116 

 

 

  

 

 

Bayesian Estimation for Two Parameters of Weibull Distribution under 

Generalized Weighted Loss Function 

 

 

 

 

  

                                                                                     

                        
 

 

 

Abstract 

     In this paper, Bayes estimators for the shape and scale parameters of Weibull distribution 

have been obtained using the generalized weighted loss function, based on Exponential 

priors. Lindley’s approximation has been used effectively in Bayesian estimation. Based on 

theMonte Carlo simulation method, those estimators are compared depending on the mean 

squared errors (MSE’s).  

Keywords: Weibull Distribution; Bayesian estimation; Exponential prior; Generalized 

weighted loss function; Lindley’s approximation. 

 

1.Introduction       

   The Weibull Distribution is a continuous, it is one of the best-known lifetime distributions. 

It is widely used in reliability, Quality Control, weather forecasting and used to describe 

different types of observed failures of components and phenomena [1]. This distribution is 

named after Wal Oddi Weibull who described it in details in 1951.    

There are some recent works and literature of Weibull distribution: Xie et al. (2002) 

suggested the three parameters of modified Weibull distribution with a hazard function 

fashioned like a bathtub [2-3] recommend the Maximum likelihood method to estimate the 

unknown parameters of Weibull distribution, and [4] presented Bayesian estimation for 

Weibull distribution under asymmetric and symmetric loss function and that of maximum 

likelihood estimation. 

Ibn Al Haitham Journal for Pure and Applied Science 

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

 

Doi: 10.30526/34.4.2709 

Abtisam J. Kadhim 

Department of Mathematics, College of 

Science, Mustansiriyah University, 

Baghdad, Iraq. 

momasaayj@gmail.com 
 

Huda A. Rasheed 

Department of Mathematics, College of 

Science, Mustansiriyah University, 

Baghdad, Iraq. 

hudamath@uomustansiriyah.edu.iq 

Article history: Received   5, July, 2021, Accepted 29, August, 2021, Published in October 2021. 

 

mailto:momasaayj@gmail.com
mailto:hudamath@uomustansiriyah.edu.iq


Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

117 

 

The probability density function (pdf) of the two parameters Weibull distribution is defined 

as:  

𝑓( 𝑥 ∣ λ, 𝜗 ) =
λ

𝜗
 (

𝑥

𝜗
)

λ−1

𝑒
−(

𝑥
𝜗

)λ
        ;            𝑥 > 0  ,      λ, ϑ >  0                                    (1) 

The corresponding cumulative distribution function (CDF) is given by: 

𝐹( 𝑥 ∣ λ, 𝜗 ) = ∫  
𝑥

0

𝜆

𝜗𝜆
 𝑡𝜆−1𝑒

−(
𝑡
𝜗

)
λ

𝑑𝑥 = 1 − 𝑒
−(

𝑥
𝜗

)
𝜆

 

The Reliability function is defined:  

R(X ; 𝜆, 𝜗) = 1 − F(𝑋 ; 𝜆, 𝜗) = 𝑒
−(

𝑥
𝜗

)
𝜆

 

2. Bayesian Estimators  

     Bayesian estimation is an estimation that aims to minimize the posterior expected value 

of a loss function. [5] In this suction, Bayesian estimators are obtained based on a new loss 

function, which is generalized weighted loss function. The Bayesian estimators are derived 

with assuming the exponential prior for each of λ and 𝜗, i.e.: 

λ~𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 (1/ 𝛿) with the following pdf   

𝑔1(𝜆) = 𝛿𝑒
−𝛿𝜆             𝛿 > 0                                                                

          

𝜗 ~𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙 (1/ 𝛾) with the pdf defined as  

𝑔2(𝜗) = 𝛾𝑒
−𝛾𝜗                   𝛾 > 0               

𝜆 and 𝜗 are independent of each other. Therefore, the joint exponential prior is:  

g (λ, ϑ) = 𝑔1(λ)  𝑔2(𝜗)  = 𝛿 𝑒
− 𝛿𝜆  𝛾𝑒−  𝛾𝜗   

Hence, the posterior distribution of 𝜗 and λ is given by  

𝜋(𝜆, 𝜗; 𝑋) =
𝑒 −(𝛾𝜗+𝛿𝜆) (

𝜆

𝜗𝜆
)

𝑛
∏  𝑛𝑖=1 𝑥𝑖

𝜆−1 exp(−
∑  

𝑛
𝑖=1 𝑥𝑖

𝜆

𝜗𝜆
)

∫  
∞

0
∫  𝑒 −(𝛾𝜗+𝛿𝜆)   

∞

0
 (

𝜆

𝜗𝜆
)

𝑛
∏  𝑛𝑖=1 𝑥𝑖

𝜆−1 exp(−
∑  𝑛

𝑖=1
𝑥

𝑖
𝜆

𝜗𝜆
) 𝑑𝜆𝑑𝜗

             

3. Bayesian Estimator under Generalized Weighted Loss Function 

[6] suggested a new loss function in estimating the scale parameter for Laplace distribution, 

which is called generalized weighted loss function and introduced as follows:  

𝐿(𝜃, 𝜃) =
(∑  𝑘𝑗=0 𝑎𝑗 𝜃

𝑗 )(𝜃 − 𝜃)2

𝜃𝜏 
              

 𝑎𝑗 > 0, 𝜃 > 0, 𝑗 = 1,2, … , 𝑘 

Where k, 𝜏  are constants             

The risk function 𝑅𝐺𝑤 (θ̂, θ)  can be derived as  

𝑅𝐺𝑤 (𝜃, 𝜃) = E [L(�̂�, 𝜃)] = ∫ [ 
∞

0

1

𝜃𝜏
(∑  𝑘𝑗=0 𝑎𝑗 𝜃

𝑗 )(𝜃 − 𝜃)2] 𝜋(𝜃 ∣ 𝑥)𝑑𝜃       

Then, Bayesian estimator under the generalized weighted error loss function minimizes the risk 

function, as follows:  



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

118 

 

�̂�𝐺𝑤 =
𝑎0𝐸 (

1
𝜃𝜏 −1

∣ 𝑥) + 𝑎1𝐸 (
1

𝜃𝜏 −2
∣ 𝑥) + ⋯ + 𝑎𝐾 𝐸 (

1
𝜃𝜏 −(𝐾+1)

