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A simulation study of the Bayes estimator of parameters in an extension of the exponential distribution

Samira Sadeghi A simulation study of the Bayes estimator of parameters in an extension of the exponential distribution

An Extension of Exponential Distribution

The two-parameter extension of Exponential distribution The three-parameter Power Generalized Weibull distribution, introduced by Nikulin and Haghighi (2006).

Density function

An Extension of Exponential Distribution

Density function Hazard function

Estimation and Fitting

Method of maximum likelihood

1 (1 ) 1 1

( , ) (1 )

i

n t i i

l t e



 

   

   

 



Estimation and Fitting

1 1 1 1 1 1

log(1 ) (1 ) log(1 ) 0 ( 1) (1 ) (1 )

n n i i i i i n n i i i i i i

n t t t n t t t t

 

        

     

           

   

Method of maximum likelihood

1 (1 ) 1 1

( , ) (1 )

i

n t i i

l t e



 

   

   

 



Estimation and Fitting

Bayes Estimator under SEL loss function

1 1 ( ) b a

e



   





1 2 ( ) d c

e



  

 



Estimation and Fitting

Bayes Estimator under SEL loss function

1 (1 ) 1 1

( , ) (1 )

i

n t i i

l t e



 

   

   

 



1 2 1 2 0 0

( , ) ( ) ( ) ( , ) ( , ) ( ) ( ) l data l d d                 

 





1 1 ( ) b a

e



   





1 2 ( ) d c

e



  

 



Estimation and Fitting

ˆ ( )

B

E T   

1 2 0 0 1 2 0 0

( , ) ( , ) ( ) ( ) ( ( , ) ) ( , ) ( ) ( ) g l d d E g T t l d d                    

   

  



Bayes Estimator under SEL loss function

Lindley’s procedure

( ) ( )

( ) ( ) ( ) ( )

L L

w e d e d

 

     

 

( ) ( ) ( ) ( )

( ) ( ) ( ( )) ( )

L L

g e d I E g t e d

     

   

 

  



( ) ( ) ( ) ( ) ln ( ( ) ) w g          

Lindley’s procedure

1 1 ˆ ˆ ˆ ˆ ˆ ˆ ( ) [ ( ) 2 ( ) ( )] ( ) ( ) 2 2

ij i j ij ijk L ij kL ij ijkL

I g g g L g              

 

On MLE point

The approximate Bayes estimators of 𝛍 , under Lindley’s procedure (both parameters are unknown)

ˆ ( )

B

E T t     ( ) g   

The approximate Bayes estimators of 𝛍 , under Lindley’s procedure (both parameters are unknown)

2 1 11 11 111 11 22 221

1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 2 I L L          

The approximate Bayes estimators of 𝛍 , under Lindley’s procedure (both parameters are unknown)

2 1 11 11 111 11 22 221

1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 2 I L L          

The approximate Bayes estimators of α , under Lindley’s procedure (both parameters are unknown)

ˆ ( )

B

E T t     ( ) g   

The approximate Bayes estimators of α , under Lindley’s procedure (both parameters are unknown)

2 22 22 211 11 222 22

1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( ) 2 I L L          

The approximate Bayes estimators of α , under Lindley’s procedure (both parameters are unknown)

2 22 22 211 11 222 22

1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( ) 2 I L L          

The approximate Bayes estimators of 𝛍 , under Lindley’s procedure (α is known)

11 1 1 11 1 11 111

1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( ) [( 2 ) ] 2 2 I g g g g L        

2 1 11 11 111

1 ˆ ˆ ˆ ˆ ˆ 2 I L       

The approximate Bayes estimators of 𝛍 , under Lindley’s procedure (α is known)

2 2 2 2 2 1 1 3 3 3 3 3 1 1 2 2 2 2 2 2 1 1

1 ˆ ˆ ( 1) (1 ) ( 1) (1 ) 2 2 ( 1)( 2) (1 ) ( 1) (1 ) 2[ ( 1) (1 ) ( 1) ] (1 )

n n i i i i i i n n i i i i i i n n i i i i i i

b a I t n t t t t n t t t t n t t t

  

