃)= ∞ and 𝑔 (𝑥; 𝑎, 𝜃) refer to the derived of g(𝑥; 𝑎, 𝜃) 

with respect to 𝑥 [9]. 

Consequently, the cumulative distribution function (c.d. f.) of X will be:          

  𝐹 𝑥, 𝛼, 𝜆 1 𝑒 ; ,                   𝑥 0                                                                  (2) 

The first who derived the reliability of a multicomponent system in stress-strength model R(s,k) 
was Bhattacharyya and Johnson (1974) as the following form,[1].  

R(s,k) =P(at least s of  the X1, X2,…, Xk exceed Y) 

Where X1,X2…, Xk with common distribution function F(x) is subjected to the common 
random stress Y with distribution Function G(y) 

R(s,k)  =∑ 1 𝐹 𝑦 𝐹 𝑦 𝑑𝐺 𝑦   

When X~ EFD(𝛼, 𝜆, 𝜃   and Y~ EFD (𝛽, 𝜆, 𝜃  then: 

 R(s,k)= ∑  1 1 𝑒 ; , 1 𝑒 ; , 𝛽𝜆�̀� 𝑦; 𝑎, 𝜃  𝑒 ; ,  1

𝑒 ; , 𝑑𝑦    

R(s,k)  =∑  1 1 𝑒 ; , 1 𝑒 ; , 𝛽𝜆�̀� 𝑦; 𝑎, 𝜃 𝑒 ; ,  1

𝑒 ; , 𝑑𝑦 

Let z=1 𝑒 ; ,   then  𝑑𝑦
 

̀ ; ,
  

∑ 𝛽𝜆 1 𝑧 𝑧  �̀� 𝑦; 𝑎, 𝜃 1 𝑧
 

̀ ; ,
=(s, k )  R  



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 

           =∑ 𝛽 1 𝑧 𝑧  𝑑𝑧  

   Let 𝑤 𝑧  → 𝑧 𝑤  → 𝑑𝑧 𝑤  𝑑𝑤  

And by simplification, R(s,k)  became:  

R(s,k)= ∑
!

!
 ∏ 𝑘 𝑗   ; 𝑘, 𝑖, 𝑗 are integers                                              (3)                

2. Estimation methods of R(s,k) 

3. Maximum Likelihood Estimator (MLE):  

  Consider a random sample x1,x2,…,xn of size n following EFD(𝛼, 2,3  ,  x(1) <x(2)<…<x(n)  , 
the order random sample of x and y1 ,y2,…,ym  is a random sample of size m follows EFD 
(𝛽, 2,3 , and y(1) <y(2)<…<y(m) is the order random sample of y. Then the likelihood functions 
will be as follows: [9] 

L= L(𝛼, 𝛽; 𝑥, 𝑦)=∏ 𝑓 𝑥 ∏ 𝑓  𝑦 )  

=∏ 𝛼𝜆�̀� 𝑥 ; 𝑎, 𝜃 𝑒 ; , 1 𝑒 ; , ∏ 𝛽𝜆�̀� 𝑦 ; 𝑎, 𝜃 𝑒 ; , 1

𝑒 ; ,  

=𝛼 𝜆 𝑒 ∑ ; , ∏ �̀� 𝑥 ; 𝑎, 𝜃 ∏ 1

𝑒 ; , 𝛽 𝑒 ∑ ; , ∏ �̀� 𝑦 ; 𝑎, 𝜃 ∏ 1 𝑒 ; ,  

ln 𝑙   𝑛𝑙𝑛𝛼 𝑛 𝑚 ln 𝜆 ∑ �̀� 𝑥𝑖; 𝑎, 𝜃 𝜆 ∑ 𝑔 𝑥𝑖; 𝑎, 𝜃 𝛼 1 ∑ ln 1 𝑒
𝜆𝑔 𝑥𝑖;𝑎,𝜃  + 𝑚𝑙𝑛𝛽