∣ 𝑥)

𝑎0𝐸 (
1

𝜃𝜏 
∣ 𝑥) + 𝑎1𝐸 (

1
𝜃𝜏 −1

∣ 𝑥) + ⋯ + 𝑎𝐾 𝐸 (
1

𝜃𝜏 −𝐾
∣ 𝑥)

                                                       (2)  

Whereas 𝑎0, 𝑎1, … , 𝑎𝑘 are constants. 

a-Bayesian Estimator for 𝜗 under Generalized Weighted Loss Function  

 Bayesian estimation for ϑ under generalized weighted loss function can be obtained as follows: 

By suppose that, k=1 and 𝜏  = 0 then, 

�̂�10 =
𝑎0𝐸(𝜗 ∣ 𝑋) + 𝑎1𝐸(𝜗

2 ∣ 𝑥)

𝑎0 + 𝑎1𝐸(𝜗 ∣ 𝑥)
                                                                                                        (3)        

Assumed that w (λ, ϑ) be any function for 𝜆, 𝜗. Therefore:  

E[w(λ, ϑ)] = ∫ ∫ w(λ, ϑ)

∞

0

∞

0

π(λ, ϑ) dλ dϑ 

                     = ∫ ∫ w(λ, ϑ)
L(x1, x2, … , xn; λ, ϑ)π(λ, ϑ) dλ dϑ 

∫ ∫ L(x1, x2, … , xn; λ, ϑ)π(λ, ϑ) dλ dϑ
∞

0

∞

0
 

∞

0

∞

0

 

 

                     =
∫ ∫ w(λ, ϑ)L(x1, x2, … , xn; λ, ϑ)π(λ, ϑ) dλ dϑ

∞

0

∞

0

∫ ∫ L(x1, x2, … , xn; λ, ϑ)π(λ, ϑ) dλ dϑ
∞

0

∞

0
 

 

E[𝜗│𝑋]  =
∫  

∞

0
∫  𝜗  

∞

0
 𝑒

−(𝛾𝜗+𝛿𝜆)
 (

𝜆

𝜗𝜆
)

𝑛

∏  𝑛𝑖=1 𝑥𝑖
𝜆−1

exp (−
∑  𝑛𝑖=1 𝑥𝑖

𝜆

𝜗𝜆
) 𝑑𝜆𝑑𝜗

∫  
∞

0
∫   𝑒−(𝛾𝜗+𝛿𝜆)   

∞

0
 (

𝜆

𝜗𝜆
)

𝑛

∏  𝑛𝑖=1 𝑥𝑖
𝜆−1

exp (−
∑  𝑛𝑖=1 𝑥𝑖

𝜆

𝜗𝜆
)  𝑑𝜆 𝑑𝜗

 

Observed that, it is difficult to obtain the solution of the ratio of two integrals. Hence, the 

solution will be approximately by using Lindley’s approximation [7], as follows: 

E[𝜗│𝑋] ≈ �̂� + 𝑝1𝑤1𝜎11 +
1

2
(𝐿30𝑤1𝜎11

2 ) +
1

2
(𝐿12𝑤1𝜎11𝜎22)                                                                               (4) 

Where,  

Assuming that 

w (λ, ϑ) = ϑ 

Thus, 

𝑤1 =
∂𝑤(λ,ϑ)

∂𝜗
=

∂

∂ϑ
(ϑ) = 1                                                                                                                                      (5) 

Lij =
∂i+j

∂ϑi ∂λj
ln L(λ, ϑ)                                                       i, j = 0,1,2,3 

       =
∂i+j

∂ϑi ∂λj
[n ln(λ) − n λ ln(ϑ) + (λ − 1) ∑  

n

i=1

ln(xi) − ∑  

n

i=1

(
xi
ϑ

)
λ

] 

𝐿12 =
∂3𝑙𝑛𝐿(𝜆, 𝜗)

∂𝜗 ∂𝜆2
=

𝜆

𝜗
∑  

𝑛

𝑖=1

(
𝑥

𝜗
)

𝜆

(ln (
𝑥𝑖
𝜗

))
2

+
2

𝜗
∑  

𝑛

𝑖=1

(
𝑥𝑖
𝜗

)
𝜆

ln (
𝑥𝑖
𝜗

)                                                                            (6)     

L20 =
∂2lnL(λ, ϑ)

∂ϑ2
=

n λ

ϑ2
−

λ(λ + 1)

ϑ2
∑  

n

i=1

(
xi
ϑ

)
λ

       ,    L02 =
∂2lnL(λ, ϑ)

∂λ2
= −

n

λ2
− ∑  

n

i=1

(
xi
ϑ

)
λ

(ln (
xi
ϑ

))
2

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

119 

 

𝐿30=
∂3 𝑙𝑛 𝐿(𝜆,𝜗)

∂𝜗3
= −

2𝑛𝜆

𝜗3
+

𝜆(𝜆+1)(𝜆+2)

𝜗3
∑  𝑛𝑖=1 (

𝑥𝑖

𝜗
)

𝜆

                                                                                                (7) 

𝜎11 = −
1

𝐿20
=

−𝜗2

𝑛𝜆 −  𝜆 (𝜆 + 1)  ∑  𝑛𝑖=1 (
𝑥𝑖
𝜗

)
𝜆

 

                                                                                                         (8) 

𝜎22 = −
1

L02
=

𝜆2

𝑛 + 𝜆2 ∑  𝑛𝑖=1 (
𝑥𝑖
𝜗

)
𝜆

(ln (
𝑥𝑖
𝜆

))
2

                                                                                                       (9) 

We have, 

g (λ, ϑ) = γ δ  𝑒−(𝛾𝜗+𝛿𝜆)  

 P = ln g (λ, ϑ) = ln γ + ln 𝛿 – γ ϑ – δ λ  

p1 =
∂p

∂𝜗
= −𝛾                                                                                                                                           (10)                                                                                       

Substituting (5), (6), (7), (8), (9) and (10) into (4), yields: 

E [ϑ|𝑋] ≈ ϑ̂ − 𝛾𝜎11 +
1

2
(L30𝜎11

2 ) +
1

2
(L12𝜎11𝜎22)                                                                 (11)     

