70 

 

Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

 

  

 

 

 

Bareq Baqe Selman                                                      Alaa M. Hamad 

 

 

 

 

 

Abstract 

      This paper deals with estimation of the reliability system in the stress- strength model of 

the shape parameter for the power distribution. The proposed approach has been including 

different estimations methods such as Maximum likelihood method, Shrinkage estimation 

methods, least square method and Moment method. Comparisons process had been carried 

out between the various employed estimation methods with using the mean square error 

criteria via Matlab software package. 

Keywords: Power distribution, (S-S) Stress-Strength Reliability, Maximum Likelihood, 

Moment method, Least Squares method, Shrinkage method. 

1. Introduction 

    The power function distribution represents one of the most important distribution approach 

in statistical process. It can be expressed by  𝑥~𝑝𝑜𝑤(𝑥, 𝛼) . Actually, it can be considered as 

a simple and flexible distribution method that can be used in different applications such as the 

estimation of reliability for electrical components [1]. This distribution has been introduced in 

(1967) by Malik H.J throughout studying the exact moment of power function distribution 

and found a precise expression for moment of power function distribution [2]. Although 

Bayesian estimation method of parameters have been used by several statisticians and 

mathematical analysts, but many researchers considered the power function distribution is 

better than a lot of distributions such as lognormal distribution, exponential distribution and 

weibull distribution. The power function distribution is as a special case for person first kind 

distribution that represents the simplicity of the moments for power function distribution [3].  

In fact blue estimation method has been introduced for the estimation of the scale and location 

parameter from the Log-gamma distribution. Many of others researchers were presented 

estimations of normal distribution parameters by using likelihood function [4-6]. 

    The aim of this work is to estimate the reliability system in the stress- strength model of 

power function distribution.  

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Doi:10.30526/32.3.2284 

Article history: Received 24 February 2019, Accepted 14 April 2019, Publish September 

2019. 

 

Department of Mathematics, College of Education for Pure Science, Ibn-Al-Haitham, 

University of Baghdad, Baghdad, Iraq. 

 bareqbaqe@gmail.com

                                  

        

alaa_073@yahoo.com 

Different Estimation Methods of the Stress-Strength Reliability Power 

Distribution 

 

mailto:bareqbaqe@gmail.com
mailto:bareqbaqe@gmail.com
mailto:alaa_073@yahoo.com


  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

     This process has been carried out with using and comparing different estimation methods 

such as Maximum likelihood estimation (MLE), Moment (MOM), least square (LS) and 

Shrinkage estimation methods (SH). The (c.d.f) of random variable X for power function 

distribution is given as in the following [7, 8]. 

 𝐹(𝑥, 𝜃, 𝛼) = 𝑥𝛼 𝜃𝛼                       0 < 𝑥 < 𝜃−1                                                                       (1) 

While the probability density function (P.d.f) of power distribution can be defined as: 

𝑓(𝑥, 𝜃, 𝛼) =  𝛼𝜃𝛼 𝑥𝛼−1                0 < 𝑥 < 𝜃−1
                                                                        

(2)
 

Where α and θ  represent the shape and scale parameters respectively. 

In special case, if   𝜃 = 1 then (c. d. f) will be as follows: 

  𝐹(𝑥, 𝛼) = 𝑥𝛼                             0 < 𝑥 < 1                                                                       (3) 

And consequently, the (p. d. f) of power function distribution will be:  

𝑓(𝑥, 𝛼) = 𝛼𝑥𝛼−1                         0 < 𝑥 < 1                                                                      (4) 

    The two random variable X and Y are following the power function distribution with 

parameters (α1, 1) and (𝛼2, 1) respectively and according to strength – stress (S-S) model.  

Hence, the 𝑥~𝑝𝑜𝑤(𝑥, 𝛼1) and 𝑦~𝑝𝑜𝑤(𝑦, 𝛼2)in the strength – stress model can be defined as 

in the following: 

𝑅 = 𝑃(𝑦 < 𝑥) 

∫ ∫ 𝑓(𝑥)  𝑓(𝑦)  𝑑𝑦 𝑑𝑥
𝑋

𝑂

1

0

= ∫ ∫ 𝛼1𝑥
𝛼1−1𝛼2𝑦

𝛼2−1𝑑𝑦 𝑑𝑥
𝑥

0

1

0

 

Where                                                                                    

  𝑝(𝑦 < 𝑥) = 𝑅 =
𝛼1

𝛼1+𝛼2
                                                                                                   (5)   

2. Estimation Methods 

2.1 Maximum Likelihood Estimator (MLE) 

    The maximum likelihood estimator is most popular method because it is approximates the 

minimum variance unbiased [12]. Let𝑥1, 𝑥2  … … 𝑥𝑛 be a random sample of𝑝𝑜𝑤(𝛼1, 1) 

and𝑦1, 𝑦2  … … 𝑦𝑚 be a random sample of 𝑝𝑜𝑤(𝛼2, 1)  

Then, the Maximum likelihood for𝛼1 and𝛼2 will be: 

 𝐿 = 𝐿(𝛼𝑖 , 𝑥𝑖 , 𝑦𝑖 ) =  ∏ 𝑓(𝑥𝑖 )
𝑛
𝑖=1 ∏ 𝑔(𝑦𝑖 )

𝑚
𝑖=1  

𝐿 = ∏ 𝛼1
𝑛

𝑛

𝑖=1

𝑥𝑖
𝛼1−1 ∏ 𝛼2

𝑚

𝑚

𝑖=1

𝑦𝑖
𝛼2−1 

Taking the logarithm of both sides, then: 