∑ �̀� 𝑦
𝑗
; 𝑎, 𝜃 𝜆 ∑ 𝑔 𝑦

𝑗
; 𝑎, 𝜃 𝛽 1 ∑ ln 1 𝑒 𝜆𝑔 𝑦𝑗;𝑎,𝜃                                             

∑ 𝑙𝑛 1 𝑒 𝜆𝑔 𝑥𝑖;𝑎,𝜃 0       

∑ 𝑙𝑛 1 𝑒 𝜆𝑔 𝑦𝑗;𝑎,𝜃 0  

Thus, the maximum likelihood estimator of the parameters 𝛼 and β will be as follows: 

α
∑ 𝑒 𝜆𝑔 𝑥𝑖;𝑎,𝜃

                                                                                                   (4)  

𝛽
∑ 1 𝑒

𝜆𝑔 𝑦𝑗;𝑎,𝜃
                                                                                                   (5)  

By substituting α , 𝛽   in equation (3), obtains the Maximum Likelihood estimator for 
R(s,k) : 

𝑅 , ∑
!

!
 ∏ 𝑘 𝑗                                                                   (6)   

                               



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 

4. Shrinkage Estimation Method (Sh1): 

     Thompson in 1968 assumed the problem of shrink was a usual estimator 𝛼  of the 
parameter 𝛼 to prior information  𝛼  using shrinkage weight factor ∅ 𝛼), such that 0
∅ 𝛼 1. Thompson estimating 𝛼 and he believed that  𝛼  was closed to the true value of 𝛼 or 
𝛼  that may be near the true value of  𝛼 .Thus, the form of Thompson - Type shrinkage 
estimator of α say 𝛼  will be[10].  

𝛼 ∅ 𝛼 𝛼 1 ∅ 𝛼 𝛼                                                                                            (7)  
  In this paper, we apply the unbiased estimator 𝛼  as a usual estimator and 𝛼 𝛼 as a prior 
estimation of α in equation (7), as we mentioned, ∅ 𝛼  denotes the shrinkage weight factor 
such that 0 ∅ 𝛼  1, which may be a function of 𝛼 ; a function of sample size (n,m) or  
may be constant (ad hoc basis) . 

Note that: [9]  

𝛼 𝛼 ,  Var (𝛼 ) =   

And,  𝛽 𝛽   , Var ( 𝛽 ) =   

4.1. The Shrinkage Weight Function (sh1)   

       The shrinkage weight factor as a function of sample sizes n and m respectively will be 
considered in this subsection and take the form below:  

i.e.  ∅ 𝛼 Beta (n,m)  and  ∅ 𝛽 Beta m, n     

Where n, and m are sample sizes, therefore the shrinkage estimator using shrinkage weight 
function of 𝛼 and β which is defined in equation (7), will take the following formula: 

𝛼 ∅ 𝛼 𝛼 1 ∅ 𝛼 𝛼                                                                                            (8) 

𝛽 ∅ 𝛽 𝛽 1 ∅ 𝛽 𝛽                                                                                            (9)   

Also, as Thompson mentioned, 𝛼  and  𝛽  are closed to the real value of 𝛼 𝑎𝑛𝑑 𝛽 
respectively.Then, the shrinkage estimation 𝑅 ,  in equation (3) using shrinkage weight 

function will be:-                     

𝑅 , ∑
!

!
 ∏ 𝑘 𝑗                                                            (10)  

4.2. Constant Shrinkage Weight Factor (Sh2): 

We suggest in this subsection constant shrinkage weight factor ∅ 𝛼 0.05, and ∅ 𝛽
0.05. Therefore, the shrinkage estimator through specific constant weight factor will be as 
follows: 

𝛼 0.05  𝛼 0.95 𝛼                                                                                                (11) 

𝛽 0.05 𝛽 0.95 𝛽                                                                                                  (12)       



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 

Substitute equation (11) and (12) in equation (3) to obtain the shrinkage estimation of R(s,k) 
using the above constant shrinkage weight factor as below: 

𝑅 , ∑
!

