Ratio Mathematica                            Volume 40, 2021, pp. 123-137 
 

123 
 

 

Reliability estimation of Weibull-

exponential distribution via Bayesian 

approach 
 

Arun Kumar Rao* 

Himanshu Pandey† 
 

 

 

Abstract 

Weibull-exponential distribution is considered. Bayesian method of 

estimation is employed in order to estimate the reliability function of 

Weibull-exponential distribution by using non-informative and beta 

priors. In this paper, the Bayes estimators of the reliability function have 

been obtained under squared error, precautionary and entropy loss 

functions.  

Keywords: Weibull-exponential distribution. Reliability. Bayesian 

method. Non-informative and beta priors. Squared error, precautionary 

and entropy loss functions. 

2010 AMS subject classification: 60E05, 62E15, 62H10, 62H12.§ 

 

 

 

 

  
* Department of Statistics, Maharana Pratap P.G. College, Jungle Dhusan, Gorakhpur, INDIA; 

arunrao1972@gmail.com. 
† Department of Mathematics & Statistics, DDU Gorakhpur University, Gorakhpur, INDIA; 
himanshu_pandey62@yahoo.com. 
§Received on January 12th, 2021. Accepted on May 12th, 2021. Published on June 30th, 2021. doi: 

10.23755/rm.v40i1.570. ISSN: 1592-7415. eISSN: 2282-8214. ©Rao and Pandey. This paper is 

published under the CC-BY licence agreement. 



A.K.Rao and H.Pandey 

136 

1. Introduction 

The Weibull-exponential distribution was proposed by Oguntunde et al. 

[1]. They obtained some of its basic Mathematical properties. This distribution is 

useful as a life testing model and is more flexible than the exponential distribution. 

The probability function ( )f x;  and distribution function ( )F x;  of Weibull-

exponential distribution are respectively given by 

 

( ) ( ) ( )
1

1 1 0
a a

x a x x
f x; a e e exp e    ; x .

  
  

−
−  = − − − 

  
                                  (1) 

( )
( )1

1 0 0

a
x

e
F x; e    ; x , .




 
−

− −
= −                (2) 

Let ( )R t  denote the reliability function, that is, the probability that a 

system will survive a specified time t comes out to be 

( )
( )1

0 0

a
t

e
R t e    ; t , .





− −

=                (3) 

And the instantaneous failure rate or hazard rate, h(t) is given by 

( ) ( )1a t th t a e e .  −= −               (4) 

From equation (1) and (3), we get 

( )( )
( )

( ) ( ) ( ) ( )
1

1
11 1

1

a
x

t

a x e
a

x
e

a
t

a e
f x; R t e log R t R t  ;  0<R t .

e











 −

−  −  − = − −       
−

      

(5) 

The joint density function or likelihood function of (5) is given by 

( )( )
( )

( )
( ) ( ) ( )

1

1

1
1

1

1

1
1

n

i a
n xi

i

ti
i

a x
n

en
a nx

e
na

t
i

a e
f x | R t e log R t R t

e











 =
=

 −
−  −  − 

=


  

= − −       
 −
         (6) 

The log likelihood function is given by 



Reliability estimation of Weibull-exponential distribution via bayesian approach 

 

137 

( )( )
( )

( )
( )

( )
( )

( ) ( )

1

1 1

1

1
1

1
1

1

i

i

n
nn

a
x

ina
t

i i

n
a

x

a
t

i

a
log f x | R t log a x log e

e

                           n log log R t e log R t
e












−
−

= =

=

 
  = + + −    −

 

+ − + −      
−

 


        

(7) 

Differentiating (7) with respect to R (t) and equating to zero, we get the maximum 

likelihood estimator of R (t) as 

( ) ( ) ( )
1

1 1i
n

aa
xt

i

R t exp n e e




=

  
= − − −  

  
 .            (8) 

 

2. Bayesian method of estimation 

 The Bayesian estimation procedure have been developed generally 

under squared error loss function 

( ) ( ) ( ) ( )
2

L R t , R t R t R t
    

= −   
   

.             (9) 

where ( )R t


 is an estimate of ( )R t . The Bayes estimator under the above loss 

function, say ( )
S

R t


 , is the posterior mean, i.e., 

( ) ( )
S

R t E R t


=    .             (10) 

The squared error loss function is often used also because it does not lead 

extensive numerical computation but several authors (Zellner [2], Basu & Ebrahimi 

[3]) have recognized the inappropriateness of using symmetric loss function. 

