Bayesian Generalized Self Method to Estimate Scale Parameter of Inverse Rayleigh Distribution


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 6(4) (2021), Pages 270-278 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: January 22, 2021 Reviewed: April 24, 20201  Accepted: April 30 , 2021 
DOI: http://dx.doi.org/10.18860/ca.v6i4.11482      
 

Bayesian Generalized Self Method to Estimate Scale Parameter of 
Inverse Rayleigh Distribution 

Ferra Yanuar1, Rahmi Febriyuni2, Izzati Rahmi HG3 

1,2,3Mathematics Department, Faculty of Mathematics and Natural Sciences,  
Universitas Andalas, Padang 

Email: ferrayanuar@sci.unand.ac.id, raahmifebriyuni@gmail.com,  
izzatirahmihg@sci.unand.ac.id  

ABSTRACT  

The purposes of this study are to estimate the scale parameter of invers Rayleigh distribution 
under MLE and Bayesian Generalized square error loss function (SELF). The posterior 
distribution is considered to use two types of prior, namely Jeffrey’s prior and exponential 
distribution. The proposed methods are then employed in the real data. Several criteria for the 
selection model are considered in order to identify the method which results in a suitable value 
of parameter estimated. This study found that Bayesian Generalized SELF under Jeffrey’s prior 
yielded better estimation values than MLE and Bayesian Generalized SELF under exponential 
distribution. 

Keywords: Bayesian Generalized SELF; exponential distribution; Inverse Rayleigh; Jeffrey’s 
prio; MLE. 

INTRODUCTION 

Rayleigh distribution is a special form of Weibull distribution, meanwhile, inverse 
Rayleigh distribution is a special form of Inverse Weibull distribution. The inverse 
Rayleigh distribution is very useful lifetime model that can be used for analyzing infant 
mortality, survival analysis, reliability and quality control. The probability density 
function (pdf) of the inverse Rayleigh distribution with scale parameter 𝜃 is defined as 
follows [1]:  

𝑓(𝑥;  𝜃) =  
2𝜃

𝑥3
exp (−

𝜃

𝑥2
)  ,      𝜃 > 0, 𝑥 > 0.      (1) 

The cumulative distribution function (CDF) of the inverse Rayleigh distribution is given 
by 

𝐹(𝑥;  𝜃) = exp (−
𝜃

𝑥2
) ,     𝜃 > 0,      𝑥 > 0 .      (2) 

Here 𝜃 is the scale parameter. The behavior of instantaneous failure rate of the inverse 
Rayleigh distribution has been increasing and decreasing failure rate patterns for 
lifetime data. 

A significant amount of work has been done related to the inverse Rayleigh 
distribution model in the classical framework but not much in a Bayesian setup, 
especially in Bayesian Generalized SELF (Squared Error Loss Function).  Several studies 

http://dx.doi.org/10.18860/ca.v6i4.11482
mailto:ferrayanuar@sci.unand.ac.id
mailto:raahmifebriyuni@gmail.com
mailto:izzatirahmihg@sci.unand.ac.id


Bayesian Generalized Self Method to Estimate Scale Parameter of Inverse Rayleigh Distribution 

Ferra Yanuar 271 

have used inverse Rayleigh distribution for several cases. Soliman and Al-Aboud [2] 
used classical method and Bayesian for parameter estimation based on a set of upper 
record values from the Rayleigh distribution. Aslam and Jun [3] derived an acceptance 
sampling plan from a truncated life test where multiple items in a group could be tested 
simultaneously by a tester when the lifetime of an item followed either an inverse 
Rayleigh or a log-logistic distribution. Soliman et al. [4]  discussed the parameter 
estimation for an inverse Rayleigh distribution based on lower record values. They 
implemented a maximum likelihood (ML) estimator of the unknown parameter and 
Bayesian analysis with informative prior used to derive these estimators and the 
predictive intervals. Ali [5] explored the modeling of the heterogeneity existing in the 
lifetime processes using the mixture of the inverse Rayleigh distribution, and the 
spotlight is the Bayesian inference of the mixture model using non-informative (the 
Jeffreys and the uniform) and informative (gamma) priors. Studied by Dey & Dey [6] 
derived Bayesian estimation of the scale parameter and reliability function of an inverse 
Rayleigh distribution. Yousef & Lafta [7] explored how to estimate the scale parameter 
for distribution of inverse Rayleigh using different methods, such as the method of 
Maximum likelihood estimator and moment method. Dey [8] obtained Bayesian 
estimates of an inverse Rayleigh distribution using squared error and LINEX loss 
functions. Meanwhile, Rasheed [9] designed some Bayesian estimators for the 
parameter scale and reliability function of the inverse Rayleigh distribution under the 
Generalized squared error loss function (SELF).  

