Abstract
Based on progressively Type-II censored samples, this paper deals with the estimation of multicomponent stress-strength reliability by assuming the Kumaraswamy distribution. Both stress and strength are assumed to have a Kumaraswamy distribution with different the first shape parameters, but having the same second shape parameter. Different methods are applied for estimating the reliability. The maximum likelihood estimate of reliability is derived. Also its asymptotic distribution is used to construct an asymptotic confidence interval. The Bayes estimates of reliability have been developed by using Lindley’s approximation and the Markov Chain Monte Carlo methods due to the lack of explicit forms. The uniformly minimum variance unbiased and Bayes estimates of reliability are obtained when the common second shape parameter is known. The highest posterior density credible intervals are constructed for reliability. Monte Carlo simulations are performed to compare the performances of the different methods, and one data set is analyzed for illustrative purposes.
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References
Baklizi A (2007) Inference about the mean difference of two non-normal populations based on independent samples: a comparative study. J Stat Comput Simul 77:613–623
Balakrishnan N, Aggarwala R (2000) Progressive censoring: theory, methods and applications. Birkhauser, Boston
Bhattacharyya GK, Johnson RA (1974) Estimation of reliability in multicomponent stress–strength model. J Am Stat Assoc 69:966–970
Birnbaum ZW (1956) On a use of Mann-Whitney statistics. In: Proceedings of the third Berkley Symposium in Mathematics, Statistics and Probability, vol. 1. University of California Press, Berkeley, pp 13–17
Cao JH, Cheng K (2006) An introduction to the reliability mathmematics. Higher Education Press, Beijing
Chen MH, Shao QM (1999) Monte Carlo estimation of Bayesian Credible and HPD intervals. J Comput Gr Stat 8:69–92
Dey S, Mazucheli J, Anis MZ (2016) Estimation of reliability of multicomponent stress–strength for a Kumaraswamy distribution. Commun Stat 46(4):1560–1572. doi:10.1080/03610926.2015.1022457
Efron B (1982) The jackknife, the bootstrap and other re-sampling plans. SIAM, CBMSNSF regional conference series in applied mathematics, No. 34, Philadelphia
Gelman A, Carlin JB, Stern HS, Rubin DB (2003) Bayesian data analysis. Chapman Hall, London
Gradshteyn IS, Ryzhik IM (1994) Table of integrals, series, and products, 5th edn. Academic Press, Boston
Hall P (1988) Theoretical comparison of bootstrap confidence intervals. Ann Stat 16:927–953
Hanagal DD (1999) Estimation of system reliability. Stat Pap 40:99–106
Kizilaslan F, Nadar, M (2016) Estimation of reliability in a multicomponent stress-strength model based on a bivariate Kumaraswamy distribution. Stati Pap. doi:10.1007/s00362-016-0765-8
Kizilaslan F, Nadar M (2016) Estimation and prediction of the Kumaraswamy distribution based on record values and inter-record times. J Stat Comput Simul 86:2471–2493
Kotz S, Lumelskii Y, Pensky M (2003) The stress-strength model and its generalization: theory and applications. World Scientific, Singapore
Kundu D, Raqab MZ (2009) Estimation of \(R = P(Y < X)\) for three parameter Weibull distribution. Stat Probab Lett 79:1839–1846
Lindley DV (1980) Approximate Bayesian methods. Trab de Estad 3:281–288
Lio YL, Tsai TR (2012) Estimation of \(\delta = P(X < Y)\) for Burr XII distribution based on the progressively first failure-censored samples. J Appl Stat 39:309–322
Nadar M, Papadopoulos A, Kizilaslan F (2013) Statistical analysis for Kumaraswamy’s distribution based on record data. Stat Pap 54:355–369
Nadar M, Kizilaslan F, Papadopoulos A (2014) Classical and Bayesian estimation of \(P(Y<X)\) for Kumaraswamy’s distribution. J Stat Comput Simul 84:1505–1529
Nadar M, Kizilaslan F (2016) Estimation of reliability in a multicomponent stress–strength model based on a Marshall–Olkin bivariate weibull Distribution. IEEE Trans Reliab 65:370–380
Rao GS (2012) Estimation of reliability in multicomponent stress–strength model based on generalized exponential distribution. Rev Colomb de Estad 35:67–76
Rao GS (2012) Estimation of reliability in multicomponent stress–strength model based on generalized inverted exponential distribution. Int J Curr Res Rev 4:48–56
Rao GS (2012) Estimation of reliability in multicomponent stress–strength model based on Rayleigh distribution. Prob Stat Forum 5:150–161
Rao GS (2013) Estimation of reliability in multicomponent stress–strength model based on inverse exponential distribution. Int J Stat Econ 10:28–37
Rao GS (2014) Estimation of reliability in multicomponent stress–strength model based on generalized Rayleigh distribution. J Mod Appl Stat Methods 13:367–379
Rao GS, Kantam RRL (2010) Estimation of reliability in multicomponent stress–strength model: log-logistic distribution. Electron J Appl Stat Anal 3:75–84
Rao GS, Kantam RRL, Rosaiah K, Reddy JP (2013) Estimation of reliability in multicomponent stress–strength model based on inverse Rayleigh distribution. J Stat Appl Probab 2:261–267
Rao GS, Aslam M, Kundu D (2015) Burr Type XII distribution parametric estimation and estimation of reliability of multicomponent stress-strength. Commun Stat 44:4953–4961
Raqab MZ, Madi MT, Kundu D (2008) Estimation of \(R = P(Y < X)\) for the 3-parameter generalized exponential distribution. Commun Stat 37:2854–2864
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Kohansal, A. On estimation of reliability in a multicomponent stress-strength model for a Kumaraswamy distribution based on progressively censored sample. Stat Papers 60, 2185–2224 (2019). https://doi.org/10.1007/s00362-017-0916-6
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DOI: https://doi.org/10.1007/s00362-017-0916-6
Keywords
- Kumaraswamy distribution
- Progressive Type-II censoring
- Multicomponent stress-strength
- Maximum likelihood estimator
- Bayesian estimator
- Monte Carlo simulation