Comparison of Bayesian and non-Bayesian estimations for Type-II censored Generalized Rayleigh distribution

Presented By
Iqra Sardar†
Syed Masroor Anwar
Prof. Dr. Muhammad Aslam
DEPARTMENT OF MATHEMATICS AND STATISTICS
RIPHAH INTERNATIONAL UNIVERSITY,
ISLAMABAD, PAKISTAN
Email: iqrahusan@gmail.com†
16th International Conference on Statistical Sciences:
At Department of Statistics
Islamia College, Peshawar Khyber Pakhtunkhwa, Pakistan
Comparison of Bayesian and non-Bayesian estimations for
Type-II censored Generalized Rayleigh distribution

ABSTRACT
In this paper, we compare Bayesian and non-Bayesian estimations for the unknown parameters of
Generalized Rayleigh distribution under Type-II censoring schemes. First we deal with non-Bayesian
method namely maximum likelihood estimation along with their asymptotic confidence intervals with a
given coverage probability. Further we consider the Bayesian estimates of unknown parameters under
different loss functions. As Bayes estimators cannot be obtained in nice closed form. We use Lindley’s
approximation. Monte Carlo simulation study is carried out to compare different methods and the
performance of the estimates is judged by the mean squared error values. All the numerically
computations are performed in R software. Finally, a real life data set analysis is performed for the
illustration purpose.
KEYWORDS
Generalized Rayleigh distribution; Type-II censoring; Bayesian and non-Bayesian estimations;
Symmetric and asymmetric loss functions; Lindley’s approximation.

Burr (1942) introduced twelve different forms of cumulative distributions
functions for modeling life time data.
Many researchers examined the single parameter Burr type X model by putting
scale parameter λ=1. Recently, the single parameter distribution of the
extended Burr type X by
Surles and Padgett (2001) introduce two parameters Burr type X
distribution and correctly named as the Generalized Rayleigh
distribution.
Applications of the Generalized Rayleigh distribution
The Generalized Rayleigh distribution can be used to:
 Life testing.
 Failure time of machines
 Communication Engineering.
 Speed of Gas Molecules
Kundu and Raqab (2005,2007) have discussed the different techniques of
estimation of the parameters and further properties of GR distribution.

(1.1)
Generalized Rayleigh distribution
1.1 Model Analysis Probability density function (pdf):
12 2( ) ( )2( ; , ) 2 1 ; , , 0
x xf xe e x

     

      
 
x
Fig. 1 The pdf of GR Distribution For
different values of α and λ

(1.2)
The Survival /Reliability function (sf):
(1.3)
Cumulative distribution function (cdf):
2( )( ; , ) 1 ; , , 0xF e x

   
     
 
x
2( )
1 1( ; , ) ; , , 0x
eS x

   
 
 
   
 
x

(1.4)
The hazard rate function (hrf):
2 2( ) ( )22 1
( ; , ) ; , , 0
2( )1 1
x xxe e
h x
xe

 
   


    
  
    
 
x
Fig. 2 The hrf of GR Distribution For
different values of α and λ

 Aims of Papers
Derive the non-Bayesian method namely maximum likelihood
estimation
Bayes estimates under different loss functions; squared error (SE)
loss function and LINEX loss function based on Type-II censoring
scheme
The comparison of the different estimators have been obtained.
The informative and non-informative priors in different Loss
functions to compute the Bayes estimators of GR parameters.
One real data set has been presented.

Let denotes type-II censored observations from a sample
of r failure units under consideration and the other (n-r) items are
functioning till the end of experiment and they are censored.
The Likelihood function is
Methods of Estimation
Maximum Likelihood Estimation
The log likelihood function is
 2
( )
2( )
1 1
2 2( ; , ) ln ! ln( )! ln 2 ln 2 ln ln
1 1
( 1) ln 1 ( )ln
1
ix x
e
r r
L n n r r r r x xi i
i i
r
re n r
i



    
  
 
        
 
               
x

 To obtain the MLE’s
Approximate Confidence Intervals
2 2
ˆ ˆˆ ˆvar( ) & var( )z z     

