BAYESIAN ESTIMATION OF SHAPE PARAMETER OF TOPP-LEONE 
DAGUM DISTRIBUTION 

Arun Kumar Rao1, Himanshu Pandey2* 
1Department of Statistics, MPPG College, Jungle Dhusan, Gorakhpur, INDIA 

2*Department of Mathematics & Statistics, DDU Gorakhpur University, Gorakhpur, INDIA 
E-mail: himanshu_pandey62@yahoo.com 

 

Abstract 
In this paper, Topp-Leone Dagum distribution is considered for Bayesian analysis. The 

expressions for Bayes estimators of the parameter have been derived under squared error, 
precautionary, entropy, K-loss, and Al-Bayyati’s loss functions by using quasi and gamma 
priors. 

Keywords 
Bayesian method, Topp-Leone Dagum distribution, quasi and gamma priors, squared 

error, precautionary, entropy, K-loss, and Al-Bayyati’s loss functions. 

1. Introduction 
Rasheed, N., [1] introduced the Topp-Leone Dagum distribution. He obtained some basic 

statistical properties, incomplete rth moments, mean deviation from mean, and reliability 
measures of the distribution. The probability density function of Topp-Leone Dagum distribution 
is given by 

       
 

     
1211

2 1 1 1 1 1 1 0
a a a

f x; a x x x x  ; x .


       


        

        
 

 

  (1) 

The joint density function or likelihood function of (1) is given by 

         
 

  11

1

2 1 1 1
n a an

i i i
i

f x; a x x x
     

     



 
    

 
    

     
1 12 2

11

1 1 1 1 1 1
n na a

i i
ii

x exp log x   

 
  



      
                   

            (2) 

The log likelihood function is given by 

         
 

  11

1

2 1 1 1
n a a

i i i
i

log f x; nlog a log x x x
     

     



 
     

 
    

     
1 12 2

11

1 1 1 1 1 1
n na a

i i
ii

log x log x   

 
  



    
               

          (3) 

Differentiating (3) with respect to θ and equating to zero, we get the maximum likelihood 
estimator of θ which is given as 

   
12

1

1 1 1
n a

i
i

n log x  


 



 
    

 
 .             (4) 

 

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2. Bayesian Method of Estimation 
The Bayesian inference procedures have been developed generally under squared error 

loss function 

 
2

L ,   
    

    
   

.                (5) 

The Bayes estimator under the above loss function, say, s


 is the posterior mean, i.e, 

   S E 


 .                 (6)  

Zellner [2], Basu and Ebrahimi [3] have recognized that the inappropriateness of using 
symmetric loss function. Norstrom [4] introduced precautionary loss function is given as 

 

2

L ,

 

 








 
 

    
 

.                (7) 

The Bayes estimator under this loss function is denoted by P


 and is obtained as

  
1

22
P E 



    .                (8) 

Calabria and Pulcini [5] points out that a useful asymmetric loss function is the entropy loss 

     1p
eL p log         

where  ,






  and whose minimum occurs at . 



 
Also, the loss function  L   has been used 

in Dey et al. [6] and Dey and Liu [7], in the original form having 1p .  Thus  L   can written 

be as 

    1eL b log ;  b>0.                       (9) 

The Bayes estimator under entropy loss function is denoted by E


 and is obtained by solving the 
following equation 

 

1
1

E E .



   

   
  

              (10) 

Wasan [8] proposed the K-loss function which is given as 

 

2

L ,

 

 
 







 
 

    
 

.              (11) 

Under K-loss function the Bayes estimator of θ is denoted by K


 and is obtained as 

 
 
 

1

2

1
K

E

E






  
  
 

.              (12) 

Al-Bayyati [9] introduced a new loss function which is given as 

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2

cL ,    
    

    
   

.             (13) 

Under Al-Bayyati’s loss function the Bayes estimator of θ is denoted by Al


 and is obtained as 

 
 
 

1c

Al
c

E

E









 .              (14) 

Let us consider two prior distributions of θ to obtain the Bayes estimators. 
(i) Quasi-prior: For the situation where we have no prior information about the parameter θ, we 
may use the quasi density as given by 

  1

1
0 0

d
g  ; , d , 


                 (15) 

where d = 0 leads to a diffuse prior and d = 1, a non-informative prior. 
(ii) Gamma prior: Generally, the gamma density is used as prior distribution of the parameter θ 
given by 

  
 

1
2 0g e  ; .


