Research Article

Bayesian Inference on the Shape Parameter and Future Observation of Exponentiated Family of Distributions

Table 1

(a) The Bayes estimates of the shape parameter under extended prior and the corresponding risk function under three different loss functions for the data set

𝑐 1 𝛼 𝐸 b q 𝑅 ( 𝛼 𝐸 b q ) 𝛼 𝐸 b s l 𝑅 ( 𝛼 𝐸 b s l ) 𝛾 = 0 . 5 𝛾 = 1 . 0 𝛾 = 1 . 5
𝛼 𝐸 b g e 𝑅 ( 𝛼 𝐸 b g e ) 𝛼 𝐸 b g e 𝑅 ( 𝛼 𝐸 b g e ) 𝛼 𝐸 b g e 𝑅 ( 𝛼 𝐸 b g e )

0.54.6070.04544.9360.04444.8810.00554.8260.02254.7710.0511
1.04.3870.04764.7170.04654.6620.00584.6070.02364.5510.0535
1.54.1680.05414.4980.05314.4430.00664.3870.02694.3320.0611
2.03.9490.06494.2780.06494.2230.00814.1680.03274.1130.0741
5.02.6320.2207 2.9620.30522.9070.03642.8520.13952.7960.3012

(b) The Bayes estimates of the shape parameter under conjugate prior and the corresponding risk function under three different loss functions for the data set

𝛼 𝑐 b q 𝑅 ( 𝛼 𝑐 b q ) 𝛼 𝑐 b s l 𝑅 ( 𝛼 𝑐 b s l ) 𝛾 = 0 . 5 𝛾 = 1 . 0 𝛾 = 1 . 5
𝛼 𝑐 b g e 𝑅 ( 𝛼 𝑐 b g e ) 𝛼 𝑐 b g e 𝑅 ( 𝛼 𝑐 b g e ) 𝛼 𝑐 b g e 𝑅 ( 𝛼 𝑐 b g e )

4.9360.05035.1010.04555.0730.42815.0461.06915.018 2.0264