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Bsma Abdul Hameed                     Abbas N. Salman 
 
                      Bayda Atiya Kalaf 

 

 

 

 

Abstract 

     The estimation of the stress ـstrength reliability of Invers Kumaraswamy distribution will 

be introduced in this paper based on the maximum likelihood, moment and shrinkage 

methods. The mean squared error has been used to compare among proposed estimators. Also 

a Monte Carlo simulation study is conducted to investigate the performance of the proposed 

methods in this paper.  
 

Keyword: Invers Kumaraswamy distribution, Maximum likelihood estimator, Moment 

estimator, Shrinkage estimator, Stress ـ Strength reliability, Monte Carlo simulation. 
 

1.  Introduction 

      In the past few years, a majority application in survey sampling, medical research, 

biological sciences, engineering sciences, econometrics and life testing problems have been 

interested to apply inverse distribution [1-2]. Abedul Fattah et al. [3]. Presented the Invers 

Kumaraswamy Distribution (IKumD) based on Kumaraswamy distribution when it is 

presented in 1980 [4]. Which endorse in varied range of applications counting test scores, 

atmospheric temperature, height of individuals and many others [5 -7]. 

Abedul Fattah, depends on the r.v. T (  
   

 
   , as a function of a r.v. X when X follows 

Kumaraswamy distribution ( KumD(      ), where         are shape parameters.  

Then, they explained how the (IKumD) affect long term reliability prediction, making 

optimistic predictions of rare events arising in the right tail of the distribution related to 

additional distributions.  

The probability density function (PDF) of r.v. X which is distributed as IKumD is 

𝑓(         (     (    (  (                                                          (1)                                                    

Where,         are shape parameters. 

And the cumulative distribution function (CDF) of X has the form below  

 𝐹(       (  (                                                                                      (2)  

    On the other hand, for the reliability (R) in the stress- strength (S-S) model was attracted 

Ibn Al Haitham Journal for Pure and Applied Science 

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

Doi: 10.30526/33.1.2375 

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

Baghdad, Baghdad, Iraq.  

 bbsmh896@gmail.com hbama75@yahoo.com abbasnajim66@yahoo.com 

On Estimation of P(Y<X) in Case Inverse Kumaraswamy Distribution 

Article history: Received 2 April 2019, Accepted 30 June 2019, Publish January 2020. 

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many statisticians for several years owing to their applicability in different and divers parts 

such as engineering, quality control, economics. In addition, in the previous thirty years, there 

have been many applications to medical problems and clinical trials [8-9]. 

The term stress-strength (S-S) refers to a component which has a random of strength 𝑋 subject 

to a random stress 𝑌 to evaluate the reliability.  The component fails if the stress applied to it 

exceeds the strength, while the component works whenever Y less than X (𝑌 𝑋 . Several 

researchers assuming various lifetime distributions for the 

stress - strength random variates.[11-14]. 

Accordingly, 𝑅  (𝑌  𝑋  is a measure of component reliability in (S-S) model. 

Consequently, this study estimates the (S-S) reliability when the stress and strength follows 

the two parameters Invers Kumaraswamy Distribution (IKumD) via different estimation 

methods.  

Now, if X and Y are two random variables follows the IKumD with parameters (      and 

(      as strength and stress respectively when the parameter   is known, then the (S-S) 

reliability is defined as below; 

 𝑅  (𝑌  𝑋  

     ∫ ∫ 𝑓(   (      
 

 

 

 
      

     ∫ 𝐹  (  𝑓(      
 

 
                                                                                                                

     ∫ (  (            
 

 
(     (    (  (                                                                                              

      
  

     
                                                                                                                                  (3) 

The rest of this paper organized as follows: Section 2 including some estimation methods of 

P(𝑌 𝑋) . Section 3 offers simulation study.  Section 4 demonstrates the effectiveness of the 

proposed method through results. Finally, a conclusion is provided in Section5.  