Canfield [4] points out that the use of symmetric loss function may be inappropriate 

in the estimation of reliability function. Norstrom [5] introduced an alternative 

asymmetric precautionary loss function and also presented a general class of 

precautionary loss function with quadratic loss function as a special case. A very 

useful and simple asymmetric precautionary loss function is 



A.K.Rao and H.Pandey 

136 

( ) ( )
( ) ( )

( )

2

R t R t

L R t , R t

R t







 
− 

   
= 

 
           (11) 

The Bayes estimator of ( )R t  under precautionary loss function is denoted by ( )
P

R t


, and is obtained by solving the following equation 

( ) ( )( )
1
22

P
R t E R t


 =
 

 .            (12) 

In many practical situations, it appears to be more realistic to express the 

loss in terms of the ratio 
( )

( )

R t

R t



 . In this case, Calabria and Pulcini [6] points out that 

a useful asymmetric loss function is the entropy loss 

( ) ( ) 1p eL p log    − −   , 

where  
( )

( )

R t
,

R t




=  and whose minimum occurs at ( ) ( )R t R t


=  when 0p ,  a 

positive error ( ) ( )R t R t
 

 
 

 causes more serious consequences than negative error, 

and vice-versa. For small | p |  value, the function is almost symmetric when both 

( ) ( )R t and  R t


 are measured in a logarithmic scale, and approximately 

( ) ( ) ( )
22

2
e e

p
L log R t log R t .

 
 −

 
 

  

Also, the loss function ( )L   has been used in Dey et al. [7] and Dey and Liu [8], 

in the original form having 1p .=  Thus ( )L   can be written as 

( ) ( ) 1 0eL b log ;     b .  = − −               (13) 

The Bayes estimator of ( )R t  under entropy loss function is denoted by E


 and is 

obtained as 



Reliability estimation of Weibull-exponential distribution via bayesian approach 

 

137 

( )
( )

1

1
E

R t E .
R t

−
   

=    
   

            (14) 

For the situation where we have no prior information about ( )R t  , we 

may use non-informative prior distribution 

( )( )
( ) ( )

( )1
1

0 1h R t ;   R t .
R t  logR t

=             (15) 

The most widely used prior distribution for ( )R t  is a beta distribution with 

parameters 0, ,     given by 

( )( )
( )

( ) ( ) ( )
1 1

2

1
1 0 1h R t R t R t ;   R t .

 B ,

 

 

− −
= −                 (16) 

 

3. Bayes estimators of ( )R t under ( )( )1h R t
 

Under ( )( )1h R t  , the posterior distribution is defined by 

( )( )
( )( ) ( )( )

( )( ) ( )( ) ( )

1

1

1

0

f x | R t h R t
f R t | x

f x | R t h R t dR t

=



          (17) 

Substituting the values of ( )( )1h R t  and ( )( )f x | R t  from equations (15) and (6) in 
(17), we get 



A.K.Rao and H.Pandey 

136 

( )( )

( )

( )
( ) ( )

( )
( ) ( )

( )

( )
( ) ( )

( )

1

1

1

1

1

1

1

1

1

1
1

1

1
1

n

i

i i

a
n xi

t
i

n

i

i i

x

n
na x

a nx

na
t

i

e

e

n
na x

a nx

na
t

i

e

a
e e log R t

e

             R t
R t  log R t

f R t | x

a
e e log R t

e

           R t





















=

=

=

−
−

=

 −
 
 − 

−
−

=

   
 − −     −
 
 

    
 

=

  
− −   

 −

  





( ) ( )

( )

1

1

0 1

1
1

a
n i

t
i e

dR t

R t  log R t


=

 −
 
 − 

 
 
 
 
 

 
 
 



 

                  

( ) ( )

( ) ( ) ( )

1

1

1
1 1

1

1 1
1 1

1

0

a
n xi

t
i

a
n xi

t
i

e
n

e

e
n

e

R t -log R t

R t -log R t dR t









=

=

 −
− − 

 − 

 −
− − 

 − 


      

=


      

  

or, ( )( )
( )

( ) ( )1
11 1 1
1

1

1

i

a
n xi

t
i

n
a

xn

t
ei n
e

e

e
f R t | x R t -log R t

n









=

 −= − − 
 − 

  −
  
 −   

=       



      (18) 

Theorem 1. Assuming the squared error loss function, the Bayes estimate of ( )R t

, is of the form 

( )
( )

( )
1

1
1

1i

n

a
t

nS a
x

i

e
R t

e





−



=

 
 −
 = +
 

− 
 


            (19) 