                    

        

                       

     

11 1 1 11 1 11 111

1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ( ) [( 2 ) ] 2 2 I g g g g L        

2 1 11 11 111

1 ˆ ˆ ˆ ˆ ˆ 2 I L       

The approximate Bayes estimators of α , under Lindley’s procedure (𝛍 is known)

2 22 2 2 22 222 2 22

1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [ 2 ] 2 2 I g g L g        

The approximate Bayes estimators of α , under Lindley’s procedure (𝛍 is known)

2 2 1 3 3 1 2 2 2 1

1 ˆ ˆ (1 ) ln (1 ) 2 (1 ) ln (1 ) 2 [ (1 ) ln (1 ) ]

n i i i n i i i n i i i

d a I n t t n t t n t t

  

          

  

             

  

2 22 2 2 22 222 2 22

1 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ [ 2 ] 2 2 I g g L g        

The approximate Bayes estimators of parameters, with MCMC method (Gibbs sampler)

With joint posterior density function of 𝛍 and α :

1

(1 ) 1 1 1 1

( , ) (1 )

n i i

n n t n d n b a c i i

data e t e



   

     



         

  



The approximate Bayes estimators of parameters, with MCMC method (Gibbs sampler)

posterior density function of α given 𝛍 : posterior density function of 𝛍 given α :

1 1

(1 ) ( ln(1 ) 1

( , )

n n i i i i

t c t n d

data e e



  

   

 

      

  

1 1 1

(1 ) ( 1) (1 ) 1

( , )

n n i i i i

t t a n b

data e e

 

   

   

  

      

  

The approximate Bayes estimators of parameters, with MCMC method (Gibbs sampler)

 start with α₀ as initial value for α  generate 𝛍₁ using π(𝛍 │α= α₀ )  generate α₁ using π(α│ 𝛍 = 𝛍₁ )

The approximate Bayes estimators of parameters, with MCMC method (Gibbs sampler)

 start with α₀ as initial value for α  generate 𝛍₁ using π(𝛍 │α= α₀ )  generate α₁ using π(α│ 𝛍 = 𝛍₁ )

Numerical Comparisons

Comparing

Under non-informative priors on both α and 𝛍 compute approximated Bayes estimators using Lindley’s approximation

Bayes estimators

Numerical Comparisons

Comparing

Under non-informative priors on both α and 𝛍 compute approximated Bayes estimators using Lindley’s approximation

Bayes estimators MLE estimators

Numerical Comparisons

Comparing

Under non-informative priors on both α and 𝛍 compute approximated Bayes estimators using Lindley’s approximation

average estimates (AE) square root of the mean squared error (RMS) Bayes estimators MLE estimators

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of α when 𝛍 is known

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of α when 𝛍 is known

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of 𝛍 when α is known

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of 𝛍 when α is known

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of 𝛍 when α is known

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of α , 𝛍 when both are unknown

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of α , 𝛍 when both are unknown

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of α , 𝛍 when both are unknown

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of α , 𝛍 when both are unknown

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of α , 𝛍 when both are unknown

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of α , 𝛍 when both are unknown

The (AE) and (RMS) for the MLE’s and the approximate Bayes estimate of α , 𝛍 when both are unknown

Data Analysis

Linhart and Zucchini (1986) The failure times of the air conditioning system of an airplane Data set: 23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95

Parameter estimations, Kolmogorov-Smirnov and Chi-squared statistics for the data set

The estimated density function by different methods

Comparing different methods

The estimated density function by different methods The estimated cumulative distribution functions and empirical distribution function

Comparing different methods

Thank you

KUNDU, D., and GUPTA , R. D. (2008): Generalized exponential distribution: Bayesian estimations. Computational Statistics & Data Analysis 52, 1873–1883. LINDLEY, D. V. (1980): Approximate Baysian method. Trabajos Estadist 31,223-237. LINHART, H., and ZUCCHINI, W. (1986): Model Selection. Wiley, New York. NADARAJAH, S., and Haghighi, F. (2009): An extension of the exponential distribution, Statistics

References: