Bayesian Estimation Based on Sequential Order Statistics for Heterogeneous Baseline Gompertz Distributions

Tzong-Ru Tsai,Hua Xin,Chiun-How Kao

doi:10.3390/math9020145

Tzong-Ru Tsai, Hua Xin + Show 1 more

Open Access

https://doi.org/10.3390/math9020145

Copy DOI

Journal: Mathematics	Publication Date: Jan 11, 2021
Citations: 2	License type: CC BY 4.0

Affiliation: Tamkang University, Northeast Petroleum University

Abstract

A composite dynamic system (CDS) is composed of multiple components. Each component failure can equally induce higher loading on the surviving components and, hence, enhances the hazard rate of each surviving component. The applications of CDS and the reliability evaluation of CDS has earned more attention in the recent two decades. Because the lifetime quality of components could be inconsistent, the lifetimes of components in the CDS is considered to follow heterogeneous baseline Gompertz distributions in this study. A power-trend hazard rate function is used in order to characterize the hazard rate of the CDS. In order to overcome the difficulty of obtaining reliable estimates of the parameters in the CDS model, the Bayesian estimation method utilizing a hybrid Gibbs sampling and Metropolis-Hasting algorithm to implement the Markov chain Monte Carlo approach is proposed for obtaining the Bayes estimators of the CDS parameters. An intensive simulation study is carried out to evaluate the performance of the proposed estimation method. The simulation results show that the proposed estimation method is reliable in providing reliability evaluation information for the CDS. An example regarding the service system of small electric carts is used for illustration.

Highlights

We proposed a hGSMH-Markov chain Monte Carlo (MCMC) method in order to obtain the BEs of the composite dynamic system (CDS) parameters when the lifetimes of components follow a Gompertz distribution with a heterogeneity condition
The hazard rate of the CDS is characterized by the power-trend hazard rate function
A Bayesian estimation method using the MCMC approach is proposed in order to overcome the difficulty during obtaining reliable estimates of the parameters in the CDS model

Summary

Literature Review

The composite dynamic system (CDS) is composed of n(≥ 1) identical components. The components fail one-by-one over time until r failures are observed (r ≤ n), and the CDS is defined as malfunctioned. Cramer and Kamps [3] investigated the maximum likelihood estimation method, uniformly minimal variance unbiased estimation method, and best linear unbiased estimation method based on SOS samples They used these three estimation methods in order to obtain the estimates of parameters in the exponential distribution. Balakrishnan, Beutner, and Kamps [14] studied the impact of using different link functions to model parameters in obtaining maximum likelihood estimates that are based on SOS samples. Cramer, and Górny [19] studied the parameter inference for type I censoring sequential CDSs. Sutar and Naik-Nimbalkar [20] modeled the load-sharing phenomenon in a CDS under the accelerated failure time model. Hashempour and Doostparast [26] studied Bayesian inference methods on multiple SOS samples for heterogeneous exponential distributions They studied using the generalized likelihood ratio test in order to test the homogeneity property. Baratnia and Doostparast [27] proposed an extension of SOS to model the system lifetimes with independent, but heterogeneous, components for the distribution family that was studied by Burkschat and Navarro [16]

Motivation and Organization

The Statistical Model

An Example

Conclusions