402 

© 2025 The Author(s). Published by College of Education for Pure Science (Ibn Al-Haitham), 

University of Baghdad. This is an open-access article distributed under the terms of the Creative Commons 

Attribution 4.0 International License 

 
 

Estimate the parameters of Exponential-Rayleigh distribution, by using 

Bayesian method 

 

Maral Mohammed1*  and  Iden Hassan2   
1, 2Department of Mathematics, College of Science for Women, University of Baghdad, Baghdad, Iraq.                                                                

*Corresponding Author. 

 

Received: 21 February 2024                   Accepted: 8 May 2024                      Published: 20 April 2025 

 

 

Abstract  

     In this paper, point estimation method for parameters 𝛼 and 𝜆 of the parameters of the 

Exponential-Rayleigh distribution which have been estimated by the use of a simulation 

technique by using two Bayesian estimation methods; the first  Bayesian method of estimation  

consists of lindely approximation estimation method and the second Bayesian estimation 

method consists Tierney and Kadane approximation estimation method to estimate all the 

unknown parameters (𝛼, 𝜆) of Exponential-Rayleigh distribution. The Bayes estimate of the 

unknown parameter is obtained by using the approximation methods of lindley (1980) and 

Tierney and Kadane (1986). Comparisons between these two methods were made by 

employing mean squares error criterion. Applying a simulation technique with different sample 

sizes, these methods are being compared. Simulation procedure is used to generate some 

sample sizes and mean squares error measure, and when we compared between the above two 

methods we find that lindely approximation estimation method has the less mean squares error. 

Keywords: Bayesian method, Exponential-Rayleigh distribution, Tierney and Kadane 

approximation Bayes estimation method, linedely approximation Bayes estimation method, 

Simulation technique, Mean squares error. 
 

1. Introduction  

Bayesian methods have become popular statistical procedures in areas from medicine to 

engineering (1). Initially, the most common objective priors were considered such as Jeffreys’ 

prior, reference priors, maximum data information prior. Bayesian estimation methods are 

usually based on the idea that the parameters to be estimated can be random variables and 

(𝑡1, 𝑡2, … , 𝑡𝑛) be random variables (2,3). Bayesian estimation methods are usually based on the 

idea the parameters(𝜃1, 𝜃2, … , 𝜃𝑛) to be estimated can be random variables (4). Several 

researchers have worked on the Bayesian Estimation. Ferreira et al. (5) proposed to make 

Bayesian inferences for the parameters of the Lomax distribution using non-informative priors. 

Mazaal et al. (6) Compared Weibull Stress – Strength Reliability Bayesian Estimators for Singly 

Type II Censored Data under Different loss Functions Awatif. Al-Baldawi (7) employed non-

informative priors to compare a few Bayesian estimation with the maximum likelihood 

doi.org/10.30526/38.2.3911 

https://creativecommons.org/licenses/by/4.0/
https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0000-0002-6492-8196
mailto:maral.bakth2203m@csw.uobaghdad.edu.iq
https://orcid.org/0000-0002-6492-8196
mailto:Idanha_math@csw.uobaghdad.edu.iq


IHJPAS. 2025,38(2) 

403 
 

estimator for the Maxwell distribution. Iden and Sara  (8) introduced the lindley approximation 

estimation method to estimate the logistic distribution's two parameters. Rasheed (9) used the 

Bayesian estimation under the quadratic loss function with no information before estimating 

the Maxwell distribution parameter. Alkanani and Salman (10) used Bayesian estimation and 

non- Bayesian estimation Methods for the Maxwell Boltzmann distribution parameter. Kalt and 

Hussein (11) used Bayesian estimation for Parameters of Modified Weibull distribution using 

Tierney and Kadane approximation method. Mohammed and Hussein (12) estimated methods 

for new mixture distribution with simulation and application. 

The Exponential Rayleigh distribution is obtained based on mixed between cumulative 

distribution function of Exponential distribution and cumulative distribution function of 

Rayleigh distributions (13).  

The probability density function of the Exponential-Rayleigh distribution is defined as follows 

(14): 

𝑓(𝑡; 𝛼, 𝜆) = (𝛼 + 𝜆𝑡)𝑒
−( 𝛼𝑡+ 𝜆

2
𝑡2)        

,   𝑡 ≥ 0 ;  𝛼, 𝜆 > 0                                                                   (1) 

The cumulative distribution function (CDF): 

𝐹(𝑡) = 1 − 𝑒−(𝛼𝑡+ 𝜆
2
𝑡2)        , 𝑡 ≥ 0 ;  𝛼, 𝜆 > 0                                                                                 (2)  

And the survival function is: 

𝑆(𝑡) = 𝑒−(𝛼𝑡+ 𝜆
2
𝑡2)         , 𝑡 ≥ 0 ;  𝛼, 𝜆 > 0                                                                                            (3) 

The hazard function is: 

ℎ(𝑡) = (𝛼 + 𝜆𝑡)     ,  𝑡 ≥ 0 ;  𝛼,  𝜆 > 0                                                                                             (4)         

 

The aim of this paper showing how to derive and estimate the two parameters (scale) in 

Exponential-Rayleigh distribution by using two Bayesian estimation methods, linedely 

approximation method and Tierney and Kadane approximation method. Therefore, finding and 

estimating survival function. Finely compare between these two methods to find the best 

method.  

The  structure  of  this  paper  is  as  follows:  In  Section  two; study of  Bayesian estimation  

method. Section three; derive Exponential-Rayleigh distribution by using the linedely 

approximation method. Section four; derive Exponential-Rayleigh distribution by using the 

Tierney and Kadane approximation method. Section five; the simulation technique is 

discussed. Section six; the main results are discussed. Finally, Section seven presents the 

conclusions. 

 

2. Materials and Methods 

2.1. Bayesian Method  

In the classical estimation Methods assuming that the parameters 𝛼 or 𝜆 of any distribution was 

be constant and fixed, but it is known to us, then these methods were as classical methods (15). 

Now describing another approach to estimate the parameters which are called Bayesian 

methods, these methods of estimation are typically predicated on the idea that the parameters 

to be estimated can be random variables as opposed to fixed values (16,17). These Bayesian 

estimation methods are based on the previous information available on the anonymous 

parameter plus the information which come from the sample observations(18). A loss function 

is a measure of the amount of loss resulting from a decision to be made depends on while the 

decision to be made depends on 𝜃 (19). The Bayesian estimation method has received a lot of 

interest recently for analyzing failure time data which has mostly been proposed as an 

alternative to that of the traditional methods. The Bayesian estimate method employs both the 

https://bsj.uobaghdad.edu.iq/do/search/?q=author%3A%22Iden%20H.%20Alkanani%22&start=0&context=38131414
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IHJPAS. 2025,38(2) 

404 
 

available data and one's prior knowledge of the parameters (20,21). The non-informative prior 

in Bayesian estimation can be used when one's prior knowledge about the parameter is 

unavailable (22). 

2.2. Lindely approximation estimation method 

This method introduced by researcher lindely in 1980. The two unknown parameters for the 

Exponential-Rayleigh distribution are being estimated using the lindely approximation method 

(23,24). 

lindely approximated the ratio of the integrals which as following form (25): 

∫𝑤(𝜃)𝑒𝐿(𝜃)𝑑𝜃

∫𝑣(𝜃)𝑒𝐿(𝜃)𝑑𝜃
                                                                                                                                                       (5) 

Where 𝜃 = (𝜃1, 𝜃2, … 𝜃𝑁)are parameters, 𝑤(𝜃) and 𝑣(𝜃) are any arbitrary functions for 

parameters. 

𝐿(𝜃) is the a logarithm of the likeihood function. 

Suppose that 𝑣(𝜃) is the prior density function of parameters 𝜃 and let 𝑤(𝜃) = 𝑢(𝜃) ∙ 𝑣(𝜃).   