                                                                                                                                                                         

2.  Estimation Methods of P(       

2. 1.  Maximum Likelihood Estimation (MLE) 

Let             be a random sample from IKumD (      and               be a random 

sample from IKumD (     , then the likelihood function is given by                                                                 

𝑙  (                                                                                                                                                                                 

  ∏ 𝑓(   
 
    ∏  (  )

 
                                                                                                                                        

  ∏    (     
 (     

   (  (     
        ∏    (    )

 (    
(      

      (    )
  

)
    

                                                                                                                                               

  

       
   

 ∏ (     
 (    ∏ (    )

 (     
   

 
    ∏ (  

 
   

(          
        ∏ (  (    )

  
)
     

                                                                                 

(4)                                                  Take ln to both sides and the equation (4) will be                                                                                                                                                                     

𝑙 𝑙  (               𝑙    (    ∑ 𝑙 (     
 
    (    ∑ 𝑙 (  

 
   

  )  (     ∑ 𝑙 (  (     
    (     ∑ 𝑙 (  (    )

  
)    

 
                                                                                                                                                                    

The partial derivative of   𝑙 with respect to          and equate the result to zero ,we 

obtained the following                                                                                                                                                                       



 

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 ∑   (  (     

                                                                                                                                             

    

   
 

 

  
 ∑   (  (    )

  
)                                                                                                               

The MLʼs estimator for the unknown shape parameters    (       is given by                                                                          

 ̂     
  

∑   (  (     
    

   

                                                                                                        (5) 

 ̂     
  

∑   (  (    )
  

)    
                                                                                (6)                            

By substituting equation (5) and (6) in equation (3) we get                                                                              

�̂�    
 ̂    

 ̂      ̂    
                                                                                                                 (7) 

2. 2.  Moments Method (MOM)   

    In this section, population moments for X and 𝑌 of IKumD were needed. Therefore, 

Moments method (MOM) was considered to estimate the parameter   for IKumD when the 

parameter   is known. 

 E(𝑋    ∑ (
 
 
)      

    (  
 

 
  )     𝑟  , 2 , …                                                                                                           

Hence, the population means of X and Y are given respectively as below                                                                                                             

E(𝑋     (  
 

 
   )                ,                                                                                                                                         

E(𝑌     (  
 

 
   )                ,                                                                                                                  

where  (     refer to Beta distribution.                                                                                                          

And when equality the sample mean with the corresponding population mean, we get                                                       

�̅�  
∑   

 
   

 
    (  

 

 
   )         and                                                                                                                     

�̅�  
∑   

 
   

 
    (  

 

 
   )                                                                                                                                

As a result , we obtained  

 

                ̂     
  ∑

  
 

 
   

 (  
 

 
     )

                                                                                                   (8)  

  And,     

                ̂     
  ∑

  

 
 
   

 (  
 

 
     )

                                                                                                   (9)  

Where     is an initial value for    (      .  

Similarly, substitution the equations (8) and (9) in the equation (3) we get                                                                        

�̂�    
 ̂    

 ̂    
  ̂    

                                                                                                           (10)                                                                                             

2. 3. Shrinkage Estimation Method 

      In 1968, Thompson proposed the shrink usual estimator  ̂ (ex. MLE or Unbiased 

estimator) of the parameter   to prior information     using shrinkage weight actor ψ( ̂), such 

that 0 ψ( ̂  1. Thompson say that "We are estimating   and we believe    is closed to the 



 

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true value of   and something bad happens if      and we do not use   ". Thus, Thompson 

gave the form of shrinkage estimator of   say  ̂   as bellow:  

 ̂   ψ( ̂  ̂   (  ψ( ̂                                                                                                (11)                                                   

Where   ̂   was an unbiased estimator and applied as usual estimator of  , ψ( ̂) may be a 

function of sample size (n,m) , a constant,  or found through  minimizing mean square error of  ̂   ( ad 

hoc basis). On the other hand,     is a very closed value of   which used as a prior information. [15, 

16, 17, 18].                                                                                                         

      There is no doubt, if our assumption to take       [    .001], as a prior information of 

    for i =1, 2 in this work. Noted that       is biased estimator, since E( ̂    )  
 

   
      , 

  refer to   or  .                                                            

 Thus  ̂    
   

 
 ̂     become unbiased estimator of    whereas ( ̂   )    .  