Proof. From equation (10), on using (18), 

( ) ( )
S

R t E R t


=     



Reliability estimation of Weibull-exponential distribution via bayesian approach 

 

137 

( ) ( )( ) ( )
1

0

R t  f R t | x dR t= 

 ( )
( )

( ) ( ) ( )1
1 11 1 1

1

0

1

1

i

a
n xi

t
i

n
a

xn

t
ei n
e

e

e
R t R t -log R t dR t

n









=

 −= − − 
 − 

  −
  
 −   

=       






 
( )

( ) ( ) ( )1
1 11 1

1

0

1

1

i

a
n xi

t
i

n
a

xn

t
ei n
e

e

e
= R t -log R t dR t

n









=

 −= − 
 − 

  −
  
 −   

      






 
( )

( )1

1

1

1

1
1

1

i

i

n
a

xn

t
i

n
a

xn

t
i

e

e n
=

n
e

e









=

=

  −
  
 −   

    −
  + 
 −    





  

or,   ( )
( )

( )
1

1
1

1i

n

a
t

nS a
x

i

e
R t

e





−



=

 
 −
 = +
 

− 
 


 . 

Theorem 2. Assuming the precautionary loss function, the Bayes estimate of ( )R t

, is of the form 

( )
( )

( )

2

1

2 1
1

1

n

i

a
t

nP a
x

i

e
R t =

e





−



=

 
 −
 +
 

−
 
 


 .           (20) 

Proof. From equation (12), on using (18), 

 
( ) ( )( )

1
22

P
R t E R t


 =
 

  



A.K.Rao and H.Pandey 

136 

 

( )( ) ( )( ) ( )

1
21

2

0

= R t f R t | x dR t
 
 
 
  

 

( )( )
( )

( ) ( ) ( )

1
2

1

1 11 1 12
1

0

1

1

i

a
n xi

t
i

n
a

xn

t
ei n
e

e

e
= R t R t -log R t dR t

n









=

 −= − − 
 − 

   −   
  −          

 
 
 
 





 

( )
( ) ( ) ( )

1
2

1

1 11 1 1
1

0

1

1

i

a
n xi

t
i

n
a

xn

t
ei n
e

e

e
= R t -log R t dR t

n









=

 −= + − 
 − 

   −   
  −          

 
 
 
 





 

( )

( )

1
2

1

1

1

1

1
2

1

i

i

n
a

xn

t
i

n
a

xn

t
i

e

e n
=  

n
e

e









=

=

 
  −     −    
     −   +   −      





 

1
2

1

1

1

1

1
2

1

i

i

n
a

xn

t
i

n
a

xn

t
i

e

e
=

e

e









=

=

 
  −     −   

    −   +   −      





 

or, ( )
( )

( )

2

1

2 1
1

1

n

i

a
t

nP a
x

i

e
R t =

e





−



=

 
 −
 +
 

−
 
 


. 



Reliability estimation of Weibull-exponential distribution via bayesian approach 

 

137 

Theorem 3. Assuming the entropy loss function, the Bayes estimate of ( )R t , is of 

the form 

( )
( )

( )
1

1
1

1i

n

a
t

nE a
x

i

e
R t =

e







=

 
 −
 −
 

−
 
 


            (21) 

Proof. From equation (14), on using (18), 

( )
( )

( )
( )( ) ( )

1

1
1

0

1

1

E
R t E

R t

          = f R t | x dR t
R t

−


−

  
=    
   

 
 
 


        

( ) ( )
( ) ( ) ( )1

1

1 11 1 1
1

0

1

11

i

a
n xi

t
i

n
a

xn

t
ei n
e

e

e
= R t -log R t dR t

R t n









=

−

 −= − − 
 − 

   −   
  −          

 
 
 
 





( )
( ) ( ) ( )1

1

1 11 2 1
1

0

1

1

i

a
n xi

t
i

n
a

xn

t
ei n
e

e

e
= R t -log R t dR t

n









=

−

 −= − − 
 − 

   −   
  −          

 
 
 
 





 

 
( )

( )

1

1

1

1

1

1
1

1

i

i

n
a

xn

t
i

n
a

xn

t
i

e

e n
=  

n
e

e









−

=

=

 
  −     −    
     −   −   −      





 



A.K.Rao and H.Pandey 

136 

 

1

1

1

1

1

1
1

1

i

i

n
a

xn

t
i

n
a

xn

t
i

e

e
=

e

e









−

=

=

 
  −     −   

    −   −   −      





  

or, ( )
( )

( )
1

1
1

1i

n

a
t

nE a
x

i

e
R t =

e







=

 
 −
 −
 

−
 
 


 . 