From integrals equation (1) getting the posterior expectation which is as follows: 

 𝐸[𝑢(𝜃)|𝑡] =
∫𝑢(𝜃)𝑣(𝜃) 𝑒𝐿(𝜃)+𝐺(𝜃)𝑑𝜃

∫𝑣(𝜃) 𝑒𝐿(𝜃)+𝐺(𝜃)𝑑𝜃
 =

∫𝑢(𝜃) 𝑒𝐿(𝜃)+𝐺(𝜃)𝑑𝜃

∫  𝑒𝐿(𝜃)+𝐺(𝜃)𝑑𝜃
 . Where 𝐺(𝜃) = log𝑒[𝑣(𝜃)]  

The likeihood function of Exponential-Rayleigh distribution is: 

𝐿(𝜆; 𝑡1,𝑡2, … , 𝑡𝑛) = ∏ (𝛼 + 𝜆𝑡𝑖) 𝑒
−( 𝛼𝑡𝑖+

𝜆
2
𝑡𝑖

2) 𝑛
𝑖=1                                                                     (6) 

The natural algorithm of likeihood function is:   

ln 𝐿(𝜆; 𝑡1,𝑡2, … , 𝑡𝑛) = ∑ (𝛼 + 𝜆𝑡𝑖) − 𝛼 ∑ 𝑡𝑖 − 𝜆

2
∑ 𝑡𝑖

2𝑛
𝑖=1

𝑛
𝑖=1

𝑛
𝑖=1                                                (7) 

To apply this method, assuming the prior density function 𝑔1(𝛼)  and 𝑔2(𝜆) for the 

parameters 𝛼 and 𝜆 respectively, getting the posterior distribution. Assuming the prior density 

function for parameters 𝛼 and 𝜆 is: 

𝑔1(𝛼) = {𝜃𝑒−𝜃𝛼                                  𝛼 > 0
0                                  0𝑡ℎ𝑒𝑟 𝑤𝑖𝑠𝑒

                                                                              (8) 

𝑔2(𝜆) = { 𝜃𝑒−𝜃𝜆                                 𝜆 > 0
0                                  0𝑡ℎ𝑒𝑟 𝑤𝑖𝑠𝑒

                                                                              (9)  

The joint prior density functions for parameters 𝛼 and 𝜆 is: 

𝑔(𝛼, 𝜆) = 𝑔1(𝛼) ∙ 𝑔2(𝜆) = 𝜃𝑒−𝜃𝛼 ∙ 𝜃𝑒−𝜃𝜆                                                                           (10) 

The natural algorithm of joint prior density functions for parameters 𝛼 and 𝜆 is:  

ln 𝑔(𝛼, 𝜆) = 2 ln 𝜃 − 𝜃𝛼 − 𝜃𝜆                                                                                              (11)  

By using the reverse Bayes rule in the integration of the prior density function with the 

likelihood function we obtain the function of the posterior distribution of the parameters 𝛼 and 

𝜆 as follows (26): 

𝐻(𝛼, 𝜆; 𝑡1,𝑡2, … , 𝑡𝑛) = ∏ (𝛼 + 𝜆𝑡𝑖) 𝑒
−𝛼 ∑ 𝑡𝑖−

𝜆
2
∑ 𝑡𝑖

2𝑛
𝑖=1

𝑛
𝑖=1 𝜃𝑒−𝜃𝛼 ∙ 𝜃𝑒−𝜃𝜆 𝑛

𝑖=1                            (12) 

By employing the quadratic loss function, then the Bayes estimator as �̂�𝐵𝑎𝑦𝑒𝑠(𝛼, 𝜆) for any 

function with respect to the parameters 𝜙(𝛼, 𝜆) is the posterior mean for this function. 

The squared error loss function is given by following:   

𝐿𝑜𝑠𝑠(�̂� − 𝛼) = (�̂� − 𝛼)2                                                                                                      (13) 

𝐿𝑜𝑠𝑠(�̂� − 𝜆) = (�̂� − 𝜆)
2
                                                                                                       (14) 

The Bayes estimators for 𝛼 and 𝜆 for Exponential-Rayleigh distribution under squared error 

loss function is the posterior which given as follows: 

�̂�𝐵𝑎𝑦𝑒𝑠 = 𝐸[𝛼, 𝜆] =
∫∫𝜙(𝛼,𝜆)𝐻(𝛼,𝜆;𝑡1,𝑡2,…,𝑡𝑛)𝑑𝛼 𝑑𝜆

∫∫𝐻(𝛼,𝜆;𝑡1,𝑡2,…,𝑡𝑛)𝑑𝛼 𝑑𝜆
                                                                   (15)  



IHJPAS. 2025,38(2) 

405 
 

�̂�𝐵𝑎𝑦𝑒𝑠 = 𝐸[𝛼, 𝜆] =
∫∫𝜙(𝛼,𝜆)∏ (𝛼+𝜆𝑡𝑖) 𝑒

−𝛼 ∑ 𝑡𝑖−
𝜆
2

∑ 𝑡𝑖
2𝑛

𝑖=1
𝑛
𝑖=1 𝜃𝑒−𝜃𝛼∙𝜃𝑒−𝜃𝜆 𝑛

𝑖=1 𝑑𝛼 𝑑𝜆

∫∫∏ (𝛼+𝜆𝑡𝑖) 𝑒
−𝛼∑ 𝑡𝑖−

𝜆
2

∑ 𝑡𝑖
2𝑛

𝑖=1
𝑛
𝑖=1 𝜃𝑒−𝜃𝛼∙𝜃𝑒−𝜃𝜆 𝑛

𝑖=1 𝑑𝛼 𝑑𝜆

                         (16) 

 Where the number of integrals is equal to the anonymous parameters. 

 The posterior density function is difficult to solve, the ratio of these integrals in Equation (15) 

does not seem to take a theoretical formula for the difficulty of calculating these integral (27). 

Then utilizing the lindely approximation to solve the posterior distribution as follows:  

 Ι(𝑥) = 𝐸(𝑢; 𝛼, 𝜆) =
∫ ∫ 𝑢(𝛼,𝜆) 𝑒𝐿(𝛼,𝜆;𝑡𝑖)+𝐺(𝛼,𝜆)𝑑𝛼𝑑𝜆

∞
0

∞
0

∫ ∫ 𝑒𝐿(𝛼,𝜆;𝑡𝑖)+𝐺(𝛼,𝜆) 𝑑𝛼𝑑𝜆
∞
0

∞
0

                                                               (17) 

Where: 𝑢(𝛼, 𝜆) is function for 𝛼 and𝜆. 𝐿(𝛼, 𝜆; 𝑡𝑖) is natural logarithm of likeihood function. 

𝐺(𝛼, 𝜆) is natural logorithm of joint prior density functions of 𝛼 and 𝜆. 

Then, we can calculate Ι(𝑥) = 𝐸(𝑢; 𝛼, 𝜆) as follows: 

Ι(𝑥) = 𝑢(�̂�, �̂�)           +
1

2
[
(�̂�𝜆𝜆 + 2�̂�𝜆�̂�𝜆)�̂�𝜆𝜆 + (�̂�𝛼𝜆 + 2�̂�𝛼�̂�𝜆)�̂�𝛼𝜆 + (�̂�𝜆𝛼 + 2�̂�𝜆�̂�𝛼)�̂�𝜆𝛼

+(�̂�𝛼𝛼 + 2�̂�𝛼�̂�𝛼)�̂�𝛼𝛼

]  

+
1

2
 [(�̂�𝜆�̂�𝜆𝜆 + �̂�𝛼�̂�𝜆𝛼)(�̂�𝜆𝜆𝜆�̂�𝜆𝜆 + �̂�𝜆𝛼𝜆�̂�𝜆𝛼 + �̂�𝛼𝜆𝜆�̂�𝛼𝜆 + �̂�𝛼𝛼𝜆�̂�𝛼𝛼)  + (�̂�𝜆�̂�𝛼𝜆 +

�̂�𝛼�̂�𝛼𝛼)(�̂�𝛼𝜆𝜆�̂�𝜆𝜆    + �̂�𝜆𝛼𝛼�̂�𝜆𝛼 +  �̂�𝛼𝜆𝛼�̂�𝛼𝜆 + �̂�𝛼𝛼𝛼�̂�𝛼𝛼)]                                               (18) 

Then applying the lindely approach as in Equation (5) we get: 

𝑢(�̂�, �̂�) = 𝛼, �̂�𝛼 =
𝜕𝑢(�̂�,�̂�)

𝜕𝛼
= 1, �̂�𝛼𝛼 = 0, �̂�𝜆 = 0, �̂�𝜆𝜆 = 0, �̂�𝜆𝛼 = 0, �̂�𝛼𝜆 = 0  

Ι(𝑥) = �̂� +
1

2
[(2�̂�𝜆)�̂�𝛼𝜆 + (2�̂�𝛼)�̂�𝛼𝛼] +

1

2
 [(�̂�𝜆𝛼)(�̂�𝜆𝜆𝜆�̂�𝜆𝜆 + �̂�𝜆𝛼𝜆�̂�𝜆𝛼 + �̂�𝛼𝜆𝜆�̂�𝛼𝜆 + �̂�𝛼𝛼𝜆�̂�𝛼𝛼)   