Similarly, Lindley’s approximation for E[𝜗2│𝑋] is given by:   

E[𝜗2│𝑋] ≈ ϑ̂2 +
1

2
(w11σ11)+p1w1σ11 +

1

2
(L30w1σ11

2 ) +
1

2
(L12w1σ11σ22)                                           (12)     

Assuming that, w (λ, ϑ) = ϑ2 

 w1 =
∂w(λ,ϑ)

∂ϑ
 = 2ϑ                                                                                                                                  (13) 

w11 =
∂2w(λ,ϑ)

∂ϑ2
= 2                                                                                                                                  (14)              

Substituting (6), (7), (8), (9), (10), (13) and (14) into (12), yields:  

E[𝜗2│𝑋] ≈ ϑ̂2 + (1 − 2γ�̂�) σ11 + �̂�[ (L30σ11
2 ) + (L12σ11σ22) ]                                                                                              (15) 

After substituting (11) and (15) into (2), yields: 

�̂�10 =  
(a0ϑ̂ + a1ϑ̂

2) + [−a0𝛾 + a1(1 − 2�̂�γ)]σ11 + (
a0
2

+ a1�̂�)[L30 σ11
2 + L12σ11σ22]

a0 + a1[ϑ̂ − 𝛾𝜎11 +
1
2

(L30𝜎11
2 ) +

1
2

(L12𝜎11𝜎22)  ]
                                        (16) 

Now, another estimator under generalized weighted loss function will be derived, when letting 

k=1 and 𝜏 =1, gives:     

ϑ̂11 =
a0+a1E(ϑ|x)

a0E(
1

ϑ
|x)+a1

                                                                                                      

Similarly, the Lindley’s approximation for E [
1

𝜗
│𝑋] is given by: 

E [
1

𝜗
│𝑋] ≈

1

ϑ̂
+

1

2
(w11σ11) + p1w1σ11 +

1

2
(L30w1σ11

2 ) +
1

2
(L12w1σ11σ22)                                                            (17) 

Assuming that, w (λ, ϑ) = 
1

ϑ
  

w1 =
∂w(λ,ϑ)

∂ϑ
 =

−1

𝜗2
                                                                                                         (18) 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

120 

 

w11 =
∂2w(λ,ϑ)

∂ϑ2
=

2

𝜗3
                                                                                                          (19) 

Substituting (6), (7), (8), (9), (10), (18) and (19) into (17), yields:   

E [
1

𝜗
│𝑋] ≈

1

ϑ̂
+ (

1

�̂�3
+

𝛾

�̂�2
) σ11 −

1

2�̂�2
[(L30σ11

2 ) + (L12σ11σ22) ]                                                                      (20)  

After substituting (11) and (20) into (2) give up, 

ϑ̂11 =
a0+a1[ϑ̂−𝛾𝜎11+

1

2
(L30𝜎11

2 )+
1

2
(L12𝜎11𝜎22)  ]

a1+a0[
1

ϑ̂
+(

1

�̂�3
+

𝛾

�̂�2
)σ11−

1

2�̂�2
[(L30σ11

2 )+(L12σ11σ22) ]]
                                                                            (21) 

Another estimator under generalized weighted loss function will be derived, when letting k=1 

and 𝜏 =2, gives:     

ϑ̂12 ≈
a0E(

1

ϑ
|x)+a1

a0E(
1

ϑ2
|x)+a1E(

1

ϑ
|x)

                                                                                                   

Similarly, the Lindley’s approximation for E [
1

𝜗
2 |𝑋] is given by:   

E [
1

𝜗2
|𝑋] ≈

1

ϑ̂2
+

1

2
(w11σ11) + p1w1σ11 +

1

2
(L30w1σ11

2 )

+
1

2
(L12w1σ11 σ22)                                                                                                                           (22) 

Assuming that, w (λ, ϑ) = 
1

ϑ2
 

w1 =
∂w(λ,ϑ)

∂ϑ
 =

−2

𝜗3
                                                                                                                (23) 

w11 =
∂2w(λ,ϑ)

∂ϑ2
=

6

𝜗4
                                                                                                              (24) 

Substituting (6), (7), (8), (9), (10), (23) and (24) into (22), yields: 

E [
1

𝜗2
|𝑋] ≈

1

ϑ̂2
+ (

3

�̂�4
+

2𝛾

�̂�3
) σ11 −

1

�̂�3
[(L30σ11

2 ) + (L12σ11σ22)]                                                                        (25)   

After substituting (20) and (25) into (2) give up, 

ϑ̂12 ≈
a1+a0[

1

ϑ̂
+(

1

�̂�3
+

𝛾

�̂�2
)σ11−

1

2�̂�2
 [ (L30σ11

2 )+(L12σ11σ22)]]

(
𝑎0
�̂�2

+
𝑎1
�̂�

)+𝑧1𝜎11−(
𝑎0
�̂�3

+
𝑎1

2�̂�2
)[(L30σ11

2 )+(L12σ11σ22)]
                                                          (26)      

 Whereas, 𝑧1 = 𝑎0 (
3

�̂�4
+

2𝛾

�̂�3
) + 𝑎1(

1

�̂�3
+

𝛾

�̂�2
). 

Another estimator under generalized weighted loss function will be derived, when letting k=1 

and  𝜏 =0, gives:     

 ϑ̂20 =
a0E(ϑ|x)+a1E(ϑ

2|x)+a2E(ϑ
3|x)

a0+a1E(ϑ|x)+a2E(ϑ
2|x)

                                                                                 

Similarly, the Lindley’s approximation for E[𝜗3|𝑋] is given by:   

 E[𝜗3|𝑋] ≈ ϑ̂3 +
1

2
(w11σ11)+p1w1σ11 +

1

2
(L30w1σ11

2 ) +
1

2
(L12w1σ11σ22)                      (27)  



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

121 

 

Assuming that, w (λ, ϑ) = 𝜗3 

w1 =
∂w(λ,ϑ)

∂ϑ
 = 3ϑ2                                                                                                              (28) 

w11 =
∂2w(λ,ϑ)

∂ϑ2
= 6ϑ                                                                                                             (29) 

Substituting (6), (7), (8), (9), (10), (28) and (29) into (27), yields: 