 

  


  


            (16) 

3. Posterior density under  1g   

The posterior density of θ under  1g  , on using (2), is given by 

 

       
 

  

     

       
 

  

11

1

1 12 2

11

11

1

2 1 1 1

1 1 1 1 1 1

2 1 1 1

1 1 1

n a an

i i i
i

n na a d
i i

ii

n a an

i i i
i

i

a x x x

x exp log x

f x

a x x x

x

  

 

  

  

   



  



     



 
   



     



  
    

  
                              

 
   

 

   







     
1 12 2

0

11

1 1 1
n na a d

i
ii

d

exp log x 



  



 
   



 
 
 
                           




 

  

 

 

12

1

12

1

1 1 1

1 1 1

0

n a

i
i

n a

i

i

log x
n d

log x
n d

 e

 e d





 

 



 











   
      

    

          
    









  

 

  
 

 
12

1

112

1 1 1
1

1 1 1

1

n a

i
i

n d
n a

i
log x

i
n d

log x

  e
n d





 









 


   
           

  
        

  


          (17) 

 

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Theorem 1. On using (17), we have 

    
 

  
12

1

1
1 1 1

1

c
n ac

i
i

n d c
E log x

n d
 






      
           
 .        (18) 

Proof.  By definition, 

   c cE f x d       

  
 

 
 

12

1

112

1 1 1
1

0

1 1 1

1

n a

i

i

n d
n a

i
log x

i n d c

log x

 e d
n d





 


 






 


                

  
        

  


   

  
 

 

  

112

1

112

1

1 1 1
1

1
1 1 1

n d
n a

i
i

n d c
n a

i
i

log x
n d c

n d
log x









 




  




  
           

     
       





  

 
 

  
12

1

1
1 1 1

1

c
n a

i
i

n d c
log x

n d







      
           
  . 

From equation (18), for 1c  , we have 

       
112

1

1 1 1 1
n a

i
i

E n d log x  






  
          
 .           (19) 

From equation (18), for 2c  , we have 

         
212

2

1

2 1 1 1 1
n a

i
i

E n d n d log x  






  
            

   
 .       (20) 

From equation (18), for 1c   , we have 

 
 

  
12

1

1 1
1 1 1

n a

i
i

E log x
n d









  
     

   
 .          (21) 

From equation (18), for 1c c  , we have 

    
 

  
 112

1

1

2
1 1 1

1

c
n ac

i
i

n d c
E log x

n d
 

 
 



      
           
 .        (22) 

4. Bayes estimators under  1g    

From equation (6), on using (19), the Bayes estimator of θ under squared error loss 
function is given by 

     
112

1

1 1 1 1
n a

S i
i

n d log x  


 



  
          
 .          (23) 

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From equation (8), on using (20), the Bayes estimator of θ under precautionary loss function is 
obtained as 

       
111 2

2

1

2 1 1 1 1
n a

P i
i

n d n d log x  


 



  
                
 .        (24) 

From equation (10), on using (21), the Bayes estimator of θ under entropy loss function is given 
by 

     
112

1

1 1 1
n a

E i
i

n d log x  


 



  
         
 .          (25) 

From equation (12), on using (19) and (21), the Bayes estimator of θ under K-loss function is 
given by 

      
1
2

112

1

1 1 1 1
n a

K i
i

n d n d log x  


 



  
               
 .        (26) 

From equation (14), on using (18) and (22), the Bayes estimator of θ under Al-Bayyati’s loss 
function comes out to be 

     
112

1

1 1 1 1
n a

Al i
i

n d c log x  


 



  
           
 .         (27) 

5. Posterior density under  2g     

Under  2g  , the posterior density of θ, using equation (2), is obtained as 

 

       
 

     

    

       
 

1211

1

12
1

1

11

2 1 1 1 1 1 1

1 1 1

2 1 1

n a a an

i i i i
i

n a

i
i

an

i i

a x x x x

                               exp log x e

f x

a x x

   


  

 

   


  




 


       




  



   

   
            
 

   
       

     







     

    

12

1

120
1

1

1 1 1 1

1 1 1

n a a

i i
i

n a

i
i

x x

d

                               exp log x e

 


  

 




  



  

 


  



   
           
 

   
       

     






 

  

  

12
1

1

12
1

10

1 1 1

1 1 1

n an
i

i

n an
i

i

 exp log x

 exp log x d

 

 

   

    


  




  



   
            
   
            





 

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 

    

12

1

1 1 1

1

12

1

1 1 1

n a

i
i

log x

n

n
n a

i
i

 e

n log x

  









  






                    









  
          



  

  
 

 
12

1

12

1 1 1
1

1

1 1 1 n a

i
i

n
n a

i log x
i

n

log x

 e
n







  



 










                      

  
         

 


         (28) 