        +(�̂�𝛼𝛼)(�̂�𝛼𝜆𝜆�̂�𝜆𝜆 + �̂�𝜆𝛼𝛼�̂�𝜆𝛼 +  �̂�𝛼𝜆𝛼�̂�𝛼𝜆 + �̂�𝛼𝛼𝛼�̂�𝛼𝛼)]                                                (19) 
Now derivate 𝑃 = ln 𝑔(𝛼, 𝜆) with respect to 𝛼 and 𝜆 in equation (11) we get: 

�̂�𝜆 =
𝜕 𝑙𝑛 𝑔(𝛼,𝜆)

𝜕𝜆
= −𝜃, �̂�𝛼 =

𝜕 𝑙𝑛 𝑔(𝛼,𝜆)

𝜕𝛼
= −𝜃 , �̂�𝛼 =

𝜕 ln𝐿

𝜕𝛼
= ∑

1

𝛼+𝜆𝑡𝑖
−𝑛

𝑖=1 ∑ 𝑡𝑖
𝑛
𝑖=1   

�̂�𝜆 =
𝜕 ln𝐿

𝜕𝜆
= ∑

𝑡𝑖

𝛼+𝜆𝑡𝑖
−

∑ 𝑡𝑖
2𝑛

𝑖=1

2

𝑛
𝑖=1  , �̂�𝜆𝜆 =

𝜕2 ln𝐿

𝜕𝜆2 = −∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1  , 

 �̂�𝛼𝛼 =
𝜕2 ln 𝐿

𝜕𝛼2 = −∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1  ,   �̂�𝜆𝛼 =

𝜕2 ln 𝐿

𝜕𝜆𝜕𝛼
= −∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1  , 

 �̂�𝛼𝜆 =
𝜕2 ln 𝐿

𝜕𝛼𝜕𝜆
= −∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1  , �̂�𝜆𝜆𝛼 =

𝜕3 ln𝐿

𝜕𝜆2𝜕𝛼
= 2∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1   

�̂�𝜆𝜆𝜆 =
𝜕3 ln 𝐿

𝜕𝜆3
= 2∑

𝑡𝑖
3

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1  , �̂�𝜆𝛼𝜆 =

𝜕3 ln 𝐿

𝜕𝜆𝜕𝛼𝜕𝜆
= 2∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1   

 �̂�𝛼𝛼𝜆 =
𝜕3 ln 𝐿

𝜕𝛼𝜕𝛼𝜕𝜆
= 2∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1  , �̂�𝛼𝜆𝜆 =

𝜕3 ln 𝐿

𝜕𝛼𝜕𝜆𝜕𝜆
= 2∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1    

 �̂�𝜆𝛼𝛼 =
𝜕3 ln 𝐿

𝜕𝜆𝜕𝛼𝜕𝛼
= 2∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1   , �̂�𝛼𝛼𝛼 =

𝜕3 ln 𝐿

𝜕𝛼3 = 2∑
1

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1   

�̂�𝛼𝜆𝛼 =
𝜕3 ln𝐿

𝜕𝛼𝜕𝜆𝜕𝛼
= 2∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1   , �̂�𝜆𝛼 = −

1

�̂�𝜆𝛼
=

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

   

�̂�𝛼𝛼 = −
1

�̂�𝛼𝛼
=

1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

  , �̂�𝜆𝜆 = −
1

�̂�𝜆𝜆
=

1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

 , �̂�𝛼𝜆 = −
1

�̂�𝛼𝜆
=

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

  

To estimate the parameter 𝛼 at case of squared error loss function (13), assuming that 𝜆 is 

information. 

�̂�𝐿𝐴𝐸 = �̂� + [(−𝜃)(
1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

) + (−𝜃)(
1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

)]  



IHJPAS. 2025,38(2) 

406 
 

+
1

2
[

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

(2∑
𝑡𝑖

3

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

) + 2∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

)   

+2∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

) +  2∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

))  

+
1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

(2∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

) + 2∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

)   

+2∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

) + 2∑
1

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

))]                                               

�̂�𝐿𝐴𝐸 = �̂� + [−
𝜃

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

−
𝜃

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

] +
1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

(
∑

𝑡𝑖
3

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

 +
∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

 

         +
∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

+
∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

)  

     + 1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

(
∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

+
∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

+
∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

+
∑

1

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

)                              (20) 

To estimate the parameter 𝜆 at case of squared error loss function (14), assuming that 𝛼 is 

information. 

We choose: (�̂�, �̂�) = 𝜆 , �̂�𝜆 =
𝜕𝑢(�̂�,�̂�)

𝜕𝜆
= 1 , �̂�𝛼 = 0, �̂�𝛼𝛼 = 0, �̂�𝜆𝜆 = 0, �̂�𝜆𝛼 = 0, �̂�𝛼𝜆 = 0  

Ι(𝑥) = �̂� +
1

2
[(2�̂�𝜆�̂�𝜆)�̂�𝜆𝜆 + (2�̂�𝜆�̂�𝛼)�̂�𝜆𝛼] +

1

2
 [(�̂�𝜆𝜆)(�̂�𝜆𝜆𝜆�̂�𝜆𝜆 + �̂�𝜆𝛼𝜆�̂�𝜆𝛼 + �̂�𝛼𝜆𝜆�̂�𝛼𝜆 +

�̂�𝛼𝛼𝜆�̂�𝛼𝛼) +(�̂�𝛼𝜆)(�̂�𝛼𝜆𝜆�̂�𝜆𝜆 + �̂�𝜆𝛼𝛼�̂�𝜆𝛼 +  �̂�𝛼𝜆𝛼�̂�𝛼𝜆 + �̂�𝛼𝛼𝛼�̂�𝛼𝛼)]                                       (21) 

�̂�𝐿𝐴𝐸 = �̂� + [(−𝜃)(
1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

) + (−𝜃)(
1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

)]   

      +
1

2
[

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

(2∑
𝑡𝑖

3

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

) + 2∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

)       

     +2∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

) + 2∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

))  

       +
1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

(2∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

) + 2∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

)  

     +2∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

) + 2∑
1

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1 (

1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

))]  

�̂�𝐿𝐴𝐸 = �̂� + [−
𝜃

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

−
𝜃

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

]  



IHJPAS. 2025,38(2) 

407 
 

 +
1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

(
∑

𝑡𝑖
3

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

 +
∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

+
∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

+
∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

)  

 +
1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

 (
∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

+
∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

+
∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

+
∑

1

(𝛼+𝜆𝑡𝑖)
3

𝑛
𝑖=1

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

)                             (22)      

2.4. Tierney and Kadane approximation Estimation method 

In (1986), Tierney and Kadane the two researchers used an approximation method to calculate 

the ratio of integrals and according to the following formula (28): 

∫𝑤(𝜃)𝑒𝐿(𝜃)𝑑𝜃

∫𝑣(𝜃)𝑒𝐿(𝜃)𝑑𝜃
                                                                                                                            (23) 

Where:  𝜃 = (𝜃1, 𝜃2, … 𝜃𝑚): represents the vector of parameters, to be estimated.𝑤(𝜃) and 

𝑣(𝜃): are optional functions in terms of the parameters. 

𝐿(𝜃): logarithm function of the Maximum likeihood. 

When 𝑣(𝜃) is the prior density function of parameters 𝜃 and let 𝑤(𝜃) = 𝜑(𝜃) ∙ 𝑣(𝜃).  