E[𝜗3|𝑋] ≈ ϑ̂3 + (3�̂� − 3γ�̂�2)σ11 +
3�̂�2

2
[(L30σ11

2 ) +

(L12σ11σ22)]                                                                        (30)     

After substituting (11), (15) and (30) into (2) give up, 

ϑ̂20 =
𝑧4 + 𝑧3σ11 + 𝑧2[(L30σ11

2 ) + (L12σ11σ22)]  

𝑧5 + 𝑧6 σ11 + (
𝑎1
2

+ 𝑎2�̂�) [(L30σ11
2 ) + (L12σ11σ22)]  

                                                                                 (31) 

Where,  

𝑧2 =
𝑎0

2
+ 𝑎1�̂� −

3𝑎2�̂�
2

2
,   𝑧3 = 𝑎0𝛾 + 𝑎1(1 − 2𝛾) + 𝑎2(3�̂� − 3𝛾�̂�

2)   

𝑧4 = 𝑎0ϑ̂ + 𝑎1ϑ̂
2 + 𝑎2ϑ̂

3 ,   𝑧5 = 𝑎0 + 𝑎1ϑ̂  +𝑎2 ϑ̂
2 ,    𝑧6 = −𝑎1𝛾 + 𝑎2(1 − 2𝛾�̂�)                                                       

Now, when k=2 and 𝜏 =1, gives:    

ϑ̂21 =
a0+a1E(ϑ|x)+a2E(ϑ

2|x)

a0E(
1

ϑ
|x)+a1+a2E(ϑ|x)

                                                                                          

After substituting (11), (15) and (20) into (2), yields: 

ϑ̂21 =
𝑧5+𝑧6σ11+(

𝑎1
2

+𝑎2�̂�)[(L30σ11
2 )+(L12σ11σ22)]  

(
𝑎0
ϑ̂

+a1+𝑎2ϑ̂)+(
𝑎0
�̂�3

+
𝑎0𝛾

�̂�2
−𝑎2𝛾)σ11+(

𝑎0
2�̂�2

+
𝑎2
2

)[(L30σ11
2 )+(L12σ11σ22)]

                                   (32)   

Where,   𝑧5 = 𝑎0 + 𝑎1ϑ̂  +𝑎2 ϑ̂
2, 𝑧6 = −𝑎1𝛾 + 𝑎2(1 − 2𝛾�̂�)      

When k=2 and 𝜏 =2, gives:    

ϑ̂22 =
a0E(

1

ϑ
|x)+a1+a2E(ϑ|x)

a0E(
1

ϑ2
|x)+a1E(

1

ϑ
|x)+a2

                                                                                                                                     

After substituting (11), (20) and (25) into (2), yields: 

ϑ̂22 =
(

𝑎0
ϑ̂

+a1+𝑎2ϑ̂)+(
𝑎0
�̂�3

+
𝑎0𝛾

�̂�2
−𝑎2𝛾)σ11+(

𝑎0
2�̂�2

+
𝑎2
2

)[(L30σ11
2 )+(L12σ11σ22)]

(
𝑎0

ϑ̂2̂
+

𝑎1
ϑ̂

+ 𝑎2)+𝑦1σ11+(
−𝑎0
�̂�3

+
𝑎1

2�̂�2
)[(L30σ11

2 )+(L12σ11σ22)]
                                         (33)  

 Where, 𝑦1 = 𝑎0 (
3

�̂�4
+

𝑎1

�̂�3
) + 𝑎1 (

1

�̂�3
+

𝛾

�̂�2
). 

 

b- Bayesian Estimation for 𝝀 under Generalized Weighted Loss Function  

 We can obtain Bayesian estimation for λ under generalized weighted loss function, when 

letting, k=1 and 𝜏 = 0, gives:    



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

122 

 

λ̂10 =  
a0E(λ|x ) + a1E(λ

2|x )

a0 + a1E(λ|x )
       

The Lindley’s approximation for E[𝜆│𝑋] is given by:   

E[𝜆│𝑋] ≈ λ̂ + p2w2σ22 +
1

2
(L03w2σ22

2 ) +
1

2
(L21w2σ11σ22)                                                                      (34)  

Where, 

Assume that, W(λ, ϑ) = λ   

w1  =
∂w(λ, ϑ)

∂ϑ
=

∂

∂ϑ
 (λ) 

        = 0 = w11                                                                                                                                                    

w2 =
∂w(λ,ϑ)

∂λ
= 1                                                                                                                                   (35)    

𝐿03 =
∂3𝐿(𝜆, 𝜗)

∂𝜆3
=

2𝑛

𝜆3
− ∑  

𝑛

𝑖=1

(
𝑥𝑖
𝜗

)
𝜆

(ln (
𝑥𝑖
𝜗

))
3

                                                                                                    (36) 

𝐿21 =
∂3𝐿(𝜆,𝜗)

∂𝜗2 ∂𝜆
=

𝑛

𝜗2
−

2𝜆+1

𝜗2
∑  𝑛𝑖=1 (

𝑥𝑖

𝜗
)

𝜆

−
𝜆(𝜆+1)

𝜗2
∑  𝑛𝑖=1 (

𝑥𝑖

𝜗
)

𝜆

ln (
𝑥𝑖

𝜗
)                                                                   (37)     

 P = ln g (λ, ϑ) = ln(γ) + ln(δ) – γ ϑ – δ λ  

p
2

=
∂p

∂λ
= −δ                                                                                                                                       (38) 

Substituting (8), (9), (35), (36), (37) and (38) into (34), gives us:                                                                          

E[𝜆│𝑋] ≈ λ̂ − δσ22 +
1

2
[(L03σ22

2 ) + (L21σ11σ22)]                                                              (39) 

Similarly, the Lindley’s approximation for E[𝜆2│𝑋] is given by:   

E[𝜆2│𝑋] ≈ λ̂2 +
1

2
(w22σ22) + p2w2σ22 +

1

2
(L03w2σ22

2 ) +
1

2
(L21w2σ11σ22)                           (40)                     

Assume that, W (λ, 𝜗) = 𝜆2 

w2 =
∂w(λ,ϑ)

∂λ
= 2λ                                                                                                              (41) 

w22 =
∂2w(λ,ϑ)