!
 ∏ 𝑘 𝑗                                                           (13) 

 Where 𝑘, 𝑖, 𝑗 are integers  

4.3. Modified Thompson type Shrinkage Weight Factor (th):  

In this paper, the shrinkage weight factor which was considered by Thompson will be modified 
to give the following formulas: 

𝛾 𝛼
 

   *0.005                                                                                         (14) 

 

𝛾 𝛽
 

  ∗ 0.005                                                                                       (15) 

     
      Therefore, the shrinkage estimator of α and β using modified shrinkage weight factor are 
respectively as bellow: 
   𝛼 𝛾 𝛼 𝛼 1 𝛾 𝛼 𝛼                                                                                           (16) 

   𝛽 𝛾 𝛽 𝛽 1 𝛾 𝛽 𝛽                                                                                          (17) 

Then the Shrinkage estimation of R(s,k) in equation (3) based on Modified Thompson type 
shrinkage weight factor will be:  

𝑅 ,
  

∑
!

!
 ∏ 𝑘

 
𝑗                                                              (18) 

 Now, we take some special case for the Exponentiated Family distribution, by the following 
table: 

Table1. special case for the Exponentiated Family distribution by different values the function 𝑔 𝑥; 𝑎, 𝜃  

Exponentiated 
Rayleigh 

distribution  

Exponentiated 
Pareto 

distribution  

Exponentiated  
Weibull 

distribution  

Exponentiated 
Lomax 

distribution  

Exponentiated 
Exponential 
distribution  

Name  

 

𝑔 𝑥; 𝑎, 𝜃 𝑥  𝑔 𝑥; 𝑎, 𝜃
ln 1 𝑥  

𝑔 𝑥; 𝑎, 𝜃 𝑥  𝑔 𝑥; 𝑎, 𝜃
ln 1 θ𝑥  

𝑔 𝑥; 𝑎, 𝜃 𝑥 Value   
𝑔 𝑥; 𝑎, 𝜃 

 

5. Simulation Experiments   

       In this section, numerical results were studied to compare the performance of the different 
estimators of reliability using different sample size = (10, 30, 50 and 100), based on 1000 
replication via MSE criteria. For this purpose, Monte Carlo simulation was employed by 
generating the random sample from the continuous uniform distribution defined on the interval 



   

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 33 (4) 2020 

(0,1) as u1,u2,…, un; v1, v2,…, vm. Transform uniform random samples to follow some 
distributions special case of the Family of Exponentiated distribution using (c. d. f.),[11].  

The following steps compute the real value of R(s,k)  in equation (3) and the value of estimation 

methods of  all the proposal methods 𝑅 , , 𝑅 , , 𝑅 ,   and  𝑅 ,  in equations (6), 

(10), (13) and (18) respectively.   

Based on (L=1000) Replication, we calculate the MSE for all proposed estimation methods of 

 𝑅(s,k) as follows: 

MSE ∑ 𝑅 , R ,   

   Suppose the reliability system of multicomponent stress- strength model for some 
distributions as the following: 

Table2.  The symbol of the reliability system of multicomponent stress- strength model for special case of 
Exponentiated Family distribution. 

Distributions  Exponentiated 
Exponential 
distribution

Exponentiated 
Lomax 
distribution

Exponentiated  
Weibull 
distribution

Exponentiated 
Pareto 
distribution

Exponentiated 
Rayleigh 
distribution

Symbol R1 ,  R2 ,  R3 ,  R4 ,  R5 ,  
 

Now, we use some special case of Exponentiated Family distribution and applied the Simulation 
to find the value of the reliability and MSE to compare with the other methods.     

Table 3. Values of the R1 ,  for Exponentiated Exponential Distribution whenR1 , 0.48485, s=2, k=4 and 

λ=2 

(n,m) 𝑅1 ,  𝑅1 , 𝒔𝒉𝟏 𝑹1 𝒔,𝒌 𝒔𝒉𝟐 𝑹1 𝒔,𝒌 𝑻𝒉 
(10,10) 0.48236 0.48489 0.48495 0.48491
(30,30) 0.48504 0.48489 0.48494 0.48490
(50,50) 0.48455 0.48489 0.48491 0.48489