In the present study, we consider the estimation of unknown parameters in an 
inverse Rayleigh distribution. The aim of this study is to estimate the scale parameter of 
inverse Rayleigh distributions using frequentist method (MLE) and the Bayesian 
approach which are employed to empirical data. The Bayesian approach here is 
Bayesian Generalized SELF under two types of priors, namely Jeffrey’s prior and 
exponential’s prior. The criteria to determine better performance of estimation method 
are based on the smallest value of Akaike Information Criteria (AIC), Akaike Information 
Criteria correction (AICc) and Bayesian Information Criteria (BIC).  

The remainder of this study is organized as follows: the maximum likelihood 
estimation (MLE), Bayesian Generalized SELF, Jeffrey’s method, exponential 
distribution, criteria for the goodness of fit of parameter estimation method are derived 
in Section 2. Estimation method using Bayesian Generalized SELF under two types of 
priors and implementation of the proposed method to the real data are discussed in 
Section 3. Finally, Section 4 as the last section provides some concluding remarks. 

 

METHODS 

In this section, we explore all methods which are implemented in this present 
study. The maximumm likelihood estimation method in the beginning and then followed 
by Bayesian approach and criteria for model selection. 

Maximum Likelihood Estimation 

In this section, we derived the classical estimator of the scale parameter for the 
inverse Rayleigh distribution represented by the maximum likelihood estimator. Let 
𝑋1, 𝑋2, … 𝑋𝑛  be a sequence of i.i.d random variables from invers Rayleigh distribution 
with scale parameter θ, written as 𝑋 ~ 𝐼𝑅𝐷 (𝜃), with probability density function of 𝑋 is 
𝑓(𝑥 ;  𝜃) as presented by Eq. (1). Thus, the maximum likelihood estimation is formulated 
as follows [10]: 



Bayesian Generalized Self Method to Estimate Scale Parameter of Inverse Rayleigh Distribution 

Ferra Yanuar 272 

𝐿(𝜃) = ∏ 𝑓(𝑥𝑖 , 𝜃) 

𝑛

𝑖=1

                                                                                                   

                 = 2𝑛  𝜃𝑛  (∏
1

𝑥𝑖
3 

𝑛
𝑖=1 ) 𝑒𝑥𝑝 (−𝜃 ∑

1

𝑥𝑖
2

𝑛
𝑖=1 )                    (3) 

To obtained the estimate for θ is derived by maximizing Eq. (3) until we have: 

𝜃𝑀�̂� =  
𝑛

∑
1

𝑥
𝑖
2

𝑛
𝑖=1

=  
𝑛

𝑇
 ,     𝑤ℎ𝑒𝑟𝑒   𝑇 =  ∑

1

𝑥𝑖
2

𝑛
𝑖=1                                                       (4) 

 
Bayesian Generalized SELF Method 

This section deals with the problem of obtaining Bayesian estimators for the 
scale parameter θ from the inverse Rayleigh distribution The Bayes method is a 
parameter estimation method based on the Bayes theorem. The basic concepts of the 
Bayes method are as follows. Suppose that  𝑋1, 𝑋2, … . , 𝑋𝑛 is a random example of the 
distribution 𝑓(𝒙 ;  𝜃) where 𝜃 is the parameter of the distribution. Estimation of 
parameter 𝜃 will be based on random example 𝑋1, 𝑋2, … . , 𝑋𝑛. The Bayes method is an 
estimation method based on combining information obtained from samples (objective 
knowledge), known as the likelihood function, with prior information regarding the 
distribution of estimated parameters [11], [12]. Multiplying the likelihood function by 
the prior distribution gives the posterior distribution. In other words, the posterior 
distribution is a conditional probability density function of a parameter θ which is given 
the observation 𝐱 = (𝑋1, 𝑋2, … . , 𝑋𝑛). The formula for defining the posterior distribution 
is stated by the following formula [13]: 

𝑓(𝜃|𝒙) =
𝑓(𝒙|𝜃)𝑓(𝜃)

∫ 𝑓(𝒙|𝜃)𝑓(𝜃)𝑑𝜃
=  

𝑓(𝒙,𝜃)

𝑓(𝒙)
    (5) 

 Meanwhile, the estimator for the scale parameter (𝜃) using the Bayes Generalized 
SELF method will be described as follows [9]: 

𝐿(𝜃 ;  𝜃) =  ∑ 𝛼𝑗 𝜃
𝑗  (𝜃 −  𝜃)

𝑛

𝑖=1

 

                               = (𝛼0  + 𝛼1𝜃 + ⋯ + 𝛼𝑘 𝜃
𝑘 ) (𝜃 −  𝜃)                                                         (6) 

The estimator for parameter 𝜃 is obtained by minimizing the expectation for 𝜃, 
which is denoted by 𝐿(𝜃 ;  𝜃). The expected value of this function can be found by 

combining 𝐿(𝜃 ;  𝜃)  and the probability density function of 𝜃, here denoted by ℎ(𝜃 |𝑥). 

Thus, the expectation of 𝜃 using the Bayes Generalized SELF is as follows: 

                 𝐸[𝐿(𝜃 ;  𝜃)] =  ∫ 𝐿(𝜃 ;  𝜃) ℎ(𝜃 |𝑥) 𝑑𝜃

∞

0

 

   𝐸[𝐿(𝜃 ;  𝜃)] =  𝛼0 𝜃
2 − 2 𝛼0 𝜃 𝐸(𝜃 | 𝑥) +  𝛼0 𝐸(𝜃

2 | 𝑥) +  𝛼1 𝜃 
2𝐸(𝜃 | 𝑥) −

                                              2 𝛼1 𝜃 𝐸(𝜃
2 | 𝑥) +  𝛼1 𝐸(𝜃

3 | 𝑥) + ⋯ +  𝛼𝑘  𝜃 
2𝐸(𝜃𝑘  | 𝑥) −

                                                2 𝛼𝑘  𝜃 𝐸(𝜃
𝑘+1 | 𝑥) + 𝛼𝑘  𝐸(𝜃

𝑘+2 | 𝑥).                                            (7) 

To obtain the estimated value for 𝜃  with the Bayesian Generalized SELF method, 

the Eq. (7) is derived on 𝜃 , so that: 

𝜃𝐵�̂� =  
𝛼0 𝐸(𝜃 | 𝑥) +  𝛼1 𝐸(𝜃

2 | 𝑥) + ⋯ +  𝛼𝑘  𝐸(𝜃
𝑘+1 | 𝑥)

𝛼0 +  𝛼1 𝐸(𝜃 | 𝑥) + ⋯ +  𝛼𝑘  𝐸(𝜃
𝑘  | 𝑥)

                                   (8) 



Bayesian Generalized Self Method to Estimate Scale Parameter of Inverse Rayleigh Distribution 

Ferra Yanuar 273 

Jeffreys’ Prior as Non-Informative Prior  

The most widely used noninformative priors in Bayesian analysis is Jeffreys’ 
prior. This method is also attractive because it is proper under mild conditions and 
requires no elicitation of hyperparameters. Jeffreys’ rule is derived from likelihood 
function then take the prior distribution to be the determinant of the square root of the 

Fisher information matrix, denoted by 𝑓(𝜃) ∝  √𝐼(𝜃). Fisher's information for the 

parameter 𝜃, defined as [13], [14] 

𝐼(𝜃) =  −𝑛 𝐸 (
𝜕2 ln(𝑓(𝑥𝑖  ;  𝜃))

𝜕𝜃2
) 

Let b is constant, thus : 