∂λ2
= 2                                                                                                             (42) 

Substituting (8), (9), (36), (37), (38), (41) and (42) into (40), yields:  

 E[𝜆2│𝑋] ≈ λ̂2 + (1 − 2δ�̂�)σ22 + �̂�[(L03σ22
2 ) + (L21σ11σ22)]                                                         (43)   

After substituting (39) and (43) into (2) give up, 

λ̂10 =  
(𝑎0λ̂+𝑎1λ̂

2)+[𝑎1(1−2𝛿�̂�)−𝑎0𝛿]σ22+(
𝑎0
2

+𝑎1�̂�)[(L03σ22
2 )+(L21σ11σ22)   ]

(a0+a1λ̂)− 𝑎1δ σ22+
𝑎1
2

[(L03σ22
2 )+(L21σ11σ22)] 

                                       (44)      

Now, another estimator under generalized weighted loss function will be derived by letting, k=1 

and 𝜏 =1, gives:    



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

123 

 

  λ̂11 =
a0+a1E(λ|x)

a0E(
1

λ
|x)+a1

       

Similarly, the Lindley’s approximation for E [
1

𝜆
│𝑋] is given by:    

E [
1

𝜆
│𝑋] ≈

1

λ̂
+

1

2
(w22σ22) + p2w2σ22 +

1

2
(L03w2σ22

2 ) +
1

2
(L21w2σ11σ22)                                                    (45)   

Assume that, W (λ, ϑ) =
1

λ
  

w2 =
∂w(λ,ϑ)

∂λ
=

−1

𝜆2
                                                                                                            (46) 

w22 =
∂2w(λ,ϑ)

∂λ2
=

2

𝜆3
                                                                                                          (47)  

Substituting (8), (9), (36), (37), (38), (46) and (47) into (45), yields:                                         

E [
1

𝜆
│𝑋] ≈

1

λ̂
+ (

1

�̂�3
+

𝛿

�̂�2
) σ22

−
1

2�̂�2
[(L03σ22

2 ) + (L21σ11σ22)]                                                                            (48)     

After substituting (39) and (48) into (2) give up, 

λ̂11 =
(a0+a1λ̂)− 𝑎1δ σ22+

𝑎1
2

[(L03σ22
2 )+(L21σ11σ22)] 

a1+
𝑎0
λ̂

+𝑎0(
1

�̂�3
+

𝛿

�̂�2
)𝜎22−

𝑎0
2�̂�2

[(L03𝜎22
2 )+(L21𝜎11𝜎22)]

                                                                 (49) 

Another estimator under generalized weighted loss function will be derived, when letting  

k=1 and 𝜏 =2, gives:    

 λ̂12 =
a0E(

1

λ
|x)+a1

a0E(
1

λ2
|x)+a1E(

1

λ
|x)

 

Similarly, the Lindley’s approximation for E [
1

𝜆
2 │𝑋] is given by:    

E [
1

𝜆2
│𝑋] ≈

1

𝜆2̂
+

1

2
(w22σ22) + p2w2σ22 +

1

2
(L03w2σ22

2 ) +
1

2
(L21w2σ11σ22)                                                (50) 

 Assume that, W (λ, ϑ) = 
1

𝜆2
 

w2 =
∂w(λ,ϑ)

∂λ
=

−2

𝜆3
                                                                                                               (51) 

w22 =
∂2w(λ,ϑ)

∂λ2
=

6

𝜆4
                                                                                                            (52) 

Substituting (8), (9), (36), (37), (38), (51) and (52) into (50), yields: 

E [
1

𝜆2
│𝑋] ≈

1

𝜆2̂
+ (

3

�̂�4
+

2𝛿

�̂�3
) σ22 −

1

�̂�3
[(L03σ22

2 ) + (L21σ11σ22)]                                                                         (53)   

After substituting (48) and (53) into (2) give up, 

λ̂12 =
a1+

𝑎0
λ̂

+𝑎0(
1

�̂�3
+

𝛿

�̂�2
)𝜎22+

𝑎0
2�̂�2

[(L03𝜎22
2 )+(L21𝜎11𝜎22)]

(
𝑎0
λ̂2

+
𝑎1
λ̂

)+𝑦2σ22−(
𝑎0
�̂�3

+
𝑎1

2�̂�2
)[(L03σ22

2 )+(L21σ11σ22)] 
                                                                     (54) 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

124 

 

Where, 𝑦2 = 𝑎0 (
3

�̂�4
+

2𝛿

�̂�3
) + 𝑎1(

1

�̂�3
+

𝛿

�̂�2
).  

Now, when K=1 and 𝜏 =0, gives:    

λ̂20 =
a0E(λ|x)+a1E(λ

2|x)+a2E(λ
3|x)

a0+a1E(λ|x)+a2E(λ
2|x)

                                                                     

Similarly, the Lindley’s approximation for E[𝜆3│𝑋] is given by:    

E[𝜆3│𝑋] ≈ λ̂
3

+
1

2
(w22σ22) + p

2
w2σ22 +

1

2
(L03w2σ22

2 ) +
1

2
(L21w2σ11σ22)                           (55)    

Assume that, w (λ, ϑ) = λ3 

w2 =
∂w(λ,ϑ)

∂λ
= 3𝜆2                                                                                                            (56) 

w22 =
∂2w(λ,ϑ)

∂λ2
= 6λ                                                                                                                                     (57)   

Substituting (8), (9), (36), (37), (38), (56) and (57) into (55), gives us: 

E[𝜆3│𝑋] ≈ λ̂3 + (3�̂� − 3δ�̂�2)σ22 +
3�̂�2

2
[(L03σ22

2 ) + (L21σ11σ22)]                                               (58) 

After substituting (39), (43) and (58) into (2) give up, 

λ̂20 = 
𝑦3+𝑦4σ22+(

𝑎0
2

+𝑎1�̂�+
3𝑎2

2
�̂�2)[(L03σ22

2 )+(L21σ11σ22)]

𝑦5+[−𝑎1𝛿+𝑎2(1−2𝛿�̂�)]σ22+(
𝑎1
2

+𝑎2�̂�)[(L03σ22
2 )+(L21σ11σ22)]

                                               (59)                                                     

Where, 𝑦3 = (𝑎0�̂� + 𝑎1λ̂
2 + 𝑎2λ̂

3), 𝑦5 = (𝑎0 + 𝑎1�̂� + 𝑎2λ̂
2), 𝑦4 = −𝑎0𝛿 + 𝑎1(1 − 2𝛿�̂�) + 𝑎2(3�̂� − 3𝛿�̂�

2.                                                                                                 

Another estimator under generalized weighted loss function will be derived, when letting     

 k=2 and 𝜏 =1, gives:                                                                                   

λ̂21 =  
a0 + a1E(λ|x) + a2E(λ

2|x)

a0E (
1
λ

|x) + a1 + a2E(λ|x)
         

After substituting (39), (43) and (48) into (2) give up, 

λ̂21 =
𝑦5+[−𝑎1𝛿+𝑎2(1−2𝛿�̂�)]σ22+(

𝑎1
2

+𝑎2�̂�)[(L03σ22
2 )+(L21σ11σ22)]

(
𝑎0
�̂�

+𝑎1+𝑎2�̂�)+[𝑎0(
1

�̂�3
+

𝛿

�̂�2
)−𝑎2𝛿]σ22+(

−𝑎0
2�̂�2

+
𝑎2
2

)[(L03σ22
2 )+(L21σ11σ22)]

                          (60)                 

 Where,  𝑦5 = (𝑎0 + 𝑎1�̂� + 𝑎2λ̂
2).  

Now, when k=2 and 𝜏 =2, gives:   

λ̂22 =
a0E(

1

λ
|x)+a1+a2E(λ|x)

a0E(
1

λ2
|x)+a1E(

1

λ
|x)+a2

                                                                              

After substituting (48) and (53) into (2) give up, 

λ̂22 =
(

a1
λ̂

+𝑎1+𝑎2�̂�)+[𝑎0(
1

�̂�3
+

𝛿

�̂�2
)−𝑎2𝑎]σ22+(

−𝑎0
2�̂�2

+
𝑎2
2

)[(L03σ22
2 )+(L21σ11σ22)]

(
𝑎0
�̅�2

+
𝛼1
�̂�

+a2)+𝒚𝟔σ22−(
𝑎0
�̂�3

+
𝑎1

2�̂�2
))[(L03σ22

2 )+(L21σ11σ22)]
                                    (61) 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

125 

 

Where, 𝑦6 = 𝑎0 (
3

�̂�4
+

2𝛿

�̂�3
) + 𝑎1(

1

�̂�3
+

𝛿

�̂�2
)  

4. Simulation Study 

 In this section, we employed the Monte – Carlo simulation to compare the performance of 

different estimates (Bayes Estimators under generalized weighted loss function) for unknown 

shape and scale parameters of WD based on the mean squared errors, which can be written as: 

MSE (𝜃) =
∑  𝑙𝑖=1 (𝜃𝑖 − 𝜃)

2

𝐼
 

 Where, I is the number of replications.  

We generated I = 5000 samples from two parameters WD with different sizes (n = 20, 50, and 

100). With assuming (ϑ = 0.7, 1.4) and (λ = 0.5, 1.5).   

 The values of the prior’s parameter of ϑ was chosen as δ = 0.5, 1.5, and for λ ' s prior parameter 

γ = 0.5, 1.5. 

5. Discussion and Conclusion 

     The expected values and (MSE's) for estimating ϑ and λ are tabulated in Tables (1-8). 

 The following points can summarize the results of the tables:  

1. The results of the two parameters of Weibull distribution shows that the expected values 

for different estimates are close to the real values for all sizes of samples. 

2. The best estimator for 𝜆 is Bayesian estimation under generalized weighted loss function 

when (k=2, 𝜏=0) ( �̂�20 ) with (𝛾 = 𝛿 = 1.5)  when n= 100 for different cases and with all 

sample sizes, from table (4). 

3. The best estimator for 𝜗 is Bayesian estimation under generalized weighted loss function 

when (k=1, 𝜏=2) ( �̂�20 ) with (𝛾, 𝛿 = 1.5)  when n=20 for different cases and with all sample 

sizes, from table (8). 

4. It is clear that the results for ϑ, 𝜆 (expected values and MSE' s) at 𝛾, δ = 1.5 are the best as 

the corresponding result when (γ, δ = 0.5). 

5. It is observed that MSE's of all shape parameter estimators increase with the increase of 

the value of the shape parameter. Also, MSE values for all scale parameter estimates are 

increasing with the scale parameter value in all cases. 

 

 

 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

126 

 

Table 1: Expected values and MSE’s for λ̂ when λ = 0.5 and ϑ = 0.7 

 

Estimate Criterion n = 20 n = 50 n = 100 

  

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5    𝛅 

=1.5 

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5    𝛅 

=1.5 

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5    𝛅 

=1.5 

�̂�𝟏𝟎 
Mean 0.503227 0.495727 0.507006 0.504095 0.508469 0.507031 

MSE 0.000025 0.000028 0.000050 0.000018 0.000072 0.000050 

�̂�𝟏𝟏 
Mean 0.489750 0.482754 0.501506 0.498679 0.505701 0.504283 

MSE 0.000113 0.000310 0.000003 0.000003 0.000033 0.000019 

�̂�𝟏𝟐 
Mean 0.478153 0.472111 0.496335 0.493675 0.503017 0.501642 

MSE 0.000494 0.000804 0.000015 0.000043 0.000009 0.000003 

�̂�𝟐𝟎 
Mean 0.518424 0.511074 0.512979 0.510084 0.511433 0.509998 

MSE 0.000380 0.000150 0.000171 0.000104 0.000131 0.000100 

�̂�𝟐𝟏 
Mean 0.504007 0.496489 0.507316 0.504403 0.508624 0.507185 

MSE 0.000032 0.000023 0.000055 0.000020 0.000075 0.000052 

�̂�𝟐𝟐 
Mean 0.490440 0.483398 0.501803 0.498966 0.505852 0.504432 

MSE 0.000100 0.000288 0.000004 0.000002 0.000034 0.000020 

 