𝐿𝑛 𝐿 =  𝑛 ln 𝛼1 + (𝛼1 + 1) ln ∑ 𝑥𝑖

𝑛

𝑖=1

+ 𝑚 ln 𝛼2 + (𝛼2 + 1) ln ∑ 𝑦𝑖

𝑚

𝑖=1

 

Taking the partial derivation for the above equations with respect to the shape parameter and 

then equating the results to zero, this will lead to: 

�̂�1𝑚𝑙𝑒 =  
−𝑛

ln ∑ 𝑥𝑖
𝑛
𝑖=1

                                                                                                                       (6) 

�̂�2𝑚𝑙𝑒 =
−𝑚

𝑙𝑛 ∑ 𝑦𝑖
𝑚
𝑖=1

                                                                                                                       (7)                  

�̂�2𝑚𝑙𝑒 in equation (5); the reliability estimation of stress – strength model using maximum 

likelihood method will become:  

�̂�𝑚𝑙𝑒 =  
�̂�1𝑚𝑙𝑒

�̂�1𝑚𝑙𝑒
+�̂�2𝑚𝑙𝑒

                                                                                                        (8) 



  

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2.2. Moment Method (MOM) 

    The moment method can be considered as the most common and a simple method that have 

been used in estimation of the parameters. It can be summarized by equating the population 

moments 𝑀𝑟 = 𝐸(𝑥
𝑟 ) as in the following [8]. 

  𝐸(𝑥) =  
𝛼1

𝛼1+1
 𝑎𝑛𝑑 𝐸(𝑦) =  

𝛼2

𝛼2+1
 

When equating the sample mean E(x) and E(y) to the corresponding population mean, then 

  
𝛼1

𝛼1+1
=  

∑ 𝑥𝑖
𝑛
𝑖=1

𝑛
                                                                                                                 (9)     

  

  
𝛼2

𝛼2+1
=  

∑ 𝑦𝑖
𝑚
𝑖=1

𝑚
                                                                                                                        (10) 

From equations (9) and (10), the estimation of parameters 𝛼1and α2 using moment method 

will be written as in the following:  

  �̂�1𝑚𝑜𝑚 =  
∑ 𝑥𝑖𝑛𝑖=1

𝑛−∑ 𝑥𝑖𝑛𝑖=1
                                                                                                                (11)      

                                                                                            

  �̂�2𝑚𝑜𝑚 =  
∑ 𝑦𝑖𝑚𝑖=1

𝑚−∑ 𝑦𝑖𝑚𝑖=1
                                                                                                           (12) 

  

Substituting the equations (11) and (12) in equation (5), then the estimation of stress – 

strength reliability with using moment method will become:  

   �̂�𝑚𝑜𝑚 =
�̂�1𝑚𝑜𝑚

�̂�1𝑚𝑜𝑚 +�̂�2𝑚𝑜𝑚
                                                                                                 (13) 

2.3 Least Square Method (LSM) [5] [9] 

      Let𝑥1, 𝑥2  … … 𝑥𝑛 be a random sample of 𝑝𝑜𝑤(𝛼1, 1) and𝑦1, 𝑦2  … … 𝑦𝑚 be a random 

sample of 𝑝𝑜𝑤(𝛼2, 1), then:  

𝐹(𝑥1) = 𝑥𝑖
𝛼1  

𝑥𝑖 = [𝐹(𝑥𝑖 )]
1

𝛼1  

Taking the logarithm for the both sides of the above equation, then:  

ln(𝑥𝑖 ) =
1

𝛼1
ln[𝐹(𝑥𝑖 )] 

𝑦 = 𝑎𝑥 + 𝑏 

𝑦 = ln 𝑥𝑖,     𝑎 =
1

𝛼𝑖
,      𝑥𝑖 = ln[𝐹(𝑥𝑖 )],      𝑏 = 0 

Hence  

𝑎 =
∑ 𝑥𝑖 𝑦𝑖

𝑛
𝑖=1 −

∑ 𝑥𝑖 𝑦𝑖
𝑛
𝑖=1

𝑛

∑ 𝑥𝑖
2𝑛

𝑖=1 −
(∑ 𝑥𝑖

𝑛
𝑖=1

)2

𝑛

 

�̂�1𝐿𝑠 =
∑ [ln 𝐹(𝑥𝑖)]

2𝑛
𝑖=1 −

[∑ ln 𝐹(𝑥𝑖)
𝑛
𝑖=1 ]

2

𝑛

∑ ln 𝐹(𝑥𝑖) ln 𝑥𝑖−
𝑛
𝑖=1

∑ ln 𝐹(𝑥𝑖) ln 𝑥𝑖
𝑛
𝑖=1

𝑛

                                                                                      (14) 

�̂�2𝐿𝑠 =
∑ [ln 𝐺(𝑦𝑖)]

2𝑚
𝑖=1 −

[∑ ln 𝐺(𝑦𝑖)
𝑚
𝑖=1 ]

2

𝑚

∑ ln 𝐺(𝑦𝑖) ln 𝑦𝑖−
𝑚
𝑖=1

∑ ln 𝐺(𝑦𝑖) ln 𝑦𝑖
𝑚
𝑖=1

𝑚

                                                                                (15) 

For stress – strength reliability �̂�𝐿𝑠using least square method as in the following: 



  

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   �̂�𝐿𝑠 =
�̂�1𝐿𝑠

�̂�1𝐿𝑠 +�̂�2𝐿𝑠
                                                                                                                    (16) 