Loss
Function
Expression of Loss
Function
Bayes Estimator
SELF 𝜃 − 𝜃∗ 2
𝐸 𝜃|𝐱(𝜃
LLF 𝑒 𝑐(𝜃∗−𝜃
− 𝑐(𝜃∗
− 𝜃 − 1
−
1
𝑐
ln 𝐸 𝜃|𝑥 𝑒−𝑐𝜃
Types of loss functions are given in the table.
Mean Square Errors
𝑀𝑆𝐸 𝜃 = 𝜃𝑖 − 𝜃
2
𝑘
𝑖=1
𝑘

We assumed the following joint density of proposed gamma
priors for α and λ
are the hyper-parameters.
(Berger and Sun, 1993; Kundu, 2008;
Wahed, 2006; Kundu and Pradhan, 2009;
Shrestha and Kumar, 2014).
Prior Distribution
1 2 1 2, , ,a a b b
1 2 2 11 1
( , ) a a b b
g e  
      

. For all the censoring schemes, we have used 𝛂 = 0.5 and 𝛌 = 1. First we considered the non-informative
prior for both  and, i.e 1 2 1 2 0a a b b   
.
In this case the priors becomes improper. We call this
prior as Prior-I. We have taken one informative priors, namely Prior-II: 1 2 1 21, 2.a a b b   

Bayesian Method of Estimation
The joint posterior density function of α and λ can be written as;
Bayes estimator using Lindley’s Approximation
the posterior expectation is expressible in the form of
ratio of integral as follow:
𝑢(𝛼, 𝜆 = is a function of α and λ only
𝐿(𝛼, 𝜆 = Log-Likelihood function and
𝐺(𝛼, 𝜆 = Log of joint prior density
1 2
1 2
0 0
( | , ) ( ) ( )
( , | )
( | , ) ( ) ( )
L g g
p
L g g d d
   
 
     
 


x
x
x
( , ) ( , )
( , ) ( , )
( , ) ( , )
( , )
( ) ( , | )
L G
L G
u e d
e d
I X E
   
   
   
 
 


  

x

 According to D.V Lindley (1980), if ML estimates of the parameters
are available and n is sufficiently large then the above ratio
integral can be approximated as:
 
   
   
  
 
ˆˆ ˆ2
ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ2 2
ˆˆ( ) , 0.5
ˆˆ ˆ ˆ ˆ2
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ
0.5
ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ ˆ
u u
u u u u
I X u
u u
u u L L L L
u u L L L L
  
      
   
          
         

   
 
  
     
    
 
 
  

  
 
  
   

    ˆ
 
 
 
 
 

 The Bayes estimator of under SELF is given by
The Bayes estimator of under SELF is given by
 Bayesian Estimation under Squared Error Loss Function

* 1
1
ˆ
1 ˆ ˆˆ ˆ ˆ ˆ ˆ( ) 0.5( ))]
ˆcSELF cMLE
a
b L L           



   

* 2
2
1ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ( ) 0.5( ))]
ˆcSELF cMLE
a
b L L            


    

Bayesian Estimation under LINEX Loss Function
 Bayesian Estimation 0f under LINEX Loss Function
 The Bayes estimator of under LLF is given by
 
*
2
2
ˆ
1
ˆ ˆ
ˆ1 2ˆ ln 1
1 ˆ ˆˆ ˆ ˆ
2
cLINEX MLE
a c
b
c
c
L L 
 
   


 

  
   
         
  
    

 
  
 
 
*
1
1
2
1
ˆ ˆ
ˆ1 2ˆ ln 1
1 ˆ ˆˆ ˆ ˆ
2
cLINEX MLE
a c
b
c
c
L L
 
   
 
 
  
   
         
  
    

 
  
 

