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
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