 𝑓(𝜃)  ∝ √𝐼(𝜃) = 𝑏 √−𝑛 𝐸 (
𝜕2 ln(𝑓(𝑥𝑖 ; 𝜃))

𝜕𝜃2
)                                                                    (9) 

For inverse Rayleigh distribution, it’s found that  
𝜕2  ln(𝑓(𝑥𝑖 ; 𝜃)) 

𝜕𝜃2
= −  

1

𝜃2
. Thus, it’s also 

obtained that  

𝐸 (
𝜕2  ln(𝑓(𝑥𝑖  ;  𝜃)) 

𝜕𝜃2
) = −  

1

𝜃2
                                                                                        (10) 

By substituting Eq. (10) into Eq. (9), then it results  

𝑓(𝜃) =  
𝑏

𝜃
 √𝑛   ,      𝜃 > 0                                                                                                           (11) 

By combining this Jeffrey’s prior and likelihood function, it yields the following posterior 
distribution : 

ℎ1(𝜃 | 𝑥1, 𝑥2, … , 𝑥𝑛 ) =  
𝑇𝑛  𝜃𝑛−1 exp(−𝜃𝑇)

г(𝑛)
                                                                 (12) 

The posterior distribution in Eq. (12) has identic form with Gamma distribution with 

scale parameter 
1

𝑇
  and shape parameter n, written as  𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 (

1

𝑇
, 𝑛). 

 
Exponential Distribution as Conjugate Prior  

We also derive the parameter estimation for 𝜃 based on Bayesian Generalized 
SELF with exponential distribution as prior. The probability distribution function for 
random variable 𝜃 which has Exponential distribution with scale parameter 𝜆, written as 
𝜃~𝐸𝑥𝑝(𝜆),  is formulated as follows: 

𝑔(𝜃) =  
1

𝜆
exp (

−𝜃

𝜆
)  , 𝜃 > 0, 𝜆 > 0                                                                                   (13) 

Eq. (13) then is combined with likelihood function in Eq. (3) until we have the posterior 
distribution as follows:  

ℎ2(𝜃 | 𝑥1, 𝑥2, … , 𝑥𝑛 ) =  
(𝑇 +

1

𝜆
)

𝑛+1

 𝜃𝑛 exp (−𝜃 (𝑇 +
1

𝜆
))

г(𝑛 + 1)
                                         (14) 



Bayesian Generalized Self Method to Estimate Scale Parameter of Inverse Rayleigh Distribution 

Ferra Yanuar 274 

This posterior distribution has a similar form with Gamma distribution with scale 

parameter is
1

𝑇+
1

𝜆

 and shape parameter is n+1, written as 𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 (
1

𝑇+
1

𝜆

, 𝑛 + 1) or 

𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 (
1

𝑃
, 𝑛 + 1) with 𝑃 = 𝑇 +

1

𝜆
. 

 
Criteria Model Selection 

The Akaike information criterion (AIC) which is widely used for statistical inference, 
is an estimator of out-of-sample prediction error and thereby the relative quality 
of statistical models for a given set of data [15]. Given several models for the data, AIC 
estimates the quality of each model, relative to each of the other models. This method 
provides a means for each model. When a statistical model is used to represent the 
process that generated the data, the representation will almost never be exact; so some 
information will be lost by using the model to represent the process. AIC estimates the 
relative amount of information lost by a given model: the less information a model loses, 
the higher the quality of that model. In estimating the amount of information lost by a 
model, AIC deals with the trade-off between the goodness of fit of the model and the 
simplicity of the model. In other words, AIC deals with both the risk of overfitting and 
the risk of underfitting. Suppose that we have a statistical model of some data. Let k be 
the number of estimated parameters in the model. Let �̂� be the maximum value of 
the likelihood function for the model. Then the AIC value of the model is the following 
[15]:  

𝐴𝐼𝐶 = 2𝑘 − 2𝑙𝑛(�̂�) 

For condition  
𝑛

𝑘
< 40 with n represent the amount of data, it’s suggested to use AICc 

(Akaike Information Criteria correction): 

𝐴𝐼𝐶𝑐 = 𝐴𝐼𝐶 +
2𝑘(𝑘 + 1)

𝑛 − 𝑘 − 1
 

Another method to estimate the quality of each model relative to each of the 
other models is Bayesian Information Criteria (BIC), which is represented by following: 

𝐵𝐼𝐶 = 𝑘𝑙𝑛(𝑛) − 2𝑙𝑛 (𝐿(𝜃)). 

