Abstract
This paper has to do with 3-component mixture of the Frechet dis- tributions when the shape parameter is known under Bayesian view point. The type-I right censored sampling scheme is considered due to its extensive use in reliability theory and survival analysis. Taking dif- ferent non-informative and informative priors, Bayes estimates of the parameter of the mixture model along with their posterior risks are derived under squared error loss function, precautionary loss function and DeGroot loss function. In case, no or little prior information is available, elicitation of hyper parameters is given. In order to study numerically, the execution of the Bayes estimators under different loss functions, their statistical properties have been simulated for different sample sizes and test termination times. A real life data example is also given to illustrate the study.
Highlights
Frechet distribution was introduced by a French mathematician named Maurice Frechet (1878, 1973) who had determined before one possible limit distribution for the largest order statistic in 1927
The Bayes estimators and posterior risks using the Uniform Prior (UP), the Jeffreys’ prior (JP) and IP for parameters β1, β2, β3, p1 and p2 under squared error loss function (SELF) are obtained with their respective marginal posterior distributions are given below: where v=1 for the UP, v=2 for the JP, v=3 for the Exponential prior (EP) and v=4 for the Inverse Levy prior (ILP)
The limiting expressions for Bayes estimators and posterior risks using the UP, the JP, the EP and the ILP under SELF, precautionary loss function (PLF) and DeGroot loss function (DLF) are given in the Tables 1-6
Summary
Frechet distribution was introduced by a French mathematician named Maurice Frechet (1878, 1973) who had determined before one possible limit distribution for the largest order statistic in 1927. Posterior distributions of parameters given data, say x, are derived using the non-informative (Uniform and Jeffreys’) and the informative (Exponential and Inverse Levy) priors. 4. Bayes estimators and posterior risks using the UP, the JP, the Exponential and Inverse Levy prior under SELF, PLF and DLF. The Bayes estimators and posterior risks using the UP, the JP and IP for parameters β1, β2, β3, p1 and p2 under SELF are obtained with their respective marginal posterior distributions are given below: where v=1 for the UP, v=2 for the JP, v=3 for the EP and v=4 for the ILP. The limiting (complete sample) expressions for Bayes estimators and posterior risks using the UP, the JP, the EP and the ILP under SELF, PLF and DLF are given in the Tables 1-6
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