2.4 Shrinkage Estimation Method (sh) [10][13].  

     The shrinkage estimation method can be considered as the Bayesian approach which has 

depended on the prior information. The basic reasons for using the prior estimation had been 

introduced by Thompson in 1968.In the shrinkage estimation method the parameter was used 

as initial value 𝛼0 from the past and usual estimator �̂�𝑢𝑏 through consideration them by 

shrinkage weight factor, ∅(�̂�)   , 0 < ∅(�̂�) < 1, which can be written as:  

 �̂�𝑠ℎ = ∅(�̂�)�̂�𝑢𝑏 +  (1 − ∅(�̂�))𝛼0                                                                                     (17) 

To find  α̂ub from �̂�𝑚𝑙𝑒  

�̂�𝑚𝑙𝑒 =  
−𝑛

ln ∑ 𝑥𝑖
𝑛
𝑖=1

 

𝑛

𝑛 − 1
�̂�𝑚𝑙𝑒 ≠ �̂�𝑚𝑙𝑒  

 �̂�𝑢𝑏 =
𝑛−1

𝑛
�̂�𝑚𝑙𝑒                                                                                                                                                             

�̂�𝑢𝑏 =
−(𝑛−1)

∑ ln 𝑥𝑖
𝑛
𝑖=1

                                                                                                                        (18) 

2.4.1 Shrinkage Weight Function ( 𝒔𝒉𝟏) [11]. 

     In this case, the shrinkage weight factor has used as a function of n, in which:  

∅(�̂�) = |
sin 𝑛

𝑛
|,   �̂�𝑢𝑏 =

−(𝑛−1)

∑ ln 𝑥𝑖
𝑛
𝑖=1

  

Substituting in equation (17), then:  

�̂�1𝑠ℎ1
= |

sin 𝑛

𝑛
| �̂�𝑢𝑏 + (1 − |

sin 𝑛

𝑛
|) 𝛼0                                                                                            (19) 

�̂�2𝑠ℎ1
= |

sin 𝑚

𝑚
| �̂�𝑢𝑏 + (1 − |

sin 𝑚

𝑚
|) 𝛼0                                                                                    (20) 

                                                                                                      

    Substituting �̂�1𝑠ℎ1
,�̂�2𝑠ℎ1

 in equation (5), then the reliability estimation of stress – strength 

model with using shrinkage weight function will become: 

 �̂�𝑠ℎ1 =
�̂�1𝑠ℎ1

�̂�1𝑠ℎ1
+�̂�2𝑠ℎ1

                                                                                                                 (21) 

2.4.2 Constant Shrinkage Factor ( 𝒔𝒉𝟐 ) [11]. 

     In constant shrinkage factor case, it can be assumed that∅(�̂�) = 𝑘, K=0.001 which 

represent the constant shrinkage weight factor. Implying (1-K=0.999), then: 

�̂�1𝑠ℎ2
= 𝑘1�̂�𝑢𝑏 + (1 − 𝑘1)𝛼0                                                                                                (22)  

 �̂�2𝑠ℎ2
= 𝑘2�̂�𝑢𝑏 + (1 − 𝑘2)𝛼0                                                                                               (23)                                                                                                           

Substituting�̂�1𝑠ℎ2
,�̂�2𝑠ℎ2

 in equation (5), then reliability estimation of stress – strength model 

with using constant shrinkage factor will become as in the following: 

    

 �̂�𝑠ℎ2 =
�̂�1𝑠ℎ2

�̂�1𝑠ℎ2
+�̂�2𝑠ℎ2

                                                                                                                (24)           

 

 

 

 



  

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2.4.3 Beta Shrinkage Factor (𝒔𝒉𝟑) [11]. 
     In this case, the assumption of∅(�̂�) =  𝛽(1, 𝑛, 𝑚) for the Beta shrinkage weight factor has 

been taken as∅(�̂�1) =  𝛽(1, 𝑛), and  ∅(�̂�2) =  𝛽(1, 𝑚) and this implies to the following 

shrinkage estimators: 

  �̂�1𝑠ℎ3
= 𝛽(1, 𝑛)�̂�𝑢𝑏 + (1 − 𝛽(1, 𝑚))𝛼0                                                                                        (25) 

  �̂�2𝑠ℎ3
= 𝛽(1, 𝑛)�̂�𝑢𝑏 + (1 − 𝛽(1, 𝑚))𝛼0                                                                                  (26) 

Substituting�̂�1𝑠ℎ3
, �̂�2𝑠ℎ3

 in equation (5), then the reliability estimation of stress – strength 

model with using Beta shrinkage factor will become as in the following:  

  �̂�𝑠ℎ3 =
�̂�1𝑠ℎ3

�̂�1𝑠ℎ3
+�̂�2𝑠ℎ3

                                                                                                                (27) 

3- Simulation Process 

     The simulation process has done with using different sample size such as (30, 50 and 100) 

and built on 1000 iteration and using the mean square error (MSE) to measure and check the 

performance as in the following steps: 

Step 1: the random sample generated for X according to the uniform distribution over the 

interval (0, 1) as𝑟1,𝑟2…………𝑟𝑛 and the random sample generated for Y according to the 

uniform distribution over the interval (0, 1), as 𝑠1,𝑠2,…………𝑠𝑚 

Step 2: transforming the above power distribution with using reliability as in the following: 

𝑅(𝑥) = 1 − 𝑥𝛼   

 𝑥𝑖 = (1 − 𝑟𝑖 )
1

𝛼                                       i=1, 2, 3 ……..n  

Using the same procedure .then 

  𝑦𝑗 = (1 − 𝑠𝑗 )
1

𝛼                                       j=1, 2, 3 ……..m 

Step 3: calculating�̂�1𝑚𝑙𝑒  and �̂�2𝑚𝑙𝑒  using equations (6) and (7) respectively. 