Likewise,  

var( ̂   )  
(   

 

(    
  and var( ̂   )  

(   
 

(    
                                                                                              

i.e.;   ̂    
   

 ∑   (  (     
    

   

                                                                      (12)                                                                      

  And   ̂    
   

 ∑   (  (    )
  

)    
                                                                   (13)     

2. 3. 1. Constant Shrinkage Weight Factor (Sh1)  

In this subsection, the constant shrinkage weight factor will be assumed as;  

𝛹 ( ̂ ) =   = 0.01, and 𝛹( ̂ ) =       , then applying to the following shrinkage 

estimators;  

  ̂    =    ̂   + (                                                                                                     (14) 

 ̂     =    ̂   + (                                                                                                      (15) 
When substitution the equation (14) and (15) in equation (3), lead to the estimation of S-S 

reliability using shrinkage estimator �̂�    as below  

 �̂�    
 ̂    

 ̂    
  ̂    

                                                                                                                 (16) 

 

2. 3. 2. Shrinkage Weight Function (Sh2) 

     In this subsection, we considered the shrinkage weight factor as a function of n and m, 

respectively in equation (11)  

 𝛹 ( ̂ ) =   = 
  , and 𝛹 ( ̂ ) =    = 

  .  

 ̂    =    ̂   + (         for i= 1, 2                     .                                                                      (17) 

The corresponding S-S reliability estimation was used using equation (17) as follows;                                                                     

�̂�    
 ̂    

 ̂    
  ̂    

                                                                                                                                (18)               

2. 3. 3. Modified Thompson Type Shrinkage Weight Factor (ShTh)  

    The shrinkage weight factor which was considered by Thompson in 1968 will be modified 

in this subsection as follows:  



 

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 ( ̂   )  
( ̂   

  ̂  )
 

( ̂   
  ̂  )

 
    ( ̂   

)
 (        for i=1, 2                                                              (19)                      

where,   𝑟( ̂   ) defined in section (2.1).  

In consequence, the modified Thompson type shrinkage estimator will be 

 ̂     ( ̂ ) ̂    (   ( ̂ ))        , for i = 1, 2.                                                             (20)  

By substituting equation (20) in the equation (3), we get the modified Thompson type 

shrinkage estimation of the S-S reliability as below  

�̂�   
 ̂   

 ̂   
  ̂   

                                                                                                                     (21) 

 

Simulation Study .3 

     Simulation is the best criteria for investigate the performance comparison between the 

different estimators of reliability which is obtained in this section. Different samples were 

used based on MSE criteria with size = (20, 30, 50 and 100), with 1000 trials. Mote Carlo 

simulation was used for this purpose as a following steps; 

Step1:  Initialize random samples, which is follows the continuous uniform distribution 

defined on the interval (0,1) as            

Step2:  Generate a random sample follows the continuous uniform distribution over the 

interval (0,1) as            

Step3:  Transform  the  above  uniform  random  samples  to a  random  samples  follows 

IKumD using  the  cumulative distribution function (CDF) as follow:   

(   (  (     
     𝐹 

   (  (     
       

   [  (   
 

  ]
 

 

 

      

As well as, by the same method, we get  

   [  (  )
 

  ]
 

 

 

      

Step4: Recall MLE estimator for reliability using equation (7). 

Step5: Moment method was computed for reliability using equation (10). 

Step6: Compute Shrinkage estimators of reliability using equation (16), (18) and (21). 

Step7: Calculate the MSE based on (L=1000) trials as follows: 

MSE 
 

 
∑ (�̂�  𝑅)

  
    

 

4. Numerical Results  
      In this section, the simulation results of the proposed estimation methods are illustrated in 

Table 1-8. Below and distinguish the following results.  

 Four samples problem size 20,30,50,100 have been implemented 1000 trials based on two 

parameters values (   ). The Mote Carlo simulation was coded using Matlab b 2010. All 
proposed estimation methods has reasonably MSE see Table 2, 4, 6, 8. Also showed that the 

shrinkage estimator based on shrinkage weight function (�̂�   ) had minimum mean square 
error for the S-S reliability estimator of the Invers Kumaraswamy distribution. This implies 

that shrinkage estimator (�̂�   ) of S-S reliability was the best than the others. While Modified 
Thompson type shrinkage estimator (�̂�    had the second rank and followed by Sh1, MOM 



 

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and MLE, respectively. At most when n fixed and m change MSE decreases. The following 

Table 1-8. Will be presents the simulation results. 
Table 1. Estimation value of 𝑅, when                     

n m 𝑅  ̂     ̂     ̂     ̂     ̂   

 

20 

20 0.333333 0.344809 0.335369 0.333362123100458 0.333277759270399 0.333284750506442 

30 0.333333 0.328533 0.331329 0.333216093650363 0.333277759265131 0.333269872853556 

50 0.333333 0.333264 0.330096 0.333293257564652 0.333277759275897 0.333275729850324 

100 0.333333 0.336435 0.331783 0.333335012065777 0.333277759271973 0.333277077788890 

 

30 

20 0.333333 0.332185 0.334541 0.333231166393347 0.333277759228006 0.333277009208634 

30 0.333333 0.337846 0.331541 0.333295628413176 0.333277759253084 0.333279323627313 

50 0.333333 0.338030 0.334136 0.333331973551988 0.333277759253085 0.333282199655278 

100 0.333333 0.338454 0.333449 0.333341266226874 0.333277759253085 0.333280477716060 

 

50 

20 0.333333 0.320632 0.331299 0.333084778976879 0.333277759213775 0.333268023990180 

30 0.333333 0.336537 0.332339 0.333271373787237 0.333277759253084 0.333280641825118 

50 0.333333 0.338671 0.333546 0.333312103469809 0.333277759253085 0.333280220606030 

100 0.333333 0.340324 0.334073 0.333353617428845 0.333277759253085 0.333282003591840 

 

100 

20 0.333333 0.332727 0.334635 0.333206494736057 0.333277759233512 0.333279135957987 

30 0.333333 0.328593 0.332474 0.333195889013743 0.333277759253083 0.333274334748396 

50 0.333333 0.336768 0.334502 0.333298953792528 0.333277759253085 0.333282567198524 

100 0.333333 0.333861 0.333008 0.333281857836376 0.333277759253085 0.333278109789984 
 

Table 2. MSE value of 𝑅          when                   . 

n m  ̂     ̂     ̂     ̂     ̂   

 

20 

20 0.005328 0.000396 0.000000606944029 0.000000003088476 0.000000006027609 

30 0.003583 0.000303 0.000000416241795 0.000000003088477 0.000000006254660 

50 0.003169 0.000278 0.000000373963720 0.000000003088476 0.000000005486328 

100 0.003641 0.000350 0.000000408208905 0.000000003088476 0.000000005655820 

 

30 

20 0.003890 0.000459 0.000000490653947 0.000000003088481 0.000000006363335 

30 0.002995 0.000336 0.000000335168930 0.000000003088478 0.000000005167804 

50 0.002506 0.000190 0.000000277353441 0.000000003088478 0.000000004467301 

100 0.002096 0.000130 0.000000245182027 0.000000003088478 0.000000004439858 

 

50 

20 0.002771 0.000237 0.000000396182755 0.000000003088483 0.000000006381154 

30 0.002947 0.000205 0.000000308374736 0.000000003088478 0.000000004792502 

50 0.002467 0.000206 0.000000263522730 0.000000003088478 0.000000004594002 

100 0.001636 0.000128 0.000000166214348 0.000000003088478 0.000000003705272 

 

