Abstract
Bayesian estimates involve the selection of hyper-parameters in the prior distribution. To deal with this issue, the empirical Bayesian and E-Bayesian estimates may be used to overcome this problem. The first one uses the maximum likelihood estimate (MLE) procedure to decide the hyper-parameters; while the second one uses the expectation of the Bayesian estimate taken over the joint prior distribution of the hyper-parameters. This study focuses on establishing the E-Bayesian estimates for the Lomax distribution shape parameter functions by utilizing the Gamma prior of the unknown shape parameter along with three distinctive joint priors of Gamma hyper-parameters based on the square error as well as two asymmetric loss functions. These two asymmetric loss functions include a general entropy and LINEX loss functions. To investigate the effect of the hyper-parameters’ selections, mathematical propositions have been derived for the E-Bayesian estimates of the three shape functions that comprise the identity, reliability and hazard rate functions. Monte Carlo simulation has been performed to compare nine E-Bayesian, three empirical Bayesian and Bayesian estimates and MLEs for any aforementioned functions. Additionally, one simulated and two real data sets from industry life test and medical study are applied for the illustrative purpose. Concluding notes are provided at the end.
Highlights
The squared error loss (SEL) function as well as two asymmetric loss functions, which contain the general entropy (GE) from Calabria and Pulcini [25] and LINEX from Pandey et al [26], will be utilized to investigate Bayesian, E-Bayesian and empirical Bayesian estimates for any function of the Lomax(α, β) shape parameter based on the adaptive type-I progressively hybrid censored sample
The simulated data set used for Example 1 and all adaptive type-I progressively hybrid censored samples used for all examples were generated by using R that is available from author on request
The E-Bayesian and empirical Bayesian methods for any function of Lomax(α, β) shape have been developed by utilizing the SEL, GE loss and LINEX loss functions
Summary
Lomax(α, β), if its probability density function (PDF) and cumulative distribution function (CDF) are respectively defined as with regard to jurisdictional claims in f ( x; α, β) published maps and institutional affiliations. All survival items will be removed at the respective terminated random time when the progressive hybrid censoring schemes are implemented. The AT-IIP HCS, discussed by Ng et al [11] and Balakrishnan and Kundu [12], implements progressive type-II scheme until Xm:n and has no survival items removed after the life test experiment passes τ (< Xm:n ). The SEL function as well as two asymmetric loss functions, which contain the general entropy (GE) from Calabria and Pulcini [25] and LINEX from Pandey et al [26], will be utilized to investigate Bayesian, E-Bayesian and empirical Bayesian estimates for any function of the Lomax(α, β) shape parameter based on the adaptive type-I progressively hybrid censored sample.
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