(100,100) 0.48634 0.484897 0.48498 0.48490
 

Table 4. Values of MSE the R1 ,  for Exponentiated Exponential Distribution when R1 , 0.48485, s=2, 

k=4  and λ=2 

(n,m) 𝑅1 ,  𝑅1 , 𝒔𝒉𝟏 𝑹1 𝒔,𝒌 𝒔𝒉𝟐 𝑹1 𝒔,𝒌 𝑻𝒉 
(10,10) 0.01521 0.239E-8 0.494E-4 0.202E-6
(30,30) 0.00576 0.239E-8 0.159E-4 0.777E-7
(50,50) 0.00349 0.239E-8 0.916E-5 0.442E-7

(100,100) 0.00165 0.239E-8 0.421E-5 0.217E-7

  

Table 5. Values of the R2 ,  for Exponentiated Lomax Distribution when R2 ,  0.48485, s=2, k=4, λ=2 and 

θ=3. 

(n,m) 𝑅2 ,  𝑅2 , 𝒔𝒉𝟏 𝑹2 𝒔,𝒌 𝒔𝒉𝟐 𝑹2 𝒔,𝒌 𝑻𝒉 
(10,10) 0.74265 0.48489 0.50888 0.48744 
(30,30) 0.78786 0.48489 0.50176 0.48651 
(50,50) 0.78828 0.48489 0.49980 0.48633 
(100,100) 0.78964 0.48489 0.49862 0.48622 



   

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Table 6. Values of the MSE R2 ,  for Exponentiated Lomax Distribution when R2 , 0.48485,  s=2, k=4 
,λ=2 and θ=3. 

(n,m) 𝑅2 ,  𝑅2 , 𝒔𝒉𝟏 𝑹2 𝒔,𝒌 𝒔𝒉𝟐 𝑹2 𝒔,𝒌 𝑻𝒉 
(10,10) 0.11060 0.247E-8 0.00242 0,549E-4
(30,30) 0.10323 0.239E-8 0.00048 0.506E-5
(50,50) 0.09969 0.239E-8 0.00031 0. 301E-5
(100,100) 0.09657 0.239E-8 0.00022 0.219E-5
 

Table 7. Values of the R3 ,  for Exponentiated Weibull Distribution when R3 ,   0.48485, s=2, k=4 ,λ=2 

and θ=3. 

(n,m) 𝑅3 ,  𝑅3 , 𝒔𝒉𝟏 𝑹3 𝒔,𝒌 𝒔𝒉𝟐 𝑹3 𝒔,𝒌 𝑻𝒉 
(10,10) 0.45181 0.48489 0.48437 0.48484

(30,30) 0.45505 0.48489 0.48451 0.48486 
(50,50) 0.45617 0.48489 0.48454 0.48486

(100,100) 0.46025 0.48489 0.48459 0.48487
 

Table 8. Values of the MSE R3 ,  for Exponentiated Weibull Distribution when R3 ,   0.48485 , s=2, k=4 , 

λ=2 and θ=3. 

(n,m) 𝑅3 ,  𝑅3 , 𝒔𝒉𝟏 𝑹3 𝒔,𝒌 𝒔𝒉𝟐 𝑹3 𝒔,𝒌 𝑻𝒉 
(10,10) 0.02444 0.239E-8 0.797E-5 0.847E-7
(30,30) 0.00955 0.239E-8 0.164E-5 0.153E-7
(50,50) 0.00596 0.239E-8 0.927E-6 0.820E-8

(100,100) 0.00317 0.239E-8 0.448E-6 0.404E-8
 

Table 9. Values of the R4 ,  for Exponentiated Pareto Distribution when R4 ,  0.48485, s=2, k=4, and λ=2.  

(n,m) 𝑅4 ,  𝑅4 , 𝒔𝒉𝟏 𝑹4 𝒔,𝒌 𝒔𝒉𝟐 𝑹4 𝒔,𝒌 𝑻𝒉 
(10,10) 0.74995 0.48489 0.51295 0.48795
(30,30) 0.78227 0.48489 0.50203 0.48654
(50,50) 0.78877 0.48489 0.49994 0.48634

(100,100) 0.78742 0.48489 0.49821 0.48618 

 

Table 10.  Values of the MSE R4 ,  for Exponentiated Pareto Distribution when R4 ,  0.48485 , s=2, k=4 
and λ=2 

(n,m) 𝑅4 ,  𝑅4 , 𝒔𝒉𝟏 𝑹4 𝒔,𝒌 𝒔𝒉𝟐 𝑹4 𝒔,𝒌 𝑻𝒉 
(10,10) 0.11724 0.248E-8 0.00375 0.105E-3
(30,30) 0.10319 0.239E-8 0.00050 0.507E-5 

(50,50) 0.09941 0.239E-8 0.00031 0.305E-5
(100,100) 0.09519 0.239E-8 0.00021 0. 203E-5
Table 11.  Values of the R5 ,  for Exponentiated Rayleigh Distribution when R5 ,  0.48485 , s=2, k=4 and 

λ=2. 

(n,m) 𝑅5 ,  𝑅5 , 𝒔𝒉𝟏 𝑹5 𝒔,𝒌 𝒔𝒉𝟐 𝑹5 𝒔,𝒌 𝑻𝒉 
(10,10) 0.45555 0.48489 0.48443 0.48485 
(30,30) 0.45467 0.48489 0.48450 0.48486 
(50,50) 0.45551 0.48489 0.48453 0.48486 
(100,100) 0.45835 0.48489 0.48458 0.48487 
 



   

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Table 12.  Values of the R5 ,  for Exponentiated Rayleigh Distribution when R5 , 0.48485 , s=2, k=4 and 
λ=2. 

(n,m) 𝑅5 ,  𝑅5 , 𝒔𝒉𝟏 𝑹5 𝒔,𝒌 𝒔𝒉𝟐 𝑹5 𝒔,𝒌 𝑻𝒉 
(10,10) 0.02510 0.239E-8 0.886E-5 0.917E-7 

(30,30) 0.00952 0.239E-8 0.163E-5 0.151E-7
(50,50) 0.00611 0.239E-8 0.956E-6 0.839E-8

(100,100) 0.0031 0.239E-8 0.441E-6 0.379E-8

 

6. Discussion Simulation results                                                                                                

      From the tables above, for n= (10,30,50,100) besides m=(10,30,50,100) we conclude that 
the minimum (MSE) for the estimator of system reliability R (s,k) , (i=1,2,3,4,5),held for 
shrinkage estimator by using shrinkage weight function .This indicates that the shrinkage 

reliability estimator (𝑅 ,  ) is the best for any n and m and follows by shrinkage estimator 

using modified Thomson weight factor. 

7. Conclusion 

      The simulation results exhibited that the shrinkage estimator method is suitable for 
estimation the parameters and system reliability for the Exponentiated Family distribution 
especially when shrinkage weight factor as a linear combination between unbiased estimator, 

and prior estimate. The result estimator 𝑅  perform well and will be the best estimator than 

the other in the sense of MSE. 

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.  

2. Pandit, P.V. ; Kantu, K.J. Reliability Estimation in Multicomponent Pareto Stress-
Strength Models. Pioneer Journal of Theoretical and Applied Statistical, 2012, 5, 1, 
35-40.    

3. Rao, G. S. ; Naidu, CH. R. Estimation of Multi-component Stress-Strength Based on 
Exponentiated Half-Logistic Distribution, Journal of Statistics: Advances in Theory 
and Applications, 2013, 9, 1, 19-35. 

4. Rao, G. S. Estimation of Reliability in Multicomponent Stress- Strength Based on 
Generalized Rayleigh Distribution, Journal of Modern Applied Statistical Methods, 
2014, 13, 24. 

5. Yakubu, A. ; Yahaya, A. Bayesian Estimation for the shape parameter of Generalized 
Rayleig1h distribution under non-informative Prior. International Journal of Advanced 
Statistics and Probability,2015,4 ,1, 1-10. 

6. Rao, G. S.; Aslam, M. ; Arif, O.H. Estimation of Reliability in Multicomponent  Stress-
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