 

4. Bayes estimators of ( )R t under ( )( )2h R t   
Under ( )( )2h R t  , the posterior distribution is defined by 

( )( )
( )( ) ( )( )

( )( ) ( )( ) ( )

2

1

2

0

f x | R t h R t
f R t | x

f x | R t h R t dR t

=



          (22) 

Substituting the values of ( )( )2h R t  and ( )( )f x | R t  from equations (16) and (6) 
in (22), we get 

( )( )

( )

( )
( ) ( )

( )
( )

( ) ( )

( )

( )
( ) ( )

( )

1

1

1

1

1

1
1 1

1

1

1

1

1

1
1

1
1

1
1

n

i

i i

a
n xi

t
i

n

i

i i

xi

t

n
na x

a nx

na
t

i

e

e

n
na x

a nx

na
t

i

e

e

a
e e log R t

e

R t R t R t
B ,

f R t | x

a
e e log R t

e

R t














 








 



=

=

=

−
−

=

 −
− − 

 − 

−
−

=

 −

−

   
 − −     −
 
 

 −           
 

=

  
− −   

 −

  





( )
( ) ( )

( )

1

1

0
1 11

1

a
n

i

dR t

R t R t
B ,

 

 
=


− − 

 
 

 
 
 
 
 

 −       
 



  



Reliability estimation of Weibull-exponential distribution via bayesian approach 

 

137 

      

( ) ( ) ( )

( ) ( ) ( ) ( )

1

1

1
1 1

1

1 1
1 1

1

0

1

1

a
n xi

t
i

a
n xi

t
i

e
n

e

e
n

e

R t -log R t R t

R t -log R t R t dR t









 

 

=

=

 −
+ − − 

 − 

 −
+ − − 

 − 


−          

=


−          

 

or, ( )( )
( ) ( ) ( )

( ) ( )

1

1
1 1

1

1
1

0 1

1

1 1
1 1 1

1

a
n xi

t
i

i

e
n

e

n
a

xn
k

t
k i

R t -log R t R t
f R t | x

e
n k

ek




 








=

 −
+ − − 

 − 

+
−

= =


−          

=
   −   −    + − + +      −        

 

   (23)

 

Theorem 4. Assuming the squared error loss function, the Bayes estimate of ( )R t

, is of the form 

( )

( )

( )

1
1

0 1

1
1

0 1

1 1
1 1 1

1

1 1
1 1

1

i

i

n
a

xn
k

t
k i

nS
a

xn
k

t
k i

e
k

ek
R t

e
k

ek















+
−

 = =

+
−

= =

   −   −   − + + +      −       =
   −   −   − + +       −       

 

 

       (24) 

Proof. From equation (10), on using (23), 

( ) ( )
S

R t E R t


=     

( ) ( )( ) ( )
1

0

R t  f R t | x dR t=   

( )
( ) ( ) ( )

( ) ( )

( )
1

1
1 1

1 1

1
10

0 1

1

1 1
1 1 1

1

a
n xi

t
i

i

e
n

e

n
a

xn
k

t
k i

R t -log R t R t
R t dR t

e
n k

ek




 








=

  − + − − 
  −  

+
−

= =


−          

=
   −   −    + − + +      −        



 

 



A.K.Rao and H.Pandey 

136 

( ) ( ) ( ) ( )

( ) ( )

1

1 1
1

1

0

1
1

0 1

1

1 1
1 1 1

1

a
n xi

t
i

i

e
n

e

n
a

xn
k

t
k i

R t -log R t R t dR t

=

e
n k

ek




 








=

  − + − 
  −  

+
−

= =


−          

   −   −    + − + +      −        



 

  

or,  ( )

( )

( )

1
1

0 1

1
1

0 1

1 1
1 1 1

1

1 1
1 1

1

i

i

n
a

xn
k

t
k i

nS
a

xn
k

t
k i

e
k

ek
R t

e
k

ek















+
−

 = =

+
−

= =

   −   −   − + + +      −       =
   −   −   − + +       −       

 

 

. 