100 

20 0.002889 0.000226 0.000000325225366 0.000000003088481 0.000000005030656 

30 0.001862 0.000144 0.000000248485615 0.000000003088478 0.000000005013088 

50 0.001453 0.000123 0.000000156553276 0.000000003088478 0.000000003699886 

100 0.007893 0.000819 0.00000084766426940 0.00000003088478396 0.00000003599008268 

 



 

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Table 3. Estimation value of 𝑅, when                       

n m 𝑅  ̂     ̂     ̂     ̂     ̂   

 

20 

20 0.571428 0.565418 0.570898 0.571436691408593 0.571469411085378 0.571468821676133 

30 0.571428 0.575999 0.571278 0.571580800458761 0.571469411136943 0.571477366116829 

50 0.571428 0.571258 0.570731 0.571538155015663 0.571469411121242 0.571470488585277 

100 0.571428 0.560984 0.570845 0.571405564640412 0.571469411087639 0.571458433287792 

 

30 

20 0.571428 0.571720 0.576066 0.571462591577377 0.571469411073979 0.571469931801052 

30 0.571428 0.568857 0.572765 0.571474641003307 0.571469411092053 0.571469814477133 

50 0.571428 0.572942 0.571064 0.571524903973974 0.571469411092054 0.571472757959768 

100 0.571428 0.572777 0.572013 0.571519136443476 0.571469411092053 0.571469148069890 

 

50 

20 0.571428 0.558760 0.570994 0.571297047177488 0.571469411053786 0.571458816445993 

30 0.571428 0.556923 0.569852 0.571313920162524 0.571469411092051 0.571459850543671 

50 0.571428 0.574199 0.570379 0.571506291371074 0.571469411092053 0.571472448384525 

100 0.571428 0.568220 0.570833 0.571457181783688 0.571469411092053 0.571466452241728 

 

100 

20 0.571428 0.560389 0.571697 0.571299432578466 0.571469411048078 0.571461064899951 

30 0.571428 0.570032 0.570274 0.571435249866571 0.571469411092051 0.571469693744469 

50 0.571428 0.568900 0.572045 0.571441022201234 0.571469411092053 0.571468347558121 

100 0.571428 0.573643 0.571769 0.571495477701192 0.571469411092053 0.571471319635579 

 

Table 4. MSE value of 𝑅           when                       

n m  ̂     ̂     ̂     ̂     ̂   

 

20 

20 0.006315 0.000448 0.000000767437031 0.000000001667878 0.000000006170575 

30 0.004956 0.000347 0.000000725951140 0.000000001667882 0.000000006621191 

50 0.004430 0.000339 0.000000606951619 0.000000001667880 0.000000005436988 

100 0.002530 0.000264 0.000000249898428 0.000000001667878 0.000000002254632 

 

30 

20 0.005278 0.000384 0.000000662992946 0.000000001667877 0.000000005517575 

30 0.005006 0.000393 0.000000571214723 0.000000001667878 0.000000005112411 

50 0.003065 0.000248 0.000000400704653 0.000000001667878 0.000000004318326 

100 0.002887 0.000227 0.000000308254345 0.000000001667878 0.000000003266297 

 

50 

20 0.004811 0.000315 0.000000664165384 0.000000001667875 0.000000004828579 

30 0.003213 0.000206 0.000000349478890 0.000000001667878 0.000000002725463 

50 0.002675 0.000180 0.000000292883062 0.000000001667878 0.000000003460735 

100 0.002007 0.000138 0.000000211130641 0.000000001667878 0.000000002648511 

 

100 

20 0.005067 0.000278 0.000000631569442 0.000000001667875 0.000000004596756 

30 0.002638 0.000198 0.000000299913294 0.000000001667878 0.000000003341709 

50 0.001422 0.000133 0.000000148485486 0.000000001667878 0.000000002331929 

100 0.001176 0.000090 0.000000128239633 0.000000001667878 0.000000002525611 

 

 

 

 



 

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Table 5. Estimation value of 𝑅, when                      

n m 𝑅  ̂     ̂     ̂     ̂     ̂   

 

20 

20 0.428571 0.426216 0.427667 0.428499176135113 0.428530588901450 0.428530671931656 

30 0.428571 0.436086 0.427873 0.428643265773266 0.428530588952929 0.428538657119244 

50 0.428571 0.431015 0.427382 0.428600534764464 0.428530588937226 0.428531934876214 

100 0.428571 0.419576 0.427517 0.428467710362867 0.428530588903618 0.428520515627275 

 

30 

20 0.428571 0.431923 0.432813 0.428524999814827 0.428530588889961 0.428532133408948 

30 0.428571 0.428914 0.429389 0.428537009362929 0.428530588907947 0.428531008897794 

50 0.428571 0.431907 0.427614 0.428587166971384 0.428530588907949 0.428534074306687 

100 0.428571 0.431603 0.428634 0.428581318295701 0.428530588907948 0.428530659761990 

 

50 

20 0.428571 0.427726 0.430724 0.428473495982272 0.428530588885358 0.428530584436506 

30 0.428571 0.433572 0.430752 0.428540048829947 0.428530588907947 0.428532667398454 

50 0.428571 0.434124 0.428382 0.428577386378101 0.428530588907947 0.428534347244274 

100 0.428571 0.422278 0.425756 0.428471308507441 0.428530588907947 0.428524467817398 

 

100 

20 0.428571 0.421956 0.427712 0.428403361576469 0.428530588863387 0.428527175934630 

30 0.428571 0.427808 0.4280613 0.428488126663042 0.428530588907947 0.428531400079066 

50 0.428571 0.426419 0.427354 0.428486850288367 0.428530588907947 0.428529425614141 

100 0.428571 0.434333 0.429112 0.428585964386336 0.428530588907947 0.428535299823111 

 

Table 6. MSE value of 𝑅          when                       

n m  ̂     ̂     ̂     ̂     ̂   

 

20 

20 0.006255 0.000450 0.000000772703228 0.000000001667879 0.000000006187307 

30 0.005167 0.000345 0.000000709137244 0.000000001667874 0.000000005218974 

50 0.004552 0.000333 0.000000596670408 0.000000001667876 0.000000005099792 

100 0.002444 0.000258 0.000000260225013 0.000000001667878 0.000000003858325 

 

30 

20 0.005241 0.000390 0.000000663812611 0.000000001667880 0.000000005462927 

30 0.005007 0.000390 0.000000570321279 0.000000001667878 0.000000004989129 

50 0.003191 0.000243 0.000000392020094 0.000000001667878 0.000000003690361 

100 0.002892 0.000218 0.000000300246386 0.000000001667878 0.000000003180100 

 

50 

20 0.004315 0.000359 0.000000516976424 0.000000001667880 0.000000004621862 

30 0.002932 0.000261 0.000000352796511 0.000000001667878 0.000000003644106 

50 0.002819 0.000221 0.000000307535973 0.000000001667878 0.000000003157659 

100 0.001554 0.000158 0.000000180345696 0.000000001667878 0.000000003242982 

 

100 

20 0.003516 0.000309 0.000000443835525 0.000000001667882 0.000000004264359 

30 0.002307 0.000212 0.000000250681322 0.000000001667878 0.000000002930059 

50 0.001744 0.000164 0.000000201750387 0.000000001667878 0.000000002876528 

100 0.001426 0.000084 0.000000153365133 0.000000001667878 0.000000002279107 

 

 

 

 

 



 

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Table 7. Estimation value of 𝑅, when                         

n m 𝑅  ̂     ̂     ̂     ̂     ̂   

 

20 

20 0.695652 0.698574 0.697915 0.695815269349488 0.695737277093191 0.695741078313743 