Step 4: calculating�̂�1𝑚𝑜𝑚  and �̂�2𝑚𝑜𝑚  using equations (9) and (10) respectively. 

Step 5: calculating�̂�1𝐿𝑠  and �̂�2𝐿𝑠   using equations (14) and (15) respectively. 

Step 6: calculating �̂�1𝑠ℎ𝑖
and �̂�2𝑠ℎ𝑖

 when i=1, 2, 3 using equations (19), (20), (22), (23), (25) and 

(26) respectively. 

Step 7: calculating�̂�𝑚𝑙𝑒  , �̂�𝑚𝑜𝑚 , �̂�𝐿𝑠 , �̂�𝑠ℎ1  , �̂�𝑠ℎ2  and�̂�𝑠ℎ3   using equations (8), (13), (16), 

(21), (24) and (27) respectively. Where, 𝑅 ̂point to the suggest estimator of reliability R. The 

outcomes results are listed in the Tables 1, 3, 5, 7, 9, 11, 13, 15, 17. respectively. 
 

4. Numerical Result 

     Some methods of goodness of fit analysis are employed here; the measurement give an 

indication of best method is mean square error (MSE) from tables for all. 

1. For all n=(30,50,100) and m=(30,50,100) in this work for minimum mean square error 

(MSE) for the stress-strength reliability estimator of power function distribution after 

noted the mean square error in tables, the result indicates that shrinkage estimator 

(Sh2) is the best .  

2. For all n=(30,50,100) and m=(30,50,100) the minimum mean square error (MSE) for 

the stress- strength reliability estimator of power function distribution ,we noticed that 

the shrinkage estimator is the best and follows by maximum likelihood estimator 

(MLE), moment estimator (MOM) and least square estimator (LS). 

3. For the various cases when (n=30 and m=30), (𝛼1=1and α2=1) then be moment 



  

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estimator (MOM) batter then maximum likelihood estimator (MLE). 

Table 1. The estimation when R=0.5, alpha1= 1, alpha2= 1. 

 

Table 2. MSE values when R = 0.5, alpha1= 1, alpha2= 1. 

 

Table 3. The estimation when R =0.333333, alpha1= 1, alpha2= 2. 

n m
 

�̂�mle �̂�mom �̂�sh1 �̂�sh2 �̂�sh3 �̂�Ls 

 

30 

30 0.490139 0.511770 0.500105 0.500003 0.500106 0.491397 

50 0.502377 0.501424 0.485568 0.500000 0.492930 0.498530 

100 0.500411 0.497019 0.485535 0.500002 0.487835 0.495937 

 

50 

30 0.497673 0.499256 0.514426 0.500001 0.507071 0.501789 

50 0.501290 0.502633 0.499992 0.499999 0.499969 0.501975 

100 0.501921 0.501610 0.499903 0.499999 0.494837 0.500422 

100 30 0.496428 0.499589 0.514526 0.500001 0.512236 0.502114 

50 0.497654 0.498059 0.500099 0.500001 0.505168 0.501268 

100 0.500210 0.500975 0.499999 0.500000 0.499997 0.498823 

n m
 

𝒎𝒔𝒆mle 𝒎𝒔𝒆mom 𝒎𝒔𝒆sh1 𝒎𝒔𝒆sh2 𝒎𝒔𝒆sh3 𝒎𝒔𝒆Ls Best 

 

30 

30 0.000519010585 0.000403849563 0.000000057885 0.000000000048 0.000000059360 0.001867492654 𝑚𝑠𝑒sh2 

50 0.003335498758 0.00404372155 0.00021100121 0.00000000367 0.00005342323 0.00596515484 𝑚𝑠𝑒sh2 

100 0.00275059025 0.00358047587 0.00021203899 0.00000000287 0.00015090743 0.00535912842 𝑚𝑠𝑒sh2 

 

50 

30 0.00338030723 0.00439697723 0.00021093108 0.00000000358 0.00005343251 0.00621983678 𝑚𝑠𝑒sh2 

50 0.00264717587 0.00349108397 0.00000007768 0.00000000277 0.00000120029 0.00505326237 𝑚𝑠𝑒sh2 

100 0.00186086110 0.00248158856 0.00000006313 0.00000000196 0.00002730713 0.00387593133 𝑚𝑠𝑒sh2 

100 30 0.00268746093 0.00362852707 0.00021403837 0.00000000301 0.00015287915 0.00474061679 𝑚𝑠𝑒sh2 

50 0.00191579777 0.00247303858 0.00000006418 0.00000000198 0.00002736419 0.00382297898 𝑚𝑠𝑒sh2 

100 0.00130008952 0.00169115906 0.00000003502 0.00000000134 0.00000013940 0.00230190166 𝑚𝑠𝑒sh2 

n m
 

�̂�mle �̂�mom �̂�sh1 �̂�sh2 �̂�sh3 �̂�Ls 

 

30 

30 0.333767 0.334263 0.333414 0.333336 0.333415 0.336289 

50 0.337382 0.334902 0.320640 0.333333 0.327086 0.336407 

100 0.338877 0.336147 0.320533 0.333332 0.322533 0.335001 

 

50 

30 0.336594 0.338421 0.346229 0.333331 0.339564 0.342261 

50 0.333468 0.332847 0.333339 0.333334 0.333357 0.334368 

100 0.335005 0.333733 0.333255 0.333334 0.328789 0.335376 



  

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Table 4. MSE values when R = 0.333333, alpha1= 1, alpha2= 2. 

 

 

Table 5. The estimation when R =0.666666, alpha1= 2, alpha2= 1. 