Table 2: Expected values and MSE’s for λ̂ when λ = 1.5 and ϑ = 0.7  

Estimate Criterion 

n = 20 n = 50 n = 100 

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5          

𝛅 =1.5 

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5        

𝛅 =1.5 

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5           

𝛅 =1.5 

�̂�𝟏𝟎 
Mean 1.484634 1.411570 1.499238 1.472202 1.504442 1.491329 

MSE 0.000332 0.007883 0.000008 0.000776 0.000021 0.000075 

�̂�𝟏𝟏 
Mean 1.442376 1.376958 1.482284 1.456403 1.496002 1.483166 

MSE 0.003336 0.015360 0.000315 0.001917 0.000016 0.000286 

�̂�𝟏𝟐 
Mean 1.406637 1.352321 1.466431 1.442446 1.487826 1.475477 

MSE 0.008800 0.022182 0.001133 0.003348 0.000149 0.000607 

�̂�𝟐𝟎 
Mean 1.531194 1.456616 1.517092 1.489821 1.513201 1.500028 

MSE 0.001342 0.001922 0.000321 0.000106 0.000178 0.000000 

�̂�𝟐𝟏 
Mean 1.485492 1.412314 1.499574 1.472524 1.504610 1.491494 

MSE 0.000310 0.007750 0.000008 0.000758 0.000022 0.000073 

�̂�𝟐𝟐 
Mean 1.443126 1.377496 1.482602 1.456688 1.496156 1.483315 

MSE 0.003250 0.015224 0.000303 0.001892 0.000015 0.000280 

 

Table 3: Expected values and MSE’s for λ̂ when λ = 0.5 and ϑ = 1.4  

Estimate Criterion 

n = 20 n = 50 n = 100 

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5         

𝛅 =1.5 

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5         

𝛅 =1.5 

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5          

𝛅 =1.5 

�̂�𝟏𝟎 
Mean 0.503216 0.495738 0.507010 0.504107 0.508471 0.507037 

MSE 0.000025 0.000028 0.000050 0.000018 0.000072 0.000050 

�̂�𝟏𝟏 
Mean 0.489778 0.482803 0.501525 0.498704 0.505712 0.504299 

MSE 0.000113 0.000309 0.000003 0.000003 0.000033 0.000019 

�̂�𝟏𝟐 
Mean 0.478211 0.472186 0.496365 0.493712 0.503036 0.501666 

MSE 0.000492 0.000800 0.000015 0.000042 0.000010 0.000003 

�̂�𝟐𝟎 
Mean 0.518368 0.511040 0.512969 0.510081 0.511426 0.509996 

MSE 0.000378 0.000150 0.000171 0.000104 0.000131 0.000100 

�̂�𝟐𝟏 
Mean 0.503993 0.496498 0.507320 0.504413 0.508626 0.507192 

MSE 0.000031 0.000023 0.000055 0.000020 0.000075 0.000052 

�̂�𝟐𝟐 
Mean 0.490466 0.483443 0.501818 0.498989 0.505862 0.504448 

MSE 0.000099 0.000287 0.000004 0.000002 0.000034 0.000020 

 

 

 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

127 

 

Table 4: Expected values and MSE’s for λ̂ when λ = 1.5 and ϑ = 1.4  

Estimate Criterion 

n = 20 n = 50 n = 100 

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5         

𝛅 =1.5 

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5         

𝛅 =1.5 

𝛄 =0.5  

𝛅=0.5 

𝛄 =1.5           

𝛅 =1.5 

�̂�𝟏𝟎 
Mean 1.484249 1.411831 1.499103 1.472299 1.504371 1.491378 

MSE 0.000341 0.007840 0.000008 0.000770 0.000020 0.000075 

�̂�𝟏𝟏 
Mean 1.442389 1.377537 1.482297 1.456638 1.496009 1.483286 

MSE 0.003334 0.015225 0.000314 0.001897 0.000016 0.000282 

�̂�𝟏𝟐 
Mean 1.406981 1.353100 1.466585 1.442799 1.487910 1.475667 

MSE 0.008740 0.021964 0.001124 0.003309 0.000147 0.000598 

�̂�𝟐𝟎 
Mean 1.530402 1.456433 1.516805 1.489758 1.513049 1.499994 

MSE 0.001288 0.001935 0.000311 0.000107 0.000174 0.000000 

�̂�𝟐𝟏 
Mean 1.485102 1.412570 1.499438 1.472615 1.504535 1.491541 

MSE 0.000318 0.007709 0.000008 0.000753 0.000022 0.000072 

�̂�𝟐𝟐 
Mean 1.443130 1.378073 1.482612 1.456920 1.496159 1.483440 

MSE 0.003249 0.015092 0.000303 0.001872 0.000015 0.000276 

 

Table 5: Expected values and MSE’s for ϑ̂ when ϑ = 0.7 and λ =0.5 

Estimate Criterion 

n = 20 n = 50 n = 100 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5         
𝛅 =1.5 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5         
𝛅 =1.5 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5         
𝛅 =1.5 

�̂�𝟏𝟎 
Mean 0.974984 0.928047 0.845343 0.815308 0.782665 0.764920 

MSE 0.086157 0.061646 0.023881 0.015392 0.007475 0.004655 

�̂�𝟏𝟏 
Mean 0.944438 0.845109 0.809432 0.767619 0.759095 0.738223 

MSE 0.080694 0.030476 0.014643 0.005823 0.003931 0.001666 

�̂�𝟏𝟐 
Mean 0.840278 0.709600 0.754818 0.709395 0.730536 0.708963 

MSE 0.043190 0.001755 0.004297 0.000292 0.001097 0.000110 

�̂�𝟐𝟎 
Mean 0.968277 0.955044 0.863843 0.846656 0.800397 0.787121 

MSE 0.076434 0.070536 0.028897 0.023547 0.010764 0.008180 

�̂�𝟐𝟏 
Mean 0.976019 0.930808 0.846594 0.817055 0.783544 0.765949 

MSE 0.086503 0.062885 0.024243 0.015826 0.007628 0.004799 

�̂�𝟐𝟐 
Mean 0.947983 0.850438 0.811464 0.769948 0.760205 0.739402 

MSE 0.082355 0.032423 0.015141 0.006200 0.004076 0.001768 

 