Table.1 Average Estimates and the associated MSEs of the MLE and the
Bayes estimates of 𝛂 = 0.5 and 𝛌=1 (Prior-I)
Scheme MLE SELF LLF
20
10% ˆ
ˆ
0.65301 (0.03878)
1.39513 (0.12710)
0.64567 (0.03653)
1.38793 (0.12353)
0.63486 (0.03302)
1.37127 (0.11519)
20% 0.71793 (0.06550)
1.70383 (0.33302)
0.71101 (0.06427)
1.69512 (0.32576)
0.69384 (0.05509)
1.66787 (0.30276)
40
10% ˆ
ˆ
0.60527 (0.01720)
1.35927 (0.08579)
0.60246 (0.01666)
1.35564 (0.08430)
0.59803 (0.01583)
1.34767 (0.08109)
20% 0.65247 (0.02872)
1.65889 (0.25491)
0.64907 (0.02782)
1.65407 (0.25142)
0.64302 (0.02622)
1.64092 (0.24196)
80
10% ˆ
ˆ
0.58227 (0.00733)
1.33685 (0.06659)
0.58106 (0.00719)
1.33503 (0.06594)
0.57905 (0.00697)
1.33117 (0.06455)
20% 0.61971 (0.01261)
1.62560 (0.21309)
0.61822 (0.01238)
1.62311 (0.21146)
0.61561 (0.01197)
1.61674 (0.20728)
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
Note: For each scheme, first entry represents average estimates and MSE within Brackets

Table .2 Average Estimates and the associated MSEs of the MLE and the
Bayes estimates of 𝛂 = 0.5 and 𝛌=1 (Prior-II)
Scheme SELF LLF
20
10% ˆ
ˆ
0.61829 (0.02801)
1.34881 (0.10441)
0.61785 (0.02815)
1.34680 (0.10373)
20% 0.67057 (0.04516)
1.63244 (0.27397)
0.66975 (0.04523)
1.63003 (0.27283)
40
10% ˆ
ˆ
0.59163 (0.01467)
1.33706 (0.07692)
0.59095 (0.01457)
1.33564 (0.07639)
20% 0.63453 (0.02405)
1.62403 (0.23006)
0.63359 (0.02386)
1.62178 (0.22857)
80
10% ˆ
ˆ
0.57622 (0.00667)
1.32604 (0.06273)
0.57585 (0.00663)
1.32525 (0.06246)
20% 0.61198 (0.01142)
1.60859 (0.20200)
0.61150 (0.01135)
1.60726 (0.20115)
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ

The Findings of the Simulation Study
 The Lindley Bayes estimates under different loss functions
performance better than non-Bayesian estimates.
 The estimated values of the parameters converge to the
true values by increasing the sample size under both cases.
 The estimates under LLF are having the best convergence
among all loss functions.
 The informative priors provide better convergence than
non-informative priors.
 By increase the censoring rate the MSEs of estimates under
both (informative and noninformative) priors increases.

The data represent the number of million revolutions before failure for
each of the 23 ball bearings in the life test (Lawless; 1982, p.228).
Gupta and Kundu (2001) have analyzed this data set and GR distribution
works affectively.
Ball bearings lifetime data set
17.88, 28.92, 33.00, 41.52, 42.12, 45.60, 48.80, 51.84, 51.96, 54.12, 55.56, 67.80,
68.64, 68.64, 68.88, 84.12, 93.12, 98.64, 105.12, 105.84, 127.92, 128.04, 173.40
One-sample Kolmogorov-Smirnov test
data: x
D = 0.1572, p-value = 0.6208
alternative hypothesis: two-sided
The fit is good at 0.05 significance level.

Table 4. MLE and Bayes estimates with respect to different loss functions when Prior-I is used.
Estimates MLE
Lindley Bayes Estimates
SELF LLF
10% ˆ
ˆ
1.95066
0.01852
1.91456
0.01846
1.84184
0.01848
20% ˆ
ˆ
2.21545
0.02089
2.17826
0.02084
2.06717
0.02085
Table 5. MLE and Bayes estimates with respect to different loss functions when Prior-II is used.
Estimates MLE
Lindley Bayes Estimates
SELF LLF
10% ˆ
ˆ
1.95066
0.01852
1.72628
0.01846
1.74294
0.01848
20% ˆ
ˆ
2.21545
0.02089
1.89572
0.02083
1.93122
0.02085

 The Lindley Bayes estimates performance better than non-
Bayesian estimates (MLE).
 The performance of Bayes estimates under LLF is better than
SELF.
 By increase the censoring rate the estimates under both
(informative and noninformative) priors increases.
 Informative prior is better prior for the estimation of shape
and scale parameters of the GR distribution.
The Findings of the Real life Data

 The informative priors are superior to non-
informative priors.
 The Bayesian estimation can be preferred
than non-Bayesian estimation.
 The LLF provides the better convergence.

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