Theorem 2. On using (28), we have 

    
 

  
12

1

1 1 1

c
n ac

i
i

n c
E log x

n


  







     
           

 .           (29) 

Proof.  By definition, 

   c cE f x d      

  
 

 
12

1

12

1 1 1
1

1

0

1 1 1 n a

i
i

n
n a

i log x
i

n c

log x

 e d
n







  



 

 








                        

  
         

 


  

  
 

 

  

12

1

12

1

1 1 1

1 1 1

n
n a

i
i

n c
n a

i
i

log x
n c

n
log x









 



 






 




  
            

    
        





  

 
 

  
12

1

1 1 1

c
n a

i
i

n c
log x

n


 







     
           

 . 

From equation (29), for 1c  , we have 

       
112

1

1 1 1
n a

i
i

E n log x    






  
          

 .             (30) 

From equation (29), for 2c  , we have 

        
212

2

1

1 1 1 1
n a

i
i

E n n log x     






  
                

 .       (31) 

From equation (29), for 1c   , we have 

 
 

  
12

1

1 1
1 1 1

1

n a

i
i

E log x
n

 
 






   
              

 .         (32) 

From equation (29), for 1c c  , we have 

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    
 

  
 112

1

1

1
1 1 1

c
n ac

i
i

n c
E log x

n


  


 
 



      
           

 .       (33) 

6. Bayes estimators under  2g    

From equation (6), on using (30), the Bayes estimator of θ under squared error loss 
function is given by 

     
112

1

1 1 1
n a

S i
i

n log x    


 



  
          

 .         (34) 

From equation (8), on using (31), the Bayes estimator of θ under precautionary loss function is 
obtained as 

       
1
2

112

1

1 1 1 1
n a

P i
i

n n log x     


 



  
                

 .       (35) 

From equation (10), on using (32), the Bayes estimator of θ under entropy loss function is given 
by 

     
112

1

1 1 1 1
n a

E i
i

n log x    


 



  
           

 .         (36) 

From equation (12), on using (30) and (32), the Bayes estimator of θ under K-loss function is 
given by 

      
1
2

112

1

1 1 1 1
n a

K i
i

n n log x     


 



  
                

 .       (37) 

From equation (14), on using (29) and (33), the Bayes estimator of θ under Al-Bayyati’s loss 
function comes out to be 

     
112

1

1 1 1
n a

Al i
i

n c log x    


 



  
           

 .          (38) 

Conclusion 
In this paper, we have obtained a number of estimators of parameter of Gompertz Fréchet 

distribution. In equation (4) we have obtained the maximum likelihood estimator of the 
parameter. In equation (23), (24), (25), (26) and (27) we have obtained the Bayes estimators 
under different loss functions using quasi prior. In equation (34), (35), (36), (37) and (38) we 
have obtained the Bayes estimators under different loss functions using gamma prior. In the 
above equations, it is clear that the Bayes estimators depend upon the parameters of the prior 
distribution. We therefore recommend that the estimator’s choice lies according to the value of 
the prior distribution which in turn depends on the situation at hand. 

 

References 

[1] Rasheed, N, (2020): “Topp-Leone Dagum distribution: Properties and its Applications”. 
Research Journal of Mathematical and Statistical Sciences, Vol. 8(1), 16-30.  

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[2] Zellner, A., (1986): “Bayesian estimation and prediction using asymmetric loss functions”. 
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[3] Basu, A. P. and Ebrahimi, N., (1991): “Bayesian approach to life testing and reliability 
estimation using asymmetric loss function”. Jour. Stat. Plann. Infer., 29, 21-31. 
[4] Norstrom, J. G., (1996): “The use of precautionary loss functions in Risk Analysis”. IEEE 
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[5] Calabria, R., and Pulcini, G. (1994): “Point estimation under asymmetric loss functions for 
left truncated exponential samples”. Comm. Statist. Theory & Methods, 25 (3), 585-600. 
[6] D.K. Dey, M. Ghosh and C. Srinivasan (1987): “Simultaneous estimation of parameters 
under entropy loss”. Jour. Statist. Plan. And infer., 347-363. 
[7] D.K. Dey, and Pei-San Liao Liu (1992): “On comparison of estimators in a generalized life    
    Model”. Microelectron. Reliab. 32 (1/2), 207-221. 
[8] Wasan, M.T., (1970): “Parametric Estimation”. New York: Mcgraw-Hill.  
[9] Al-Bayyati, H.N., (2002): “Comparing methods of estimating Weibull failure models using 
simulation”. Ph.D. Thesis, College of Administration and Economics, Baghdad University, Iraq. 

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