30 0.695652 0.691805 0.693986 0.695752268068552 0.695737277099320 0.695735422932987 

50 0.695652 0.690957 0.697203 0.695755093068527 0.695737277086360 0.695734322497301 

100 0.695652 0.693813 0.695859 0.695787596389732 0.695737277088173 0.695734619568542 

 

30 

20 0.695652 0.695140 0.697120 0.695740494751221 0.695737277062622 0.695737475508695 

30 0.695652 0.690159 0.697153 0.695693017315603 0.695737277076990 0.695732707125729 

50 0.695652 0.696888 0.695996 0.695795180678449 0.695737277076991 0.695738795459388 

100 0.695652 0.691288 0.695656 0.695741969015759 0.695737277076990 0.695733471720934 

 

50 

20 0.695652 0.677644 0.694686 0.695543285102023 0.695737277037347 0.695725927625687 

30 0.695652 0.694141 0.696227 0.695734633716281 0.695737277076989 0.695739339129245 

50 0.695652 0.694247 0.694868 0.695731670078937 0.695737277076988 0.695736576753283 

100 0.695652 0.699558 0.696755 0.695799211433752 0.695737277076988 0.695740317699564 

 

100 

20 0.695652 0.682233 0.695121 0.695584190595241 0.695737277045690 0.695730583868550 

30 0.695652 0.694780 0.695037 0.695720177095366 0.695737277076990 0.695737213007891 

50 0.695652 0.692752 0.696609 0.695708826932018 0.695737277076988 0.695736010915983 

100 0.695652 0.693149 0.695976 0.695717387439208 0.695737277076988 0.695735769739067 

 

Table 8. MSE value of 𝑅          when                         

n m  ̂     ̂     ̂     ̂     ̂   

 

20 

20 0.004723 0.000359 0.000000576797968 0.000000007242551 0.000000010986471 

30 0.003827 0.000242 0.000000427720564 0.000000007242552 0.000000009327424 

50 0.003017 0.000267 0.000000378836873 0.000000007242550 0.000000009035571 

100 0.002962 0.000220 0.000000343245951 0.000000007242550 0.000000008697600 

 

30 

20 0.003800 0.000218 0.000000499424739 0.000000007242546 0.000000010446698 

30 0.002959 0.000200 0.000000360304775 0.000000007242549 0.000000008652092 

50 0.002627 0.000190 0.000000323750433 0.000000007242549 0.000000009427161 

100 0.002320 0.000148 0.000000241786558 0.000000007242549 0.000000008204097 

 

50 

20 0.003917 0.000205 0.000000468561104 0.000000007242542 0.000000008183235 

30 0.002502 0.000151 0.000000278982601 0.000000007242549 0.000000009102433 

50 0.001971 0.000124 0.000000217789936 0.000000007242549 0.000000008333216 

100 0.001224 0.000086 0.000000160735972 0.000000007242549 0.000000008692135 

 

100 

20 0.002748 0.000235 0.000000304691320 0.000000007242543 0.000000007794913 

30 0.001921 0.000115 0.000000220162503 0.000000007242549 0.000000008479394 

50 0.001183 0.000116 0.000000129626103 0.000000007242549 0.000000007717318 

100 0.008696 0.000760 0.000000908710753 0.000000072425485 0.000000075162844 

 

 

 

 



 

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5. Conclusion  

       Stress ـstrength reliability for two parameters based on Invers Kumaraswamy distribution 

was introduced in this paper by using different estimation methods namely; Maximum 

likelihood Estimation, Moments method, Constant shrinkage weight factor (Sh1), Shrinkage 

weight function (Sh2), and Modified Thompson type shrinkage weight factor.  After that, 

Monte Carlo Simulation was exhibited to analyses(compare) and investigate these methods. 

Based on the results, the performance of shrinkage weight function (�̂�   ) was appropriate 

behavior and it is efficient estimator than the others in the sense of MSE. 

 

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