 

 

 

 

 

 

100 30 0.328662 0.329458 0.346444 0.333337 0.344379 0.334704 

50 0.332007 0.332608 0.333424 0.333335 0.337963 0.334676 

100 0.335370 0.335221 0.333327 0.333332 0.333320 0.335713 

n m
 

𝒎𝒔𝒆mle 𝒎𝒔𝒆mom 𝒎𝒔𝒆sh1 𝒎𝒔𝒆sh2 𝒎𝒔𝒆sh3 𝒎𝒔𝒆Ls Best 

 

30 

30 0.00333045942 0.00407349011 0.00000456676 0.00000000366 0.00000468648 0.00602348672 𝑚𝑠𝑒sh2 

50 0.00272675676 0.00317425319 0.00016326720 0.00000000289 0.00004172719 0.00491135012 𝑚𝑠𝑒sh2 

100 0.00235315660 0.00304112894 0.00016601546 0.00000000241 0.00011895110 0.00417668486 𝑚𝑠𝑒sh2 

 

50 

30 0.00253964500 0.00309479408 0.00016860628 0.00000000272 0.00004151959 0.00486156577 𝑚𝑠𝑒sh2 

50 0.0020245050 0.00236857438 0.00000005918 0.00000000211 0.00000091460 0.00367715958 𝑚𝑠𝑒sh2 

100 0.00155424555 0.00194082119 0.00000004951 0.00000000158 0.00002113966 0.00293098875 𝑚𝑠𝑒sh2 

100 30 0.00224635487 0.00265176602 0.00017436372 0.00000000250 0.00012458560 0.00424344399 𝑚𝑠𝑒sh2 

50 0.00149852589 0.00180930296 0.00000005236 0.00000000161 0.00002197596 0.00292503967 𝑚𝑠𝑒sh2 

100 0.00105247266 0.00125820431 0.00000002811 0.00000000107 0.00000011190 0.00203644507 𝑚𝑠𝑒sh2 

n m
 

�̂�mle �̂�mom �̂�sh1 �̂�sh2 �̂�sh3 �̂�Ls 

 

30 

30 0.662728 0.663078 0.666702 0.666668 0.666702 0.664002 

50 0.663332 0.663325 0.653800 0.666670 0.660460 0.658256 

100 0.667976 0.666353 0.653665 0.666666 0.655736 0.664065 

 

50 

30 0.662408 0.663825 0.679385 0.666668 0.672938 0.661386 

50 0.666429 0.665317 0.666658 0.666665 0.666634 0.664104 

100 0.666855 0.665989 0.666582 0.666666 0.662053 0.663561 

100 30 0.660560 0.663617 0.679517 0.666669 0.677516 0.667085 

50 0.665754 0.666582 0.666742 0.666665 0.671203 0.667305 

100 0.66579 0.665672 0.666668 0.666667 0.666668 0.665879 



  

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Table 6. MSE values when R = 0.666666, alpha1= 2, alpha2= 1. 

Table7. The estimation when R =0.2500000, alpha1= 1, alpha2= 3. 

 Table 8. MSE values when R = 0.2500000, alpha1= 1, alpha2= 3. 

n m
 

�̂�mle �̂�mom �̂�sh1 �̂�sh2 �̂�sh3 �̂�Ls 

 

30 

30 0.251955 0.249262 0.250052 0.250001 0.250053 0.251874 

50 0.253903 0.253116 0.239376 0.250000 0.244759 0.253825 

100 0.253407 0.252539 0.239330 0.250001 0.241002 0.253077 

 

50 

30 0.251326 0.252295 0.260959 0.250000 0.255319 0.256074 

50 0.251116 0.249421 0.250005 0.250001 0.250020 0.251890 

100 0.250869 0.250132 0.249939 0.250002 0.246185 0.251854 

100 30 0.248037 0.248544 0.261095 0.250002 0.259334 0.252563 

50 0.250922 0.250898 0.250069 0.250000 0.253894 0.254081 

100 0.250477 0.250285 0.250003 0.250000 0.250005 0.250916 

n m
 

𝒎𝒔𝒆mle 𝒎𝒔𝒆mom 𝒎𝒔𝒆sh1 𝒎𝒔𝒆sh2 𝒎𝒔𝒆sh3 𝒎𝒔𝒆Ls Best 

 

30 

30 0.00328250122 0.00373222061 0.00000425693 0.00000000342 0.00000436837 0.00606980612 𝑚𝑠𝑒sh2 

50 0.00259447656 0.00324478044 0.00016767398 0.00000000266 0.00004106095 0.00517968821 𝑚𝑠𝑒sh2 

100 0.00206071212 0.00242868274 0.00017126775 0.00000000222 0.00012180771 0.00389842610 𝑚𝑠𝑒sh2 

 

50 

30 0.00273481415 0.00325502994 0.00016399215 0.00000000291 0.00004210945 0.00532038893 𝑚𝑠𝑒sh2 

50 0.00185242541 0.00225935108 0.00000005461 0.00000000194 0.00000084415 0.00353063805 𝑚𝑠𝑒sh2 

100 0.00157365989 0.00185528680 0.00000005292 0.00000000166 0.00002184280 0.00302255985 𝑚𝑠𝑒sh2 

100 30 0.00239094911 0.00285689399 0.00016762461 0.00000000253 0.00012031708 0.00390968491 𝑚𝑠𝑒sh2 

50 0.00138303657 0.00166057068 0.00000004423 0.00000000141 0.00002102378 0.00268810353 𝑚𝑠𝑒sh2 

