Estimation of reliability in a multicomponent stress-strength model for inverted exponentiated Rayleigh distribution und
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Estimation of reliability in a multicomponent stress‑strength model for inverted exponentiated Rayleigh distribution under progressive censoring Amulya Kumar Mahto1 · Yogesh Mani Tripathi1 Accepted: 7 May 2020 © Operational Research Society of India 2020
Abstract We consider estimation of the multicomponent stress-strength reliability for inverted exponentiated Rayleigh distributions under progressive Type II censoring. It is assumed that stress and strength variables follow inverted exponentiated Rayleigh distributions with a common scale parameter. Point and interval estimates of the reliability are obtained using maximum likelihood and Bayesian approaches when common parameter is unknown. Bayes estimates are derived using Lindley approximation and Markov chain Monte Carlo methods. The case of known common parameter is also considered. Then uniformly minimum variance unbiased estimator of the reliability is derived. We have also computed the exact Bayes estimates under the squared error loss function. The asymptotic and HPD intervals of the reliability are constructed under this case also. Proposed methods are compared numerically using simulations and comments are obtained. Finally, a real data set is analyzed for illustration purposes. Keywords Bayes estimate · HPD interval · Lindley method · Maximum likelihood estimate · Multicomponent stress-strength model Mathematics Subject Classification 62F10 · 62F15 · 62N02
* Yogesh Mani Tripathi [email protected] Amulya Kumar Mahto [email protected] 1
Department of Mathematics, Indian Institute of Technology Patna, Bihta 801103, India
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1 Introduction In many life testing experiments, we often need to derive inferences for multicomponent systems. The various components of such a system can be arranged in series, parallel or some hybrid combinations. Many examples of multicomponent systems abound in practice such as hanging bridges, V-8 engine cars, airplanes with multiple engines, electronic equipment etc. The study of stress-strength model is initially discussed in Birnbaum [1] and Birnbaum and MaCarty [2]. Some useful references on stress-strength reliability estimation are Kundu and Gupta [3], Hussian [4], Nadar and Kizilaslan [5]. In reliability context, a stress-strength system fails provided the stress X applied to the system exceeds the strength Y of the system under consideration and R = P(X < Y) is referred as the reliability of the system. Kotz et al. [6] have discussed several applications of stress-strength models. We also refer to Ghitany et al. [7], Yadav et al. [8], Alizadeh et al. [9], Kumari et al. [10] for some recent results on this topic. Bhattacharyya and Johnson [11] initially studied the problem of estimating stressstrength reliability in a multicomponent system. A multicomponent system with k independent strength components and with a common stress component functions properly if at least s (1 ≤ s ≤ k) out of these k strength components operate simultaneously. Such systems are commonly referre
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