Abstract
The Bayes estimators of the shape parameter of exponentiated family of distributions have been derived by considering extension of Jeffreys′ noninformative as well as conjugate priors under different scale‐invariant loss functions, namely, weighted quadratic loss function, squared‐log error loss function and general entropy loss function. The risk functions of these estimators have been studied. We have also considered the highest posterior density (HPD) intervals for the parameter and the equal‐tail and HPD prediction intervals for future observation. Finally, we analyze one data set for illustration.
Highlights
Let X be a random variable whose cumulative distribution function cdf and probability density function pdf are given byG x; α, θ Fα x; θ, 1.1 g x; α, θ αFα−1 x; θ f x; θ, 1.2 respectively
X is said to be belonging to the exponentiated family of distributions abbreviated as EFD or the proportional reversed hazard family
A (i) (ii) (iii) devoted to study the behavioral patterns of the parameters of the generalized exponential distribution using both classical and Bayesian framework, and a very good summary of this work can be found in Gupta and Kundu 1–4, Raqab 5, Raqab and Ahsanullah 6, Zheng 7, Raqab and Madi 8, Alamm et al 9, Singh et al 10, Dey 11, and the references cited there for some recent developments on GE distribution
Summary
F ·, θ is the continuous baseline distribution function with the corresponding probability density function f x; θ , and θ may be vector valued, and α is the positive shape parameter. Gupta and Gupta 15 have shown that positively skewed data can be analyzed very well for normal baseline distribution.
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