100 0.00096451894 0.00119679615 0.00000002584 0.00000000099 0.00000010287 0.00200269936 𝑚𝑠𝑒sh2 

n m
 

𝒎𝒔𝒆mle 𝒎𝒔𝒆mom 𝒎𝒔𝒆sh1 𝒎𝒔𝒆sh2 𝒎𝒔𝒆sh3 𝒎𝒔𝒆Ls Best 

 

30 

30 0.00238794868 0.002815755086 0.000003239263 0.000000002596 0.000003324242 0.004110057260 𝑚𝑠𝑒sh2 

50 0.00179231243 0.002194623563 0.000114274591 0.000000001810 0.000029189762 0.003461646653 𝑚𝑠𝑒sh2 

100 0.00154622400 0.002007092701 0.000115260720 0.000000001531 0.000082459639 0.003073759490 𝑚𝑠𝑒sh2 

 

50 

30 0.00194448207 0.002350645135 0.000121745483 0.000000002092 0.000030292392 0.003608076782 𝑚𝑠𝑒sh2 

50 0.00148586648 0.001741949682 0.00000004300 0.00000000153 0.00000066465 0.00273533263 𝑚𝑠𝑒sh2 

100 0.0010774882 0.00133269966 0.00000003359 0.00000000109 0.00001488988 0.00219412488 𝑚𝑠𝑒sh2 

100 30 0.00152834454 0.001698567219 0.00012497442 0.00000000173 0.00008905011 0.00286844206 𝑚𝑠𝑒sh2 

50 0.00112174712 0.001257705394 0.00000003779 0.00000000120 0.00001556871 0.00215713682 𝑚𝑠𝑒sh2 

100 0.00072228004 0.000834738472 0.00000001933 0.00000000074 0.00000007694 0.00145595257 𝑚𝑠𝑒sh2 



  

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Table 9. The estimation when R =0.5000000, alpha1= 2, alpha2= 2. 

 

 

Table 10. MSE values when R = 0.5000000, alpha1= 2, alpha2= 2. 

Table 11.The estimation when R =0.750000, alpha1= 3, alpha2= 1. 

n m
 

�̂�mle �̂�mom �̂�sh1 �̂�sh2 �̂�sh3 �̂�Ls 

 

30 

30 0.500752 0.500283 0.499984 0.500000 0.499984 0.498106 

50 0.501153 0.500582 0.485629 0.500000 0.492974 0.497582 

100 0.502017 0.500986 0.485552 0.500001 0.487843 0.498328 

 

50 

30 0.497610 0.498436 0.514437 0.500001 0.507076 0.501432 

50 0.498133 0.498306 0.500010 0.500002 0.500040 0.495669 

100 0.502684 0.501383 0.499900 0.499999 0.494824 0.500963 

100 30 0.497646 0.499452 0.514483 0.499999 0.512190 0.500630 

50 0.499560 0.500307 0.500088 0.499999 0.505127 0.502454 

100 0.500073 0.500067 0.500000 0.500000 0.499999 0.501504 

n m
 

𝒎𝒔𝒆mle 𝒎𝒔𝒆mom 𝒎𝒔𝒆sh1 𝒎𝒔𝒆sh2 𝒎𝒔𝒆sh3 𝒎𝒔𝒆Ls Best 

 

30 

30 0.004128967671 0.004604492710 0.000005672908 0.000000004557 0.000005821534 0.007466248540 𝑚𝑠𝑒sh2 

50 0.003355245833 0.003784728713 0.000209383584 0.000000003620 0.000052863372 0.006185754049 𝑚𝑠𝑒sh2 

100 0.002539072228 0.002862493900 0.000211322115 0.000000002679 0.000150482732 0.004649858803 𝑚𝑠𝑒sh2 

 

50 

30 0.003222287571 0.003616390312 0.0002112752765 0.000000003490 0.000053445804 0.006250665745 𝑚𝑠𝑒sh2 

50 0.002543357768 0.002951662489 0.000000077652 0.000000002770 0.000001200648 0.005053937844 𝑚𝑠𝑒sh2 

100 0.001813708137 0.001971894924 0.000000061788 0.000000001892 0.000027409095 0.003494129231 𝑚𝑠𝑒sh2 

100 30 0.002623859568 0.003039695366 0.000212392743 0.000000002745 0.000151354885 0.004670769245 𝑚𝑠𝑒sh2 

50 0.001796372291 0.001975020089 0.000000057684 0.000000001826 0.000026875471 0.003170938343 𝑚𝑠𝑒sh2 

100 0.001240136539 0.001388572811 0.000000033491 0.000000001284 0.000000133296 0.002447561555 𝑚𝑠𝑒sh2 

n m
 

�̂�mle �̂�mom �̂�sh1 �̂�sh2 �̂�sh3 �̂�Ls 

 

30 

30 0.747445 0.747676 0.749984 0.750000 0.749984 0.742832 

50 0.748323 0.748968 0.739063 0.750000 0.744691 0.743426 

100 0.749614 0.749635 0.739003 0.750001 0.740768 0.745187 

 

50 

30 0.745730 0.747290 0.760672 0.750000 0.755268 0.746282 

50 0.746656 0.747272 0.750008 0.750001 0.750029 0.742852 

100 0.750657 0.749959 0.749925 0.749999 0.746097 0.748081 

100 30 0.746228 0.748209 0.760705 0.750000 0.759030 0.746915 

50 0.748316 0.749575 0.750066 0.749999 0.753825 0.749445 

100 0.748535 0.748006 0.750002 0.750000 0.750005 0.749047 



  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

Table 12. MSE values when R = 0.750000, alpha1= 3, alpha2= 1. 

Table 13. The estimation when R =0.400000, alpha1= 2, alpha2= 3. 

Table 14. MSE values when R = 0.400000, alpha1= 2, alpha2= 3. 

n m
 

𝒎𝒔𝒆mle 𝒎𝒔𝒆mom 𝒎𝒔𝒆sh1 𝒎𝒔𝒆sh2 𝒎𝒔𝒆sh3 𝒎𝒔𝒆Ls Best 

 

30 

30 0.002384459481 0.002851628898 0.000003192342 0.000000002563 0.000003275999 0.004521397784 𝑚𝑠𝑒sh2 

50 0.001948985166 0.002244248987 0.000121340373 0.000000002036 0.000030219158 0.003732969877 𝑚𝑠𝑒sh2 

100 0.001431149872 0.001606728294 0.000122472102 0.000000001507 0.000086823942 0.002762765254 𝑚𝑠𝑒sh2 

 

50 

30 0.001920772013 0.002309071883 0.000115391937 0.000000001963 0.000029585622 0.003739290580 𝑚𝑠𝑒sh2 

50 0.001497007264 0.001814282137 0.000000043673 0.000000001558 0.000000675002 0.003051262077 𝑚𝑠𝑒sh2 

100 0.001020052304 0.001169978602 0.000000034777 0.000000001064 0.000015586022 0.002017961664 𝑚𝑠𝑒sh2 

100 30 0.001547120432 0.001996286322 0.000115999905 0.000000001544 0.000083023582 0.002746512587 𝑚𝑠𝑒sh2 

50 0.001026872629 0.001229979595 0.000000032430 0.000000001027 0.000014956803 0.001827470288 𝑚𝑠𝑒sh2 

100 0.000702148814 0.000841172097 0.000000018293 0.000000000701 0.000000072800 0.001371989596 𝑚𝑠𝑒sh2 

n m
 

�̂�mle �̂�mom �̂�sh1 �̂�sh2 �̂�sh3 �̂�Ls 

 

30 

30 0.404506 0.403784 0.399881 0.399997 0.399879 0.403496 

50 0.404397 0.402729 0.386249 0.399998 0.393222 0.399588 

100 0.402856 0.400909 0.386181 0.400001 0.388360 0.398094 

 

50 

30 0.398661 0.399451 0.413926 0.400001 0.406817 0.406414 

50 0.403676 0.403770 0.399987 0.399997 0.399947 0.407139 

100 0.401190 0.400025 0.399914 0.400001 0.395066 0.403155 

100 30 0.397284 0.397751 0.414066 0.400001 0.411838 0.402213 

50 0.401166 0.401576 0.400081 0.399998 0.404928 0.402906 

100 0.401270 0.401548 0.399996 0.399999 0.399992 0.403123 

n m
 

𝒎𝒔𝒆mle 𝒎𝒔𝒆mom 𝒎𝒔𝒆sh1 𝒎𝒔𝒆sh2 𝒎𝒔𝒆sh3 𝒎𝒔𝒆Ls Best 

 

30 

30 0.003968719837 0.004339244098 0.000005400944 0.000000004337 0.000005542471 0.007343458817 𝑚𝑠𝑒sh2 

50 0.002871054638 0.003168491061 0.000191532791 0.000000003008 0.000048857539 0.005249976869 𝑚𝑠𝑒sh2 

100 0.002560059256 0.002759431582 0.000193533534 0.000000002729 0.000138181811 0.004430761642 𝑚𝑠𝑒sh2 

 

50 

30 0.003057266048 0.003368884673 0.000196646030 0.000000003292 0.000049708350 0.005687576189 𝑚𝑠𝑒sh2 

50 0.002354827003 0.002572130065 0.000000068179 0.000000002433 0.000001053332 0.004695426152 𝑚𝑠𝑒sh2 

100 0.001758583317 0.001851572994 0.000000057177 0.000000001821 0.000024931773 0.003154647116 𝑚𝑠𝑒sh2 

100 30 0.002509204252 0.002634167448 0.000200568619 0.000000002757 0.000142971140 0.004464657451 𝑚𝑠𝑒sh2 

50 0.001776285618 0.001939439022 0.000000056642 0.000000001834 0.000024872335 0.003451614797 𝑚𝑠𝑒sh2 

100 0.001103485740 0.001185957806 0.000000029215 0.000000001120 0.000000116266 0.002211734727 𝑚𝑠𝑒sh2 



  

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Table 15. The estimation when R =0.600000, alpha1= 3, alpha2= 2. 

Table 16. MSE values when R = 0.600000, alpha1= 3, alpha2= 2. 

Table 17. The estimation when R =0.500000, alpha1= 3, alpha2= 3. 

n m
 

�̂�mle �̂�mom �̂�sh1 �̂�sh2 �̂�sh3 �̂�Ls 

 

30 

30 0.598078 0.599079 0.600008 0.600000 0.600008 0.597839 

50 0.599919 0.598806 0.586143 0.600000 0.593246 0.594149 

100 0.602922 0.601586 0.585963 0.599999 0.588188 0.597739 

 

50 

30 0.597608 0.597803 0.613749 0.600000 0.606745 0.594849 

50 0.598873 0.599349 0.600002 0.600000 0.600008 0.598037 

100 0.599726 0.598941 0.599914 0.600001 0.595049 0.596674 

100 30 0.595986 0.597631 0.613846 0.600000 0.611669 0.598814 

50 0.600409 0.601163 0.600076 0.599998 0.604884 0.602419 

100 0.598798 0.598789 0.600004 0.600001 0.600008 0.599235 

n m 𝒎𝒔𝒆mle 𝒎𝒔𝒆mom 𝒎𝒔𝒆sh1 𝒎𝒔𝒆sh2 𝒎𝒔𝒆sh3 𝒎𝒔𝒆Ls Best 

 

30 

30 0.004102721803 0.004522527871 0.000005539858 0.000000004450 0.000005685018 0.006919868113  𝑚𝑠𝑒sh2 

50 0.002991823998 0.003292540439 0.000194592389 0.000000003159 0.000048701124 0.005449357116 𝑚𝑠𝑒sh2 

100 0.002314174187 0.002431988951 0.000199450713 0.000000002484 0.000142040001 0.004537099349 𝑚𝑠𝑒sh2 

 

50 

30 0.003193844527 0.003419488337 0.000191806526 0.000000003451 0.000048845005 0.005550811660 𝑚𝑠𝑒sh2 

50 0.002287353696 0.002455944918 0.000000066626 0.000000002377 0.000001029306 0.004372047489 𝑚𝑠𝑒sh2 

100 0.001805434274 0.001965307620 0.000000059644 0.000000001910 0.000025136702 0.003323110300 𝑚𝑠𝑒sh2 

100 30 0.002590582075 0.002917038767 0.000194259147 0.000000002723 0.000138841425 0.004619688057 𝑚𝑠𝑒sh2 

50 0.001637625023 0.001733549825 0.000000051603 0.000000001680 0.000024385304 0.003120969456 𝑚𝑠𝑒sh2 

100 0.001157990638 0.001271271929 0.000000031146 0.000000001194 0.000000123958 0.002418378384 𝑚𝑠𝑒sh2 

n m
 

�̂�mle �̂�mom �̂�sh1 �̂�sh2 �̂�sh3 �̂�Ls 

 

30 

30 0.497664 0.497663 0.500090 0.500003 0.500091 0.499114 

50 0.501338 0.501712 0.485595 0.500001 0.492961 0.498385 

100 0.502409 0.501254 0.485551 0.500000 0.487837 0.495447 

 

50 

30 0.498054 0.498944 0.514417 0.500000 0.507064 0.501263 

50 0.501699 0.501955 0.499991 0.499998 0.499964 0.501553 

100 0.499867 0.498715 0.499915 0.500001 0.494868 0.498597 

100 30 0.496367 0.497546 0.514469 0.500001 0.512188 0.499887 

50 0.500860 0.501257 0.500083 0.499998 0.505135 0.502239 

100 0.499098 0.499134 0.500005 0.500001 0.500010 0.498614 



  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

Table 18.  MSE values when R = 0.500000, alpha1= 3, alpha2= 3. 

 

5. Conclusion 

     In this absence of real data, we study the performance of the estimator obtained from 

simulated and the tables, that to estimate the reliability of shrinkage estimator method special 

constant type shrinkage (Sh2) is the best performance. 
 

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2. Akhter, A.S. Methods for Estimating the Parameters of the Power Function Distribution. 

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3. Shakeel, M. Comparison of two new robust parameter estimation methods for the power 

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4. Al-mutairianed, O.; Low, H.C. Bayesian Estimate for Shape Parameter from Generalized 

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distribution: A generalization of the power distribution. International Journal of 

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6.  Malik, H.J. Exact moments of order statistics from a power-function 

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7. Salman, A.N.; Hamad, A. Pretest Single Stage Shrinkage Estimator for the Shape 

Parameter of the Power Function Distribution. International Journal of Mathematics 

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8. Dhanya, M.; Jeevavand, E.S. Stress-Strength Reliability of Power Function Distribution 

Based on Records. Journal of statistics Application probability.2018, 7, 1, 39-48. 

9. Kundu, D.; Raqab, M.Z. Generalized Rayleigh distribution: different methods of 

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University of Baghdad, College of education for pure science, Ibn AL-Haitham,2017. 

n m
 

𝒎𝒔𝒆mle 𝒎𝒔𝒆mom 𝒎𝒔𝒆sh1 𝒎𝒔𝒆sh2 𝒎𝒔𝒆sh3 𝒎𝒔𝒆Ls Best 

 

30 

30 0.004060583137 0.004327864388 0.000005599720 0.000000004497 0.000005746462 0.007686836923 𝑚𝑠𝑒sh2 

50 0.003215899017 0.003479527896 0.000210304289 0.000000003546 0.000052923876 0.006368191686 𝑚𝑠𝑒sh2 

100 0.002691692024 0.002822200371 0.000211674187 0.000000002885 0.000150965205 0.004745640879 𝑚𝑠𝑒sh2 

 

50 

30 0.003106889720 0.003372070208 0.000210555251 0.000000003353 0.000053185081 0.005623109954 𝑚𝑠𝑒sh2 

50 0.002496078235 0.002679722628 0.000000074148 0.000000002646 0.000001145840 0.004620145844 𝑚𝑠𝑒sh2 

100 0.001836797867 0.001944809665 0.000000059352 0.000000001906 0.000026943768 0.003330357287 𝑚𝑠𝑒sh2 

100 30 0.002900030258 0.003015179856 0.000212289915 0.000000003071 0.000151620014 0.004919492350 𝑚𝑠𝑒sh2 

50 0.002003087543 0.002106729457 0.000000066242 0.000000002166 0.000027084341 0.003655290108 𝑚𝑠𝑒sh2 

100 0.001207869505 0.001273452513 0.000000032502 0.000000001246 0.000000129359 0.002575201627 𝑚𝑠𝑒sh2 



  

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Ibn Al-Haitham Jour. for Pure & Appl. Sci. 32 (3) 2019 

 

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