Table 6: Expected values and MSE’s for ϑ̂ when ϑ = 0.7 and λ =1.5 

Estimate Criterion 

n = 20 n = 50  n = 100 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5           
𝛅 =1.5 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5           
𝛅 =1.5 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5           
𝛅 =1.5 

�̂�𝟏𝟎 
Mean 0.767471 0.756788 0.733466 0.728896 0.721592 0.719300 

MSE 0.004935 0.003493 0.001154 0.000857 0.000470 0.000375 

�̂�𝟏𝟏 Mean 0.754188 0.741865 0.727286 0.722462 0.718422 0.716069 

MSE 0.003213 0.001904 0.000764 0.000515 0.000342 0.000259 

�̂�𝟏𝟐 Mean 0.738110 0.724932 0.720667 0.715726 0.715148 0.712769 

MSE 0.001593 0.000669 0.000435 0.000250 0.000230 0.000163 

�̂�𝟐𝟎 Mean 0.777992 0.769400 0.739284 0.735089 0.724739 0.722537 

MSE 0.006512 0.005171 0.001592 0.001268 0.000618 0.000513 

�̂�𝟐𝟏 Mean 0.767971 0.757360 0.733709 0.729151 0.721721 0.719429 

MSE 0.005006 0.003563 0.001171 0.000873 0.000476 0.000380 

�̂�𝟐𝟐 Mean 0.754805 0.742533 0.727550 0.722733 0.718554 0.716203 

MSE 0.003286 0.001965 0.000779 0.000528 0.000346 0.000264 

 

 

 

 

 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

128 

 

Table 7: Expected values and MSE’s for ϑ̂ when ϑ = 1.4 and λ =0.5 

Estimate Criterion 

n = 20 n = 50 n = 100 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5        
𝛅 =1.5 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5        
𝛅 =1.5 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5         
𝛅 =1.5 

�̂�𝟏𝟎 
Mean 1.891493 1.661508 1.649716 1.518772 1.536577 1.463327 

MSE 0.281091 0.094864 0.071919 0.018216 0.020733 0.004713 

�̂�𝟏𝟏 
Mean 1.778901 1.396064 1.567025 1.404391 1.487024 1.405830 

MSE 0.200647 0.004210 0.035395 0.000653 0.008776 0.000127 

�̂�𝟏𝟐 
Mean 1.528850 1.176655 1.454463 1.299242 1.429922 1.350981 

MSE 0.044197 0.053117 0.005332 0.010630 0.001209 0.002500 

�̂�𝟐𝟎 
Mean 1.908679 1.837298 1.697597 1.615853 1.575174 1.517195 

MSE 0.278134 0.222254 0.096621 0.053875 0.033146 0.015314 

�̂�𝟐𝟏 
Mean 1.893409 1.666984 1.651239 1.521140 1.537546 1.464519 

MSE 0.282716 0.098366 0.072711 0.018885 0.021015 0.004884 

�̂�𝟐𝟐 
Mean 1.783570 1.401668 1.569262 1.406779 1.488186 1.407020 

MSE 0.204673 0.004534 0.036263 0.000725 0.009003 0.000149 

 

Table 8: Expected values and MSE’s for ϑ̂ when ϑ = 1.4 and λ =1.5 

Estimate Criterion 

n = 20 n = 50 n = 100 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5    𝛅 
=1.5 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5    𝛅 
=1.5 

𝛄 =0.5  
𝛅=0.5 

𝛄 =1.5    𝛅 
=1.5 

�̂�𝟏𝟎 
Mean 1.512297 1.469568 1.451659 1.433722 1.430547 1.421613 

MSE 0.013801 0.005302 0.002773 0.001172 0.000945 0.000471 

�̂�𝟏𝟏 
Mean 1.484740 1.437263 1.439344 1.420686 1.424305 1.415198 

MSE 0.007943 0.001524 0.001603 0.000436 0.000597 0.000232 

�̂�𝟏𝟐 
Mean 1.452524 1.403687 1.426311 1.407464 1.417894 1.408746 

MSE 0.003071 0.000048 0.000710 0.000058 0.000322 0.000077 

�̂�𝟐𝟎 
Mean 1.534550 1.498874 1.463174 1.446452 1.436663 1.428020 

MSE 0.019556 0.010645 0.004150 0.002237 0.001364 0.000794 

�̂�𝟐𝟏 
Mean 1.512832 1.470226 1.451908 1.433990 1.430673 1.421744 

MSE 0.013931 0.005403 0.002800 0.001191 0.000953 0.000477 

�̂�𝟐𝟐 
Mean 1.485386 1.437965 1.439612 1.420960 1.424438 1.415334 

MSE 0.008064 0.001581 0.001625 0.000448 0.000603 0.000236 

 

 

References 

1. Lai, C. D., Generalized Weibull Distributions, Springer Briefs in Statistics, 2014,18, 3, 

293-297, DOI: 10.1007/978-3-642-39106-4-1. 

2. Xie, M.; Tang, Y.;Goh, T. N.A Modified Weibull extension with bathtub-shaped failure 

rate function, Reliability Engineering and System Safety, 2002, 765,3,279- 285. 

http://dx.doi.org/10.1016/S0951-8320(02)00022-4.  

3.Ivana, P. ; Zuzana, S. Comparison of Four Methods for Estimating the Weibull distribution 

Parameters, Applied Mathematical Sciences, 2014, 8, 83,4137-4149. 
http://dx.doi.org/10.12988/ams.2014.45389.   

4. Chris, B. G. ; Noor, A. I. Approximate Bayesian Estimates of Weibull parameters with 

Lindley 's method, Sains Malaysians, 2012, 43, 9, 1433-1437.     

5. Babacan, E. K., ; Kaya, S. Comparison of parameter estimation methods in Weibull 

Distribution, Sigma Journal of Engineering and Natural sciences. 2020, 38, 2, 1609- 1621. 



Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34(4)2021 
 

129 

 

6. Rasheed, H. A., ; AL-Shareefi, E. F.Bayes Estimator for the Scale Parameter of Laplace 

distribution under a Suggested Loss Function. International Journal of Advanced Research, 

2015, 3, 3, 788-796.  

7.Lindley, D.V.Approximate Bayesian Methods, Trabajos de Estadistica Y de Investigacion 

Operative, 1980, 31, 1,223-245.https://doi.org/10.1007/BF02888353.