Given a set of candidate models for the data, the preferred model is the one with the 
minimum AIC, AICc and BIC value.  

 

RESULTS AND DISCUSSION 

In this current study, we then employ the Bayesian Generalized SELF under non 
informative prior that is Jeffreys prior to estimate the scale parameter of invers 
Rayleight distribution. We also consider the Bayesian SELF under informative prior 
namely an exponential distribution. Both methods as well as MLE are employed to the 
empirical data then. The most suitable method to be implemented is determined based 
on the smallest values of AIC, AICc and BIC.  

https://en.wikipedia.org/wiki/Statistical_inference
https://en.wikipedia.org/wiki/Estimator
https://en.wikipedia.org/wiki/Out-of-sample
https://en.wikipedia.org/wiki/Statistical_model
https://en.wikipedia.org/wiki/Goodness_of_fit
https://en.wikipedia.org/wiki/Overfitting
https://en.wikipedia.org/wiki/Statistical_model
https://en.wikipedia.org/wiki/Statistical_parameter
https://en.wikipedia.org/wiki/Likelihood_function


Bayesian Generalized Self Method to Estimate Scale Parameter of Inverse Rayleigh Distribution 

Ferra Yanuar 275 

Bayesian Generalized SELF Under Jeffrey’s Prior 

Based on Eq. (12) is obtained that 𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 (
1

𝑇
, 𝑛). It can be derived that  

𝐸 (𝜃 | 𝑥) =
𝑛

𝑇
  and in general we also obtain  𝐸 (𝜃𝑚 | 𝑥) =  

г (𝑛+𝑚)

г(𝑛) 𝑇𝑚
 . These expected values 

then be substituted into Eq. (8) to derive the formula to estimate 𝜃, under Jeffrey’s prior, 
denoted here as 𝜃�̂� : 

𝜃�̂� =  
𝛼0 (

𝑛

𝑇
) +  𝛼1 (

(𝑛+1)𝑛

𝑇2
) + ⋯ +  𝛼𝑘 (

(𝑛+𝑘)(𝑛+𝑘−1)…(𝑛+1)𝑛

𝑇𝑘+1
)

𝛼0 +  𝛼1 (
𝑛

𝑇
) + ⋯ +  𝛼𝑘 (

(𝑛+𝑘−1)(𝑛+𝑘−2)…(𝑛+1)𝑛

𝑇𝑘
)

 

In this study, we choose first polynomial until fourth polynomial to be applied to 
estimate 𝜃: 

𝜃𝐽1̂ =  
𝛼0 (

𝑛

𝑇
) +  𝛼1 (

(𝑛+1)𝑛

𝑇2
)

𝛼0 +  𝛼1 (
𝑛

𝑇
)

                                                                                    (15) 

𝜃𝐽2̂ =  
𝛼0 (

𝑛

𝑇
) +  𝛼1 (

(𝑛+1)𝑛

𝑇2
) + 𝛼2 (

(𝑛+2)(𝑛+1)𝑛

𝑇3
)

𝛼0 +  𝛼1 (
𝑛

𝑇
) + 𝛼2 (

(𝑛+1)𝑛

𝑇2
)

                                                 (16) 

𝜃𝐽3̂ =  
𝛼0 (

𝑛

𝑇
) +  𝛼1 (

(𝑛+1)𝑛

𝑇2
) + ⋯ +  𝛼3 (

(𝑛+3)(𝑛+2)(𝑛+1)𝑛

𝑇4
)

𝛼0 +  𝛼1 (
𝑛

𝑇
) + ⋯ +  𝛼3 (

(𝑛+2)(𝑛+1)𝑛

𝑇3
)

                             (17) 

𝜃𝐽4̂ =  
𝛼0 (

𝑛

𝑇
) +  𝛼1 (

(𝑛+1)𝑛

𝑇2
) + ⋯ +  𝛼4 (

(𝑛+4)(𝑛+3)…(𝑛+1)𝑛

𝑇5
)

𝛼0 +  𝛼1 (
𝑛

𝑇
) + ⋯ +  𝛼4 (

(𝑛+3)(𝑛+2)(𝑛+1)𝑛

𝑇4
)

                           (18) 

 

 
Bayesian Generalized SELF Under Exponential Distribution 

It has been proved that 𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 (
1

𝑇+
1

𝜆

, 𝑛 + 1) or 𝜃 | 𝑥 ~ 𝐺𝑎𝑚𝑚𝑎 (
1

𝑃
, 𝑛 +

1) with 𝑃 = 𝑇 +
1

𝜆
. Then, it can be proved that  

𝐸 (𝜃𝑚 | 𝑥) =  
г (𝑛 + 1 + 𝑚)

г(𝑛 + 1) 𝑃𝑚
                                                                                      (19) 

By substituting 𝑚 = 1,2, . . . , 𝑘 to Eq. (19), we then derive the estimate formula for 
scale parameter, 𝜃 under exponential prior, denoted here as 𝜃�̂� : 

𝜃�̂� =
𝛼0 (

𝑛+1

𝑃
) + 𝛼1 (

(𝑛+2)(𝑛+1)

𝑃2
) + ⋯ + 𝛼𝑘 (

(𝑛+𝑘+1)…(𝑛+1)

𝑃𝑘+1
)

𝛼0 +  𝛼1 (
𝑛+1

𝑃
) + ⋯ +  𝛼𝑘 (

(𝑛+𝑘)(𝑛+𝑘−1)…(𝑛+1)

𝑃𝑘
)

                              (20) 

In this study, we choose first polinomial until fourth polinomial based Eq. (20) to 
be used to estimate  𝜃�̂�  . 



Bayesian Generalized Self Method to Estimate Scale Parameter of Inverse Rayleigh Distribution 

Ferra Yanuar 276 

𝜃𝐸1̂ =
𝛼0 (

𝑛+1

𝑃
) + 𝛼1 (

(𝑛+2)(𝑛+1)

𝑃2
)

𝛼0 +  𝛼1 (
𝑛+1

𝑃
)

                                                                             (21) 

𝜃𝐸2̂ =
𝛼0 (

𝑛+1

𝑃
) + 𝛼1 (

(𝑛+2)(𝑛+1)

𝑃2
) + 𝛼2 (

(𝑛+3)(𝑛+2)(𝑛+1)

𝑃3
)

𝛼0 +  𝛼1 (
𝑛+1

𝑃
) + 𝛼2 (

(𝑛+2)(𝑛+1)

𝑃2
)

                                   (22) 

𝜃𝐸3̂ =
𝛼0 (

𝑛+1

𝑃
) + 𝛼1 (

(𝑛+2)(𝑛+1)

𝑃2
) + ⋯ + 𝛼3 (

(𝑛+4)…(𝑛+1)

𝑃4
)

𝛼0 +  𝛼1 (
𝑛+1

𝑃
) + ⋯ +  𝛼3 (

(𝑛+3)(𝑛+2)(𝑛+1)

𝑃3
)

                                (23) 

𝜃𝐸4̂ =
𝛼0 (

𝑛+1

𝑃
) + 𝛼1 (

(𝑛+2)(𝑛+1)

𝑃2
) + ⋯ + 𝛼4 (

(𝑛+5)…(𝑛+1)

𝑃5
)

𝛼0 +  𝛼1 (
𝑛+1

𝑃
) + ⋯ +  𝛼4 (

(𝑛+4)…(𝑛+1)

𝑃4
)

                                 (24) 

 
Implementation of Proposed Method to Real Data 

The result of analytical study above then implemented to real data. The real data 
set represents the 72 exceedances for the years 1958–1984 (rounded to one decimal 
place) of flood peaks (in m3/s) of the Wheaton River near Carcross in Yukon Territory, 
Canada [16]. The data are as follows: 

 
1.7 2.2 14.4 1.1 0.4 20.6 5.3 0.7 1.9 13.0 12.0 9.3 
1.4 18.7 8.5 25.5 11.6 14.1 22.1 1.1 2.5 14.4 1.7 37.6 
0.6 2.2 39.0 0.3 15.0 11.0 7.3 22.9 1.7 0.1 1.1 0.6 
9.0 1.7 7.0 20.1 0.4 2.8 14.1 9.9 10.4 10.7 30.0 3.6 
5.6 30.8 13.3 4.2 25.5 3.4 11.9 21.5 27.6 36.4 2.7 64.0 
1.5 2.5 27.4 1.0 27.1 20.2 16.8 5.3 9.7 27.5 2.5 27.0 

 

In this present study, we fix several values for 𝛼0=100, 𝛼1=50, 𝛼2=10, 𝛼3=8, 𝛼4=7 
and  𝜆=0.8 to be applied to estimate the scale parameter 𝜃. Based on this real data, we 
calculate that: 

𝑇 =  ∑
1

𝑥𝑖
2

= 138,99     ∑ 𝑙𝑛 (
1

𝑥1
3

)

72

𝑖=1

72

𝑖=1

=  −388,38       𝑃 = ∑
1

𝑥𝑖
2

72

𝑖=1

+
1

𝜆
= 140,24 

We then employ both proposed methods and MLE to this empirical data. The 
comparison of criteria for model selection based on three methods are provided in Table 
1.  

Table 1. Estimated Values for AIC, AICc, and BIC 

Prior Mean 
Criteria Model Selection 

AIC AICc BIC 

MLE 0.5180 917.6820 917.7391 919.9587 

Jeffrey’s Prior 
𝜃𝐽1̂ 0.5195 917.6778 917.7349 919.9545 

𝜃𝐽2̂ 0.5198 917.6748 917.7319 919.9515 



Bayesian Generalized Self Method to Estimate Scale Parameter of Inverse Rayleigh Distribution 

Ferra Yanuar 277 

𝜃𝐽3̂ 0.5200 917.6728 917.7299 919.9495 

𝜃𝐽4̂ 0.5201 917.6718 917.7289 919.9485 

Exponential’s 

Prior 

𝜃𝐸1̂ 0.5220 917.6816 917.7387 919.9583 

𝜃𝐸2̂ 0.5223 917.6786 917.7357 919.9553 

𝜃𝐸3̂ 0.5225 917.6766 917.7337 919.9533 

𝜃𝐸4̂ 0.5226 917.6756 917.7327 919.9523 

 

Table 1 informs us that this present study yielded almost similar values for 
estimated mean for all three methods (in the third column). Based on the criteria model 
selection, this study found that Jeffrey’s prior as noninformation prior, tends to result 
smaller values than MLE and exponential ‘s prior for all four polynomials.  The smallest 
values for these criteria are at Jeffrey’s prior at fourth polynomial (𝜃𝐽4̂).  

These results inform us that the method to estimate scale parameter of invers 
Rayleigh distribution using Bayesian Generalized SELF under Jeffrey’s prior tends to 
result better values than MLE and Bayesian Generalized SELF under exponential’s prior. 
This present study proved it by employing all proposed method to real data with size 
sample is relatively moderate, n = 72. 

 

CONCLUSIONS 

This study employed the MLE, Bayesian Generalized SELF under Jeffrey’s prior 
and Bayesian Generalized SELF under exponential’s prior to estimate the scale 
parameter of invers Rayleight distribution of a real data. All 72 sample of flood peaks 
data in Canada are involved in this study. This real data has invers Rayleigh distribution. 
This study found that estimation mean of scale parameter from invers Rayleigh 
distribution based on MLE, Bayesian Generalized SELF under Jeffrey’s prios and 
Bayesian Generalized SELF under exponential’s prior tend to result similar values. Based 
on criteria of selection model, this study proved that Bayesian Generalized SELF under 
Jeffrey’s prior tend to result the smallest value of AIC, AICc and BIC. 

 

ACKNOWLEDGMENTS  

This research was partly funded by DRPM, the Deputy for Strengthening Research and 
Development of the Ministry of Research and Technology / National Research and 
Innovation Agency of Indonesia, in accordance with Contract Number 
123/SP2H/AMD/LT/DRPM/2020. 
 

 



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Ferra Yanuar 278 

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