Estimation of reliability of multicomponent stress-strength inverted exponentiated Rayleigh model

ABSTRACT

In this paper, we consider the estimation of the multicomponent reliability by assuming the inverted exponentiated Rayleigh distribution. Both stress and strength are assumed to have an inverted exponentiated Rayleigh distribution with common scale parameter. The random variable $Y$ representing the stress experienced by the system and $X$ representing the strength of system available to overcome the stress. The system works flawlessly only if at least $s$ out of $k$ $(1\le s\le k)$ strength variables exceed the random stress. The multicomponent reliability of the system is given by $R_{s,k}=P(\mathrm{a}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{t}s\mathrm{o}\mathrm{f}(X_1,X_2,\dots ,X_k)\mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{e}\mathrm{d}Y)$. We estimate $R_{s,k}$ by using frequentist and Bayesian approaches. Bayes estimates of $R_{s,k}$ have been obtained by using Markov Chain Monte Carlo methods since joint posteriors of the parameters does not have the explicit forms. We also construct asymptotic and highest probability density credible intervals for $R_{s,k}$. The behavior of the proposed estimators is studied on the basis of estimated risks through Monte Carlo simulations. Finally, a data set is analyzed for illustrative purposes.

Abbreviations: PDF: Probability Density Function; CDF: Cumulative Density Function; IER: Inverted Exponentiated Rayleigh; MLE: Maximum likelihood estimators; HPD: Highest Posterior Density; UBT: Upside down Bathtub; HNC: Head and Neck Cancer data; MCMC: Monte-Carlo Markov Chains; CP: Coverage Probability; KS: Kolmogorov - Smirnov.

KEYWORDS:

• Inverted exponentiated Rayleigh distribution
• Bayesian estimation
• maximum likelihood estimation
• multicomponent stress-strength reliability

Acknowledgments

The authors would like to thank the Referees and the Editor for several helpful comments which had improved the earlier versions of the manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Vikas Kumar Sharma

Dr. Vikas Kumar Sharma did M.Sc. and M.Phil. in statistics from CCSU, Meerut, and Ph.D. from Banaras Hindu University, Varanasi. He received Vice Chancellor Gold Medal in M.Phil. in 2011 and UGC-J.R.F. in 2012. He has six years of Teaching experience as an assistant professor. Presently, Dr. Sharma is working at IITRAM, Ahmedabad. He has published more than 25 research articles in international journals of repute. His research interest belongs to the parametric Bayesian Inference and distribution theory.

Sanku Dey

Dr. Sanku Dey He is currently working as an Associate Professor in the Department of Statistics, St. Anthony’s College, Shillong, Meghalaya, India. He did his M.Sc. in Statistics in the year of 1991 from Gauhati University, Guwahati, India and Ph.D. in Statistics (reliability theory) in the year 1998 from the same university. He has published more than 160 research papers in national and international journals of repute. He reviewed more than 300 research papers for various well reputed international journals. He is an associate editor of American Journal of Mathematical and Management Sciences and also the member of editorial board of several national and international journals of repute. He is a renowned researcher and has a good number of contributions in almost all fields of Statistics viz, distribution theory, discretization of continuous distribution, reliability theory, multicomponent stress-strength reliability, survival analysis, Bayesian inference, Record Statistics, Statistical quality control, order statistics, lifetime performance index based on classical and Bayesian approach as well as different types of censoring schemes etc.