parameters in an extension of the exponential distribution A - PowerPoint PPT Presentation

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

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

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

An Extension of Exponential Distribution Density function Hazard function

Estimation and Fitting Method of maximum likelihood n              1 1 (1 t ) l ( , ) (1 t ) e i i  i 1

Estimation and Fitting Method of maximum likelihood n              1 1 (1 t ) l ( , ) (1 t ) e i i  i 1 n n n             log(1 t ) (1 t ) log(1 t ) 0  i i i   i 1 i 1 n n n                1 1 ( 1) t (1 t ) t (1 t ) 0  i i i i   1 1 i i

Estimation and Fitting Bayes Estimator under SEL loss function               d 1 c b 1 a 2 ( ) e 1 ( ) e

Estimation and Fitting Bayes Estimator under SEL loss function               d 1 c b 1 a 2 ( ) e 1 ( ) e n              1 1 (1 t ) ( , ) (1 ) l t e i i  i 1       l ( , ) ( ) ( )     1 2 ( , data )            l ( , ) ( ) ( ) d d 1 2 0 0

Estimation and Fitting Bayes Estimator under SEL loss function ˆ    E ( T ) B               g ( , ) ( l , ) ( ) ( ) d d 1 2    0 0 E g ( ( , ) T t )            l ( , ) ( ) ( ) d d 1 2 0 0

Lindley’s procedure     L ( )       w ( ) e d ( )   L ( ) ( ) g ( ) e d ( )    I E g ( ( t ))      L ( )  ( ) e d ( )      L ( ) ( ) e d ( )      w ( ) ( ) g ( )      ( ) ln ( ( ) )

Lindley’s procedure 1 1   ˆ ˆ ˆ ˆ ˆ ˆ               I g ( ) [ g ( ) 2 g ( ) ( )] L ( ) g ( ) ij i j ij ijk L ij kL 2 2 ij ijkL On MLE point

The approximate Bayes estimators of 𝛍 , under Lindley’s procedure ( both parameters are unknown ) ˆ g        ( ) ( ) E T t B

The approximate Bayes estimators of 𝛍 , under Lindley’s procedure ( both parameters are unknown ) 1 1 ˆ ˆ ˆ           ˆ ˆ ˆ ˆ ˆ 2 I L L 1 11 11 111 11 22 221 2 2

The approximate Bayes estimators of 𝛍 , under Lindley’s procedure ( both parameters are unknown ) 1 1 ˆ ˆ ˆ           ˆ ˆ ˆ ˆ ˆ 2 I L L 1 11 11 111 11 22 221 2 2

The approximate Bayes estimators of α , under Lindley’s procedure ( both parameters are unknown )     g    ˆ E ( T t ) ( ) B

The approximate Bayes estimators of α , under Lindley’s procedure ( both parameters are unknown ) 1 ˆ ˆ           ˆ ˆ ˆ ˆ ˆ ˆ I ( L L ) 2 22 22 211 11 222 22 2

The approximate Bayes estimators of α , under Lindley’s procedure ( both parameters are unknown ) 1 ˆ ˆ           ˆ ˆ ˆ ˆ ˆ ˆ I ( L L ) 2 22 22 211 11 222 22 2

The approximate Bayes estimators of 𝛍 , under Lindley’s procedure ( α is known ) 1 1 ˆ ˆ         ˆ ˆ ˆ ˆ ˆ ˆ I g ( ) [( g 2 g ) ] g L 11 1 1 11 1 11 111 2 2 1 ˆ ˆ        ˆ ˆ ˆ 2 I L 1 11 11 111 2

The approximate Bayes estimators of 𝛍 , under Lindley’s procedure ( α is known ) 1 1 ˆ ˆ         ˆ ˆ ˆ ˆ ˆ ˆ I g ( ) [( g 2 g ) ] g L 11 1 1 11 1 11 111 2 2 1 ˆ ˆ        ˆ ˆ ˆ 2 I L 1 11 11 111 2   b 1 a ˆ  ˆ    I 2 n n n t              2 2 i ( 1) t (1 t ) ( 1)    i i 2 2 (1 t )   i 1 i 1 i 3 n n 2 t 2 n                3 3 i ( 1)( 2) t (1 t ) ( 1)    i i 3 3 (1 t )    i 1 i 1 i 2 n n n t              2 2 2 i 2[ ( 1) t (1 t ) ( 1) ]    i i 2 2 (1 t )   i 1 i 1 i

The approximate Bayes estimators of α , under Lindley’s procedure ( 𝛍 is known ) 1 ˆ 1         ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 I [ g 2 g ] L g 22 2 2 22 222 2 22 2 2

The approximate Bayes estimators of α , under Lindley’s procedure ( 𝛍 is known ) 1 ˆ 1         ˆ ˆ ˆ ˆ ˆ ˆ ˆ 2 I [ g 2 g ] L g 22 2 2 22 222 2 22 2 2   d 1 a  ˆ    ˆ I n n        2 (1 t ) ln (1 t )  i i 2  i 1 n 2 n        3 (1 t ) ln (1 t )  i i 3   i 1 2 n n        2 2 [ (1 t ) ln (1 t ) ]  i i 2  i 1

The approximate Bayes estimators of parameters, with MCMC method ( Gibbs sampler ) With joint posterior density function of 𝛍 and α : n      (1 ) n n t  i                   n d 1 n b 1 1 a c  ( , data ) e (1 t ) e i 1 i  i 1

The approximate Bayes estimators of parameters, with MCMC method ( Gibbs sampler ) posterior density function of α given 𝛍 : n n            (1 t ) ( c ln(1 t ) i i        n d 1   ( , data ) e e i 1 i 1 posterior density function of 𝛍 given α : n n               1 (1 t ) ( 1) (1 t ) a i i        n b 1   ( , data ) e e i 1 i 1

The approximate Bayes estimators of parameters, with MCMC method ( Gibbs sampler )  start with α ₀ as initial value for α  generate 𝛍 ₁ using π ( 𝛍 │α= α ₀ )  generate α ₁ using π(α│ 𝛍 = 𝛍 ₁ )

The approximate Bayes estimators of parameters, with MCMC method ( Gibbs sampler )  start with α ₀ as initial value for α  generate 𝛍 ₁ using π ( 𝛍 │α= α ₀ )  generate α ₁ using π(α│ 𝛍 = 𝛍 ₁ )

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

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

Numerical Comparisons compute approximated Bayes estimators using Bayes estimators Lindley’s approximation Under non- informative priors on both α and 𝛍 Comparing MLE estimators average estimates (AE) square root of the mean squared error (RMS)

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