Which is a function in terms of the parameter 𝜃 where: 

𝐸[𝜑(𝜃)|𝑡] =
∫𝜑(𝜃) 𝑒(𝐿(𝜃)+𝜌(𝜃))𝑑𝜃

∫  𝑒(𝐿(𝜃)+𝜌(𝜃))𝑑𝜃
=

∫𝜑(𝜃) 𝑒(Λ(𝜃))𝑑𝜃

∫  𝑒(Λ(𝜃))𝑑𝜃
                                                                 (24) 

Where: 

𝜌(𝜃) = ln 𝜈(𝜃) , Λ(𝜃) = 𝐿(𝜃) + 𝜌(𝜃) = 𝐿(𝜃) + ln 𝜈(𝜃)  

Tierney and Kadane were managed to arrive at an estimate value 𝐸[𝜑(𝜃)|𝑡] in equation (24) 

by using Taylor's Series to approximate the maximum of likeihood the parameter𝜃, using 

independent computations for the integration of the denominator and the integration of the 

numerator separately and then  the division of the output (29), taking into account the following 

two possibilities: 

Τ =
𝐿(𝜃 𝑡⁄ )+ln𝜈(𝜃)

𝑛
                                                                                                                    (25) 

Τ∗ =
𝑙𝑛 𝜑(𝜃)𝐿(𝜃 𝑡⁄ )+𝑙𝑛 𝜈(𝜃)

𝑛
                                                                                                        (26) 

Find the posterior distribution of the function by Based on equation (25) and (26), then the 

Equation (24) becomes as follows: 

𝐸[𝜑(𝜃)|𝑡] =
∫𝑒(𝑛Τ∗)𝑑𝜃

∫  𝑒(nΤ))𝑑𝜃
                                                                                                           (27)  

Then the approximate Bayes estimator by using the Tierney and Kadane method of the 

function 𝜙(𝜃) is as follows:  

�̂�𝐵𝑦𝑠𝑒 = 𝐸[𝜑(𝜃)|𝑡] = [
|𝐻∗|

|𝐻|
]
1
2
exp (𝑛{𝛵∗(�̂�∗

𝑚𝑎𝑥) − 𝑇( �̂�𝑚𝑎𝑥)})                                (28) 

Where: 

�̂�𝐵𝑦𝑠𝑒(𝑡): Estimation of the function 𝜙(𝜃) based on the method of researchers Tierney and 

Kadane 

𝜃, 𝜃∗: The values that maximize Τ and Τ∗ respectively. 

𝐻,𝐻∗: Negative inverse matrix Hessian for both,Τ , Τ∗at 𝜃and 𝜃∗ respectively of class 𝑚 × 𝑚, 

𝑚 the number of parameters to be estimated. 

As a result, it needs to find values T and Τ∗ in Equations (25, 26) as follows: 

Τ =
ln[ℎ(𝜃;𝑡1,𝑡2,…,𝑡𝑛)]

𝑛
                                                                                                                (29) 

 Τ∗ =
ln[𝜙(𝜃)ℎ(𝜃;𝑡1,𝑡2,…,𝑡𝑛)]

𝑛
                                                                                                       (30) 



IHJPAS. 2025,38(2) 

408 
 

Estimation of parameter (𝛼)when the parameter (𝜆) is knows. Assume that 𝜙(𝛼, 𝜆) = 𝛼 in 

equation (27) and then 

 �̂�𝐵𝑦𝑠𝑒 = 𝐸(𝛼|𝑡). 

Where �̂�𝐵𝑦𝑠𝑒 denote the Bayes estimator of 𝛼 according to Tierney and Kadane 

approximation.  

Then 

�̂�𝐵𝑦𝑠𝑒 =
∫ 𝜙(𝛼) 𝑒Ι(𝛼,𝜆) 𝑒ln(𝑣)𝑑𝛼

∞
0

∫ 𝑒Ι(𝛼,𝜆)𝑒ln(𝑣)𝑑𝛼
∞
0

 =
∫ 𝜙(𝛼) 𝑒Ι(𝛼,𝜆) 𝑣 𝑑𝛼

∞
0

∫ 𝑒Ι(𝛼,𝜆) 𝑣 𝑑𝛼
∞
0

 , Where 𝑣 = 𝑔1(𝛼)  

�̂�𝐵𝑦𝑠𝑒 =
𝜃  𝑒

−
𝜆
2

∑ 𝑡𝑖
2𝑛

𝑖=1 ∫ 𝛼 ∏ (𝛼+𝜆𝑡𝑖) 𝑒
−𝛼 ∑ 𝑡𝑖

𝑛
𝑖=1𝑛

𝑖=1 𝑒−𝜃𝛼 𝑑𝛼
∞
0

𝜃  𝑒
−

𝜆
2

∑ 𝑡𝑖
2𝑛

𝑖=1 ∫ ∏ (𝛼+𝜆𝑡𝑖) 𝑒
−𝛼∑ 𝑡𝑖

𝑛
𝑖=1𝑛

𝑖=1 𝑒−𝜃𝛼 𝑑𝛼
∞
0

                                             (31) 

So 

Τ =
𝑙𝑛

𝑛
[∏ (𝛼 + 𝜆𝑡𝑖) 𝑒

−𝛼 ∑ 𝑡𝑖
𝑛
𝑖=1𝑛

𝑖=1 𝑒−𝜃𝛼] =
∑ ln(𝛼+𝜆𝑡𝑖)

𝑛
𝑖=1

𝑛
−

𝛼 ∑ 𝑡𝑖
𝑛
𝑖=1

𝑛
−

𝜃𝛼

𝑛
                                (32) 

Τ∗ =
𝑙𝑛

𝑛
[𝛼 ∏ (𝛼 + 𝜆𝑡𝑖) 𝑒

−𝛼 ∑ 𝑡𝑖
𝑛
𝑖=1𝑛

𝑖=1 𝑒−𝜃𝛼] =
ln(𝛼)

𝑛
+

∑ ln(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1

𝑛
−

𝛼 ∑ 𝑡𝑖
𝑛
𝑖=1

𝑛
−

𝜃𝛼

𝑛
              (33) 

To find 𝛼𝑚𝑎𝑥  and 𝛼∗
𝑚𝑎𝑥 which are the values that maximize Τ and Τ∗respectively: 

𝜕Τ

𝜕𝛼
=

1

𝑛
∑

1

(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1 −

∑ 𝑡𝑖
𝑛
𝑖=1

𝑛
−

𝜃

𝑛
                                                                                                                           

𝜕Τ∗

𝜕𝛼
=

1

𝑛𝛼
+

1

𝑛
∑

1

(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1 −

∑ 𝑡𝑖
𝑛
𝑖=1

𝑛
−

𝜃

𝑛
                                                                                                                 

�̂�𝑚𝑎𝑥 = ∑
1

(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1 − ∑ 𝑡𝑖

𝑛
𝑖=1 − 𝜃                                                                                                                     

�̂�∗
𝑚𝑎𝑥 =

1

𝛼
+ ∑

1

(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1 − ∑ 𝑡𝑖

𝑛
𝑖=1 − 𝜃                                                                                                            

�̂�𝑚𝑎𝑥 and �̂�∗
𝑚𝑎𝑥 are maximum since:  

𝜕2Τ

𝜕2𝛼
=

−1

𝑛
∑

1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1 < 0  

𝜕2Τ∗

𝜕2𝛼
=

−1

𝑛
(

1

𝛼2 + ∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1 ) < 0 

Necessary and sufficient is that second derivate respect to 𝛼 is negative (30). The determinant 

of the negative of the inverse Hessian of Τ at �̂�𝑚𝑎𝑥 and Τ∗ at �̂�∗
𝑚𝑎𝑥 respectively: 

𝐻 = −[
𝜕2Τ

𝜕2𝛼
]
−1

𝛼=𝛼𝑚𝑎𝑥

 = − [
−1

𝑛
∑

1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1 ]

−1

 =
𝑛

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

                                              (34) 

𝐻∗ = − [
𝜕2𝛵∗

𝜕2𝛼
]
−1

𝛼=𝛼∗
𝑚𝑎𝑥

= −[
−1

𝑛
(

1

𝛼2 + ∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1 )]

−1

 =
𝑛

1

𝛼2+∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

                   (35) 

�̂�𝐵𝑦𝑠𝑒 = [
|𝐻∗|

|𝐻|
]
1
2
exp (𝑛{𝛵∗(�̂�∗

𝑚𝑎𝑥) − 𝑇( �̂�𝑚𝑎𝑥)})                                                                                            

�̂�𝐵𝑦𝑠𝑒 = [

𝑛
1

𝛼2+∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

𝑛

∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

]

1
2

 exp (𝑛 {
ln(𝛼)

𝑛
+

∑ ln(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1

𝑛
−

𝛼 ∑ 𝑡𝑖
𝑛
𝑖=1

𝑛
−

𝜃𝛼

𝑛
(

1

𝛼
+ ∑

1

(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1 ∑ 𝑡𝑖

𝑛
𝑖=1 − 𝜃)    

          − (
∑ ln(𝛼+𝜆𝑡𝑖)

𝑛
𝑖=1

𝑛
−

𝛼 ∑ 𝑡𝑖
𝑛
𝑖=1

𝑛
−

𝜃𝛼

𝑛
(∑

1

(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1 − ∑ 𝑡𝑖

𝑛
𝑖=1 − 𝜃))})                        (36) 

�̂�𝐵𝑦𝑠𝑒 = [
∑

1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

1

𝛼2+∑
1

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

]

1
2

  (1 + 𝛼 ∑
1

(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1 − 𝛼 ∑ 𝑡𝑖

𝑛
𝑖=1 − 𝛼𝜃)𝑒𝑥𝑝(

∑ 𝑙𝑛(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1

𝛼
−

∑ 𝑡𝑖
𝑛
𝑖=1 − 𝜃

)                (37) 

 

 

 



IHJPAS. 2025,38(2) 

409 
 

Estimation of parameter 𝜆 when the parameter 𝛼 is knows. Assume that 𝜙(𝛼, 𝜆) = 𝜆 in 

Equation (27) and then �̂� = 𝐸(𝜆|𝑡), Where �̂�𝐵𝑦𝑠𝑒 denote the Bayes estimator of 𝜆 according to 

Tierney and Kadane approximation method.     

�̂�𝐵𝑦𝑠𝑒 =
𝜃  𝑒−𝛼 ∑ 𝑡𝑖

𝑛
𝑖=1 ∫ 𝜆 ∏ (𝛼+𝜆𝑡𝑖) 

𝑛
𝑖=1  𝑒

−𝜆(
∑ 𝑡𝑖

2𝑛
𝑖=1

2
+𝜃)

 𝑑𝜆
∞
0

𝜃𝑒−𝛼∑ 𝑡𝑖
𝑛
𝑖=1   ∫ ∏ (𝛼+𝜆𝑡𝑖) 

𝑛
𝑖=1  𝑒

−𝜆(
∑ 𝑡𝑖

2𝑛
𝑖=1

2
+𝜃)

 𝑑𝜆
∞
0

                                                              (38) 

And 

Τ =
𝑙𝑛

𝑛
[∏ (𝛼 + 𝜆𝑡𝑖) 𝑒

−𝜆(
∑ 𝑡𝑖

2𝑛
𝑖=1

2
+𝜃)

𝑛
𝑖=1 ] =

∑ ln(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1

𝑛
−

𝜆(
∑ 𝑡𝑖

2𝑛
𝑖=1

2
+𝜃)

𝑛
                                  (39) 

Τ∗ =
𝑙𝑛

𝑛
[𝜆 ∏ (𝛼 + 𝜆𝑡𝑖) 

𝑛
𝑖=1  𝑒

−𝜆(
∑ 𝑡𝑖

2𝑛
𝑖=1

2
+𝜃)

] =
ln(𝜆)

𝑛
+

∑ ln(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1

𝑛
−

𝜆(
∑ 𝑡𝑖

2𝑛
𝑖=1

2
+𝜃)

𝑛
                (40) 

To find 𝜆𝑚𝑎𝑥 and 𝜆∗
𝑚𝑎𝑥 which are the values that maximize Τ and Τ∗respectively: 

𝜕Τ

𝜕𝜆
=

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1

𝑛
−

(
∑ 𝑡𝑖

2𝑛
𝑖=1

2
+𝜃)

𝑛
                                                                                                     

  
𝜕Τ∗

𝜕𝜆
=

1

𝑛𝜆
+

∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1

𝑛
−

(
∑ 𝑡𝑖

2𝑛
𝑖=1

2
+𝜃)

𝑛
                                                                                        

�̂�𝑚𝑎𝑥 = ∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1 − (

∑ 𝑡𝑖
2𝑛

𝑖=1

2
+ 𝜃)                                                                                                                     

�̂�∗
𝑚𝑎𝑥 =

1

𝜆
+ ∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
− (

∑ 𝑡𝑖
2𝑛

𝑖=1

2
+ 𝜃)𝑛

𝑖=1                                                                                                              

�̂�𝑚𝑎𝑥 𝑎𝑛𝑑 �̂�∗
𝑚𝑎𝑥

  is maximum since: 

𝜕2Τ∗

𝜕2𝜆
=

−1

𝑛
(

1

𝜆2 + ∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1 ) < 0 and 

𝜕2Τ

𝜕2𝜆
=

−1

𝑛
∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1 < 0 

Necessary and sufficient is that second derivate respect to 𝜆 is negative. 

The determinant of the negative of the inverse Hessian of Τ at �̂�𝑚𝑎𝑥 and Τ∗ at �̂�∗
𝑚𝑎𝑥 

respectively: 

𝐻 = −[
𝜕2Τ

𝜕2𝜆
]
−1

 𝜆=𝜆𝑚𝑎𝑥
= −[

−1

𝑛
∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1 ]

−1

 =
𝑛

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

                                              (41) 

�̂�𝐵𝑦𝑠𝑒 =

[
 
 
 
 

𝑛

1

𝜆2+∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

𝑛

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1 ]

 
 
 
 

1
2

  exp (𝑛 {
ln(𝜆)

𝑛
+

∑ ln(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1

𝑛
−

𝜆(
∑ 𝑡𝑖

2𝑛
𝑖=1

2
+𝜃)

𝑛
(

1

𝜆
+ ∑

𝑡𝑖

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

−(
∑ 𝑡𝑖

2𝑛
𝑖=1

2
+ 𝜃)

)    

           −(
∑ ln(𝛼+𝜆𝑡𝑖)

𝑛
𝑖=1

𝑛
−

𝜆(
∑ 𝑡𝑖

2𝑛
𝑖=1

2
+𝜃)

𝑛
(∑

𝑡𝑖
2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1 − (

∑ 𝑡𝑖
2𝑛

𝑖=1

2
+ 𝜃)))})                   (42) 

�̂�𝐵𝑦𝑠𝑒 =

[
 
 
 
 

𝑛

1

𝜆2+∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1

𝑛

∑
𝑡𝑖

2

(𝛼+𝜆𝑡𝑖)
2

𝑛
𝑖=1 ]

 
 
 
 

1
2

 (1 + 𝜆 ∑
𝑡𝑖

(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1 − 𝜆 (

∑ 𝑡𝑖
2𝑛

𝑖=1

2
+ 𝜃))    𝑒𝑥𝑝 (

∑ 𝑙𝑛(𝛼+𝜆𝑡𝑖)
𝑛
𝑖=1

𝜆
− (

∑ 𝑡𝑖
2𝑛

𝑖=1

2
+ 𝜃))      (43) 

 

 

 

 

 

 



IHJPAS. 2025,38(2) 

410 
 

3. Simulation technique  

The simulation procedure is now generally used in many branches of statistics. They can be 

used to evaluate the behavior of models as well as for some random variables. A simulation is 

defined as a numerical scientific method that uses logical mathematical methods to describe 

the behavior of a certain approved system. Different techniques have been conducted.             
 We select 𝑛 = 10,20,30,50 

 We select 𝛼 = 0.25,0.5,0.75  

 We select 𝜆 = 0.5,1,1.5 , These values were selected at random. 

  N=represent the Replicate of Experiment then N=1000 

 𝐹(𝑡) = 1 − 𝑆(𝑡) = 1 − 𝑒−(𝛼𝑡+ 𝜆
2
𝑡2)

 

𝑢 = 1 − 𝑒−(𝛼𝑡+ 𝜆
2
𝑡2)  

1 − 𝑢 = 𝑒−(𝛼𝑡+ 𝜆
2
𝑡2)  

ln(1 − 𝑢) = − (𝛼𝑡 +  𝜆
2
𝑡2)  

ln(1 − 𝑢) +𝛼𝑡 +  𝜆
2
𝑡2 = 0  

𝑎 =
𝜆

2
  , 𝑏 = 𝛼  ,     𝑐 = ln(1 − 𝑢)   

𝑡1,2 =
−𝑏±√𝑏2−4𝑎𝑐

2𝑎
  

𝑡 =
−𝛼±√𝛼2−2𝜆 ln(1−𝑢)

𝜆
  

 We find Absolut error value as follows: 

𝜙𝛼 = 𝑀𝑖𝑛|�̂�𝑖 − 𝛼| , 𝜙𝜆 = 𝑀𝑖𝑛|�̂�𝑖 − 𝜆|  

 We find mean squares errors for parameter( 𝛼) and parameter(𝜆). 

𝑀𝑆𝐸[𝑆(𝑡)] = ∑
[�̂�(𝑡𝑖)−𝑆(𝑡𝑖)]

2

𝑁
𝑁
𝐼=1  , 𝑀𝑆𝐸[𝛼] = ∑

[�̂�𝑖−𝛼]2

𝑁
𝑁
𝐼=1 , 𝑀𝑆𝐸[𝜆] = ∑

[�̂�𝑖−𝜆]
2

𝑁
𝑁
𝐼=1  

 
4. Numerical Results 

The numerical results for the simulation approach are exhibited in the following tables after 

the simulation procedure has been used to determine the parameter estimation, reliability 

function, and mean square error technique. With a 1000 iteration rate, the MATLAB software 

employed the Monte Carlo methodology for the simulation and the Newton-Raphson method 

for the numerical solution. As shown in table (1, 2, 3, 4, 5). Noting from the Table 1 that 

Absolut error values of parameter (𝛼)for lindely approximation estimate values method are the 

best for the Tierney and Kadane approximation Estimation method. Noting from the Table 2 

that Absolut error values of parameter (𝜆)for lindely approximation estimate values method 

are the best for the Tierney and Kadane approximation Estimation method. Noting from the 

Table 3 that mean square error (𝛼) for lindely approximation estimate values method are the 

best for the Tierney and Kadane approximation Estimation method. Noting from the Table 4. 

that mean square error of survival function for lindely approximation estimate values method 

are the best for the Tierney and Kadane approximation Estimation method. Noting from the 

Table 5 that mean square error of parameter (𝜆) for lindely approximation estimate values 

method are the best for the Tierney and Kadane approximation Estimation method 

 
 

 

 

 

 

 

 

 

 

 



IHJPAS. 2025,38(2) 

411 
 

Table 1. Represent the estimate for parameter( 𝛼) of all Bayesian method with Absolut error values 

       𝛼  𝜆  𝑛 �̂�𝐿𝐴𝐸  �̂�𝑇𝐾𝐴𝐸 ∅      Best 

0.25 0.5 10 0.232955 0.36954 0.017045 1 

0.25 0.5 20 0.24949 0.253797 0.00051 1 

0.25 0.5 30 0.220153 0.25038 0.00038 2 

0.25 0.5 50 0.250018 0.249896 1.77E-05 1 

0.25 1 10 0.259541 0.358864 0.009541 1 

0.25 1 20 0.248875 0.340926 0.001125 1 

0.25 1 30 0.259211 0.251351 0.001351 2 

0.25 1 50 0.25004 0.257901 3.95E-05 1 

0.25 1.5 10 0.267571 0.199231 0.017571 1 

0.25 1.5 20 0.247656 0.26011 0.002344 1 

0.25 1.5 30 0.249608 0.248725 0.000392 1 

0.25 1.5 50 0.250008 0.258761 7.72E-06 1 

0.5 0.5 10 0.533735 0.757909 0.033735 1 

0.5 0.5 20 0.50599 0.766192 0.00599 1 

0.5 0.5 30 0.49971 0.500022 2.15E-05 2 

0.5 0.5 50 0.499999 0.599998 8.39E-07 1 

0.5 1 10 0.484835 0.445992 0.015165 1 

0.5 1 20 0.503185 0.740497 0.003185 1 

0.5 1 30 0.499308 0.495959 0.000692 1 

0.5 1 50 0.500012 0.454503 1.22E-05 1 

0.5 1.5 10 0.534858 0.697942 0.034858 1 

0.5 1.5 20 0.495619 0.591782 0.004381 1 

0.5 1.5 30 0.500308 0.481262 0.000308 1 

0.5 1.5 50 0.499961 0.622744 3.91E-05 1 

0.75 0.5 10 0.696886 0.379964 0.053114 1 

0.75 0.5 20 0.749098 0.743101 0.000902 1 

0.75 0.5 30 0.750103 0.749712 0.000103 1 

0.75 0.5 50 0.745595 1.027192 0.004405 1 

0.75 1 10 0.72881 1.082477 0.02119 1 

0.75 1 20 0.750642 0.80692 0.000642 1 

0.75 1 30 0.749779 0.91954 0.000221 1 

0.75 1 50 0.750033 1.029041 3.27E-05 1 

0.75 1.5 10 0.736458 0.729153 0.013542 1 

0.75 1.5 20 0.750101 0.941367 0.000101 1 

0.75 1.5 30 0.750304 0.741131 0.000304 1 

0.75 1.5 50 0.749938 0.731215 6.23E-05 1 

 

 

 

 

 

 



IHJPAS. 2025,38(2) 

412 
 

Table 2. Represent the estimated for parameter( 𝜆) of all Bayesian method with Absolut error values: 

 𝛼  𝜆 𝑛 �̂�𝐿𝐴𝐸 �̂�𝑇𝐾𝐴𝐸 ∅ 

0.25 0.5 10 0.509598 0.462907 0.009598 

0.25 0.5 20 0.499007 0.443457 0.000993 

0.25 0.5 30 0.500283 0.433694 0.000283 

0.25 0.5 50 0.500042 0.469098 4.2E-05 

0.25 1 10 1.026021 0.973806 0.026021 

0.25 1 20 0.998455 1.032778 0.001545 

0.25 1 30 0.999999 0.923849 5.62E-07 

0.25 1 50 1.000025 1.116261 2.54E-05 

0.25 1.5 10 1.484813 0.85661 0.015187 

0.25 1.5 20 1.499029 1.676104 0.000971 

0.25 1.5 30 1.500487 1.395173 0.000487 

0.25 1.5 50 1.499997 1.467975 2.5E-06 

0.5 0.5 10 0.48869 0.59474 0.01131 

0.5 0.5 20 0.493391 0.500723 0.000723 

0.5 0.5 30 0.49978 0.504399 0.00022 

0.5 0.5 50 0.499963 0.508847 3.67E-05 

0.5 1 10 1.016357 0.963378 0.016357 

0.5 1 20 1.004066 1.010583 0.004066 

0.5 1 30 1.000015 1.066669 1.47E-05 

0.5 1 50 0.999995 1.110389 5.04E-06 

0.5 1.5 10 1.543717 1.476047 0.023953 

0.5 1.5 20 1.495319 1.536033 0.004681 

0.5 1.5 30 1.500462 1.220021 0.000462 

0.5 1.5 50 1.500013 1.543239 1.34E-05 

0.75 0.5 10 0.52599 0.370504 0.02599 

0.75 0.5 20 0.50068 0.551145 0.00068 

0.75 0.5 30 0.499906 0.489555 9.44E-05 

0.75 0.5 50 0.499992 0.499552 8.31E-06 

0.75 1 10 1.0467 1.436493 0.0467 

0.75 1 20 0.995807 0.783461 0.004193 

0.75 1 30 1.000245 1.196793 0.000245 

0.75 1 50 0.999999 0.990902 1.16E-06 

0.75 1.5 10 1.545809 1.351652 0.045809 

0.75 1.5 20 1.498723 1.640125 0.001277 

0.75 1.5 30 1.500127 1.439778 0.000127 

0.75 1.5 50 1.499983 1.298486 1.71E-05 

. 

 

 

 

 

 

 

 

 



IHJPAS. 2025,38(2) 

413 
 

Table 3. Represent the MSE for parameter( 𝛼) of all Bayesian method 

 𝛼 𝜆 𝑛 𝑀𝑆𝐸�̂�𝐿𝐴𝐸
 𝑀𝑆𝐸�̂�𝑇𝐾𝐴𝐸

 min 

0.25 0.5 10 0.061014 0.07584 0.17413 

0.25 0.5 20 4.11E-06 1.91E-05 4.11E-06 

0.25 0.5 30 2.53E-07 1.5E-07 9.37E-08 

0.25 0.5 50 7.47E-10 2.04E-08 7.47E-10 

0.25 1 10 0.002126 0.042557 0.002126 

0.25 1 20 8.36E-06 0.018076 8.36E-06 

0.25 1 30 2.22E-07 0.004964 2.22E-07 

0.25 1 50 2.49E-09 0.00025 2.49E-09 

0.25 1.5 10 0.000907 0.002617 0.000907 

0.25 1.5 20 1.24E-05 0.000428 1.24E-05 

0.25 1.5 30 1.7E-07 4.21E-06 1.7E-07 

0.25 1.5 50 8.12E-11 0.000194 8.12E-11 

0.5 0.5 10 0.002175 0.151484 0.002175 

0.5 0.5 20 4.75E-05 0.137914 4.75E-05 

0.5 0.5 30 2.09E-07 1.4E-08 1.4E-08 

0.5 0.5 50 7.9E-11 0.019993 7.9E-11 

0.5 1 10 0.000705 0.01733 0.000705 

0.5 1 20 1.62E-05 0.073333 1.62E-05 

0.5 1 30 5.35E-07 1.63E-05 5.35E-07 

0.5 1 50 4.62E-10 0.003877 4.62E-10 

0.5 1.5 10 0.001242 0.058183 0.001242 

0.5 1.5 20 3.03E-05 0.016625 3.03E-05 

0.5 1.5 30 1.59E-07 0.000364 1.59E-07 

0.5 1.5 50 4.42E-09 0.015201 4.42E-09 

0.75 0.5 10 0.002838 0.309147 0.002838 

0.75 0.5 20 2.4E-06 0.000105 2.4E-06 

0.75 0.5 30 1.28E-07 1.64E-07 1.28E-07 

0.75 0.5 50 7.23E-11 0.078898 7.23E-11 

0.75 1 10 0.000452 0.135392 0.000452 

0.75 1 20 1.31E-05 0.04142 1.31E-05 

0.75 1 30 7.36E-08 0.061872 7.36E-08 

0.75 1 50 1.18E-09 0.084636 1.18E-09 

0.75 1.5 10 0.000532 0.147029 0.000532 

0.75 1.5 20 1.74E-08 0.040548 1.74E-08 

0.75 1.5 30 3.63E-07 0.000119 3.63E-07 

0.75 1.5 50 4.55E-09 0.000387 4.55E-09 

 

 

 

 

 



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Table 4. Represent the MSE for parameter( 𝜆) of all Bayesian method 

  𝛼 𝜆 𝑛 𝑀𝑆𝐸�̂�𝐿𝐴𝐸
 𝑀𝑆𝐸�̂�𝑇𝐾𝐴𝐸

 𝑚𝑖𝑛 

0.25 0.5 10 0.000858 0.031695 0.000858 

0.25 0.5 20 1.5E-06 0.01405 1.5E-06 

0.25 0.5 30 1.69E-07 0.006272 1.69E-07 

0.25 0.5 50 2.85E-09 0.000979 2.85E-09 

0.25 1 10 0.002652 0.171412 0.002652 

0.25 1 20 5.36E-06 0.009031 5.36E-06 

0.25 1 30 1.7E-07 0.005808 1.7E-07 

0.25 1 50 1.43E-09 0.025105 1.43E-09 

0.25 1.5 10 0.000791 0.72097 0.000791 

0.25 1.5 20 1.1E-06 0.031248 1.1E-06 

0.25 1.5 30 4.33E-07 0.239611 4.33E-07 

0.25 1.5 50 3.21E-10 0.010335 3.21E-10 

0.5 0.5 10 0.005259 0.009682 0.005259 

0.5 0.5 20 4.92E-05 0.000478 4.92E-05 

0.5 0.5 30 8.87E-08 0.012252 8.87E-08 

0.5 0.5 50 1.48E-09 0.00043 1.48E-09 

0.5 1 10 0.004654 0.002195 0.002195 

0.5 1 20 3.26E-05 0.031042 3.26E-05 

0.5 1 30 3.42E-07 0.012912 3.42E-07 

0.5 1 50 8.28E-11 0.014289 8.28E-11 

0.5 1.5 10 0.004483 0.092447 0.004483 

0.5 1.5 20 3.71E-05 0.014191 3.71E-05 

0.5 1.5 30 4.08E-07 0.11564 4.08E-07 

0.5 1.5 50 2.63E-10 0.008968 2.63E-10 

0.75 0.5 10 0.003688 0.031815 0.003688 

0.75 0.5 20 2.55E-05 0.00287 2.55E-05 

0.75 0.5 30 8.31E-08 0.000111 8.31E-08 

0.75 0.5 50 8.9E-11 0.000294 8.9E-11 

0.75 1 10 0.002613 0.233707 0.002613 

0.75 1 20 2.49E-05 0.076475 2.49E-05 

0.75 1 30 6.02E-08 0.038741 6.02E-08 

0.75 1 50 5.13E-11 0.006712 5.13E-11 

0.75 1 100 9.85E-12 0.027358 9.85E-12 

0.75 1.5 10 0.002863 0.031182 0.002863 

0.75 1.5 20 2.1E-06 0.143916 2.1E-06 

0.75 1.5 30 1.06E-07 0.010499 1.06E-07 

0.75 1.5 50 4.03E-10 0.07206 4.03E-10 

 

 

 

 

 

 

 

 



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Table 5. Represent the MSE for survival function methods of all Bayesian method. 

 𝛼  𝜆   𝑛     𝑆𝑀𝑆𝐸𝐿𝐴𝐸
     𝑆𝑀𝑆𝐸𝑇𝐾𝐴𝐸

 min 

0.25 0.5 10 1.79E-05 0.001218 1.78826E-05 

0.25 0.5 20 6.69E-08 4.93E-05 6.68907E-08 

0.25 0.5 30 0.000186 0.000428 0.00018582 

0.25 0.5 50 2.42E-10 5.92E-05 2.42355E-10 

0.25 1 10 3.46E-05 0.000936 3.45513E-05 

0.25 1 20 3.87E-07 0.001142 3.87255E-07 

0.25 1 30 6.94E-06 0.000106 6.9434E-06 

0.25 1 50 2.22E-10 0.000422 2.21642E-10 

0.25 1.5 10 8.16E-06 0.013263 8.15897E-06 

0.25 1.5 20 3.59E-07 0.000367 3.59237E-07 

0.25 1.5 30 1.46E-09 8.85E-05 1.46397E-09 

0.25 1.5 50 2.01E-12 2.45E-06 2.00982E-12 

0.5 0.5 10 5.21E-05 0.004192 5.21315E-05 

0.5 0.5 20 9.98E-07 0.003911 9.97916E-07 

0.5 0.5 30 1.57E-08 5.9E-07 1.57214E-08 

0.5 0.5 50 2E-11 0.000716 2.00342E-11 

0.5 1 10 4.54E-06 0.000492 4.54484E-06 

0.5 1 20 1.58E-06 0.003329 1.57777E-06 

0.5 1 30 2.07E-08 3.85E-05 2.06807E-08 

0.5 1 50 3.8E-12 1.11E-05 3.79964E-12 

0.5 1.5 10 0.000107 0.001286 0.000107102 

0.5 1.5 20 1.55E-06 0.000418 1.55265E-06 

0.5 1.5 30 8.5E-09 0.000836 8.49529E-09 

0.5 1.5 50 5.85E-11 0.000875 5.84854E-11 

0.75 0.5 10 5.55E-05 0.010901 5.55124E-05 

0.75 0.5 20 1.69E-08 8.48E-06 1.69141E-08 

0.75 0.5 30 2.77E-10 1.64E-06 2.77344E-10 

0.75 0.5 50 1.12E-06 0.003646 1.12329E-06 

0.75 1 10 1.16E-06 0.006332 1.16403E-06 

0.75 1 20 2.29E-08 4.14E-05 2.29383E-08 

0.75 1 30 3.09E-10 0.001051 3.09264E-10 

0.75 1 50 4.93E-11 0.002993 4.92781E-11 

0.75 1.5 10 1.98E-06 0.000167 1.98306E-06 

0.75 1.5 20 3.31E-09 0.0011 3.31358E-09 

0.75 1.5 30 2.99E-09 1.7E-05 2.98614E-09 

0.75 1.5 50 9.37E-11 0.000217 9.37347E-11 

 

5. Conclusion 

     In this paper, we have considered the Bayesian estimation of the unknown parameters of 

the two-parameter Exponential-Rayleigh distribution. Simulation procedure using Monte-

Carlo technique and through previous tables show us that the estimated values and two 

distribution parameters are influenced by both (sample size, real value for estimator and 

estimation method), we find that the lindely approximation estimation method is the best for 

the parameters (α and λ). Bayes estimators were obtained using lindley approximation and 

Tierney and Kadane approximation while MLE were obtained using Newton-Raphson method, 

simulation study was conducted to examine and compare the lindely approximation estimation 

method with Tierney and Kadane approximation for different sample sizes with different 

values showing that the lindely approximation estimation method is the best when we use mean 

squares error for the parameter (α and λ). Noting that the lindely approximation estimation 

method is the best when we use mean squares error for survival function. 

 



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416 
 

Acknowledgment 

      The referees' insightful feedback and recommendations for enhancing the work are much 

appreciated by the writers. 

 

Conflict of Interest 

     The authors claim they have no rival interests. 

 

Funding 

     In order to prepare for publication, there is no funding help. 

 

References  

1. Ramos PL, Rodrigues FA, Ramos E, Dey DK, Louzada F. Power laws distributions in objective 

priors. Statistica Sinica. 2023;33:1959-1984. https://doi.org/10.5705/ss.202020.0521 

2. Hussein LK, Hussein IH, Rasheed HA. An estimation of survival and hazard rate functions of 

exponential Rayleigh distribution. Ibn AL-Haitham Journal for Pure and Applied Sciences. 

2021;34:93-107. https://doi.org/10.30526/34.4.2706 

3. Shatti RN, Al-Kinani IH. Estimating the parameters of exponential-Rayleigh distribution under 

Type-I censored data. Baghdad Science Journal. 2024;21:146-150. 

https://doi.org/10.21123/bsj.2023.79627 

4. Isaac RS, Mehta NB. Efficient computation of multivariate Rayleigh and exponential distributions. 

IEEE Wireless Communications Letters. 2019;8:456-459. 

https://doi.org/10.1109/LWC.2018.2875999 

5. Ferreira P, Gonzales J, Tomazella V, Ehlers R, Louzada F, Silva E. Objective Bayesian analysis 

for the Lomax distribution. Statistical Papers. 2016;1-19. 

https://doi.org/10.1016/j.spl.2019.108677 

6. Mazaal AR, Karam NS, Karam GS. Comparing Weibull stress-strength reliability Bayesian 

estimators for singly type II censored data under different loss functions. Baghdad Science Journal. 

2021;18:306-314. https://doi.org/10.21123/bsj.2021.18.2.0306 

7. Al-Baldawi. Tasnim H.K. Comparison of maximum likelihood and some Bayes estimators for 

Maxwell distribution based on non-informative priors. Baghdad Science Journal. 2013;10(2):480-

488. https://doi.org/10.21123/bsj.10.2.480-488 

8. Iden H., Sara S. Taher. Bayesian and non-Bayesian methods to estimate the two parameters of 

logistic distribution. Baghdad Science Journal. 2014;11(4):1612-1623. 

https://doi.org/10.21123/bsj.2014.11.4.1612-1623 

9. Rasheed HA. Bayes estimators for the Maxwell distribution under quadratic loss function using 

different priors. Baghdad Science Journal. 2016. 

10. Alkanani IH, Salman SG. Bayes and non-Bayes estimation methods for the parameter of Maxwell-

Boltzmann distribution. Baghdad Science Journal. 2017;14:808-812. 

https://doi.org/10.21123/bsj.2017.14.4.0808 

11. Kalt HG, Hussein IH. Bayesian estimation for parameters and the survival function of modified 

Weibull distribution using Tierney and Kadane’s approximation. Journal of Advanced Research in 

Dynamical and Control Systems. 2018;10:1960-1970. 

12. Mohammed MJ, Hussein IH. Some estimation methods for new mixture distribution with 

simulation and application. IOP Conference Series: Materials Science and Engineering. 2019;571. 

https://doi.org/10.1088/1757-899X/571/1/012014 

13. Fatima K, Ahmad SP. Statistical properties of exponential Rayleigh distribution and its 

applications to medical science and engineering. Journal of Statistical Distributions and 

Applications. 2013;2:491-506. 

14. Hussein LK, Rasheed HA, Hussein IH. A class of exponential Rayleigh distribution and new 

modified weighted exponential Rayleigh distribution with statistical properties. Ibn AL-Haitham 

Journal for Pure and Applied Sciences. 2023;36:390-406. https://doi.org/10.30526/36.2.3044 

https://doi.org/10.5705/ss.202020.0521
https://doi.org/10.30526/34.4.2706
https://doi.org/10.21123/bsj.2023.79627
https://doi.org/10.1109/LWC.2018.2875999
https://doi.org/10.21123/bsj.10.2.480-488
https://bsj.uobaghdad.edu.iq/do/search/?q=author%3A%22Iden%20H.%20Alkanani%22&start=0&context=38131414
https://bsj.uobaghdad.edu.iq/do/search/?q=author%3A%22Sara%20S.%20Taher%22&start=0&context=38131414
https://doi.org/10.21123/bsj.2014.11.4.1612-1623
https://doi.org/10.1088/1757-899X/571/1/012014
https://doi.org/10.30526/36.2.3044


IHJPAS. 2025,38(2) 

417 
 

15. Asparouhov T, Muthén B. Bayesian analysis using Mplus: Technical implementation. Mplus 

Journal. 2010;1-38. 

16. Kruschke JK. What to believe: Bayesian methods for data analysis. Trends in Cognitive Sciences. 

2010;14:293-300. https://doi.org/10.1016/j.tics.2010.05.001 

17. Al-noor NH. Non-Bayes, Bayes and empirical Bayes estimators for the shape parameter of Lomax 

distribution. Journal of Statistical Theory and Practice. 2015;5:17-28. 

18. Virbickaite A, Ausín MC, Galeano P. Bayesian inference methods for univariate and multivariate 

GARCH models: A survey. Journal of Economic Surveys. 2015;29:76-96. 

https://doi.org/10.1111/joes.12046 

19. Guure CB, Ibrahim NA, Ahmed AOM. Bayesian estimation of two-parameter Weibull distribution 

using extension of Jeffreys’ prior information with three loss functions. Mathematical Problems in 

Engineering. 2012;2012. https://doi.org/10.1155/2012/589640 

20. Rasheed HA, Abd MN. Bayesian estimation for two parameters of exponential distribution under 

different loss functions. Ibn AL-Haitham Journal for Pure and Applied Sciences. 2023;36:289-300. 

https://doi.org/10.30526/36.2.2946 

21. Pradhan B, Kundu D. Bayes estimation and prediction of the two-parameter gamma distribution. 

Journal of Statistical Computation and Simulation. 2011;81:1187-1198. 

https://doi.org/10.1080/00949651003796335 

22. Asparouhov T, Muthen B. Bayesian analysis of latent variable models using Mplus. Technical 

Report. Version 5. 2021;1-60. https://doi.org/10.3758/s13423-016-1016-7 

23. Preda V, Panaitescu E, Constantinescu A. Bayes estimators of modified-Weibull distribution 

parameters using Lindley’s approximation. WSEAS Transactions on Mathematics. 2010;9:539-

549. 

24. Singh PK, Singh SK, Singh U. Bayes estimator of inverse Gaussian parameters under general 

entropy loss function using Lindley’s approximation. Communications in Statistics - Simulation 

and Computation. 2008 ;37 :175 0-17 62.  https://doi.org/10.1080/03610910701884054 

25. Sana S, Faizan M. Bayesian estimation using Lindley’s approximation and prediction of 

generalized exponential distribution based on lower record values. Journal of Statistics 

Applications and Probability. 2021;10:61-75. https://doi.org/10.18576/jsap/100107 

26. Sharma VK, Singh SK, Singh U. Classical and Bayesian methods of estimation for power Lindley 

distribution with application to waiting time data. Communications for Statistical Applications and 

Methods. 2017;24:193-209. https://doi.org/10.5351/CSAM.2017.24.3.193 

27. Guure CB, Ibrahim NA. Approximate Bayesian estimates of Weibull parameters with Lindley’s 

method. Sains Malaysiana. 2014;43:1433-1437. 

28. Bastan F, Mirmostafaee SMTK. Approximating Bayes estimates by means of the Tierney Kadane, 

importance sampling and Metropolis-Hastings within Gibbs methods in the Poisson-Exponential 

distribution: A comparative study. 2019;10:62-77. 

29. Kazmi MSA, Aslam M, Ali S. On the Bayesian estimation for two component mixture of Maxwell 

distribution, assuming Type I censored data. International Journal of Applied Science and 

Technology. 2012;2:197-218. 

30. Ari Y. Bayesian estimation of GARCH (1,1) model using Tierney-Kadane’s approximation. 2022. 

https://doi.org/10.1007/978-3-030-02194-8 

 

https://doi.org/10.1080/00949651003796335