Theorem 5. Assuming the precautionary loss function, the Bayes estimate of ( )R t

, is of the form 

( )

( )

( )

1
21

1

0 1

1
1

0 1

1 1
1 1 2

1

1 1
1 1

1

i

i

n
a

xn
k

t
k i

nP
a

xn
k

t
k i

e
k

ek
R t

e
k

ek















+
−

 = =

+
−

= =

   −   −   − + + +      −       =
   −   −   − + +       −       

 

 

        (25) 

Proof. From equation (12), on using (23), 

 
( ) ( )( )

1
22

P
R t E R t


 =
 

 

( )( ) ( )( ) ( )

1
21

2

0

= R t f R t | x dR t
 
 
 


 



Reliability estimation of Weibull-exponential distribution via bayesian approach 

 

137 

( )( )
( ) ( ) ( )

( ) ( )

( )

1
2

1

1
1 1

1 1
2

1
10

0 1

1

1 1
1 1 1

1

a
n xi

t
i

i

e
n

e

n
a

xn
k

t
k i

R t -log R t R t
= R t dR t

e
n k

ek




 








=

  − + − − 
  −  

+
−

= =

 
 
 

−           
 

    −   −    + − + +       −          



 

 

( ) ( ) ( ) ( )

( ) ( )

1
2

1

1 1
1 1

1

0

1
1

0 1

1

1 1
1 1 1

1

a
n xi

t
i

i

e
n

e

n
a

xn
k

t
k i

R t -log R t R t dR t

=

e
n k

ek




 








=

  − + + − 
  −  

+
−

= =

 
 

 −          
 
 

    −   −    + − + +       −          



 

 

or,  ( )

( )

( )

1
21

1

0 1

1
1

0 1

1 1
1 1 2

1

1 1
1 1

1

i

i

n
a

xn
k

t
k i

nP
a

xn
k

t
k i

e
k

ek
R t

e
k

ek















+
−

 = =

+
−

= =

   −   −   − + + +      −       =
   −   −   − + +       −       

 

 

. 

Theorem 6. Assuming the entropy loss function, the Bayes estimate of ( )R t , is of 

the form 

( )

( )

( )

1
1

0 1

1
1

0 1

1 1
1 1

1

1 1
1 1 1

1

i

i

n
a

xn
k

t
k i

nE
a

xn
k

t
k i

e
k

ek
R t

e
k

ek















+
−

 = =

+
−

= =

   −   −   − + +      −       =
   −   −   − + − +       −       

 

 

       (26) 

Proof. From equation (14), on using (23), 



A.K.Rao and H.Pandey 

136 

( )
( )

1

1
E

R t E
R t

−
   

=    
     

( )
( )( ) ( )

1
1

0

1
= f R t | x dR t

R t

−
 
 
 


( )

( ) ( ) ( )

( ) ( )

( )
1

1

1
1 1

1 1

1
10

0 1

11

1 1
1 1 1

1

a
n xi

t
i

i

e
n

e

n
a

xn
k

t
k i

R t -log R t R t
= dR t

R t
e

n k
ek




 








=

−

  − + − − 
  −  

+
−

= =

 
 
 

−           
 

    −   −    + − + +       −          



 

( ) ( ) ( ) ( )

( ) ( )

1

1

1 1
2 1

1

0

1
1

0 1

1

1 1
1 1 1

1

a
n xi

t
i

i

e
n

e

n
a

xn
k

t
k i

R t -log R t R t dR t

=

e
n k

ek




 








=

−

  − + − − 
  −  

+
−

= =

 
 

 −          
 
 

    −   −    + − + +       −          



 

( )

( )

1
1

1

0 1

1
1

0 1

1 1
1 1 1

1

1 1
1 1

1

i

i

n
a

xn
k

t
k i

n
a

xn
k

t
k i

e
k

ek

e
k

ek















−
+

−

= =

+
−

= =

   −   −   − + − +      −       =
   −   −   − + +       −       

 

 
 

or, ( )

( )

( )

1
1

0 1

1
1

0 1

1 1
1 1

1

1 1
1 1 1

1

i

i

n
a

xn
k

t
k i

nE
a

xn
k

t
k i

e
k

ek
R t

e
k

ek















+
−

 = =

+
−

= =

   −   −   − + +      −       =
   −   −   − + − +       −       

 

 

 . 

 



Reliability estimation of Weibull-exponential distribution via bayesian approach 

 

137 

5. Conclusion 

We have obtained a number of Bayes estimators of reliability function ( )R t  of 

Weibull-exponential distribution. In equations (19), (20), and (21), we have 

obtained the Bayes estimators by using non-informative prior and in equations (24), 

(25), and (26), under beta prior. From the above said equation, it is clear that the 

Bayes estimators of ( )R t  depend upon the parameters of the prior distribution. In 

this case the risk function and corresponding Bayes risks do not exist. 

 

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