Estimation and Prediction for Flexible Weibull Distribution Based on Progressive Type II Censored Data
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Estimation and Prediction for Flexible Weibull Distribution Based on Progressive Type II Censored Data O. M. Bdair1 · R. R. Abu Awwad2 · G. K. Abufoudeh2 · M. F. M. Naser1 Received: 30 August 2018 / Revised: 25 September 2018 / Accepted: 27 December 2018 © School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this work, we consider the problem of estimating the parameters and predicting the unobserved or removed ordered data for the progressive type II censored flexible Weibull sample. Frequentist and Bayesian analyses are adopted for conducting the estimation and prediction problems. The likelihood method as well as the Bayesian sampling techniques is applied for the inference problems. The point predictors and credible intervals of unobserved data based on an informative set of data are computed. Markov Chain Monte Carlo samples are performed to compare the so-obtained methods, and one real data set is analyzed for illustrative purposes. Keywords Flexible Weibull distribution · Progressive censoring data · Bayes estimation · Bayes prediction · Gibbs sampling · Simulation Mathematics Subject Classification 62N02 · 62N01 · 62G30 · 62F15
1 Introduction Over the last four decades, thousands of papers dealing with various extensions of the Weibull distribution and their applications have been proposed to enhance the Weibull distribution’s capability to fit diverse lifetime data. A common factor among these generalized models is that the Weibull distribution is a special case of theirs. Either the distribution function F(t) or the hazard rate function h(t) of these modified models are related to the corresponding function of the Weibull distribution in someway. Although the hazard rate functions h(t) of these distributions are more capable to model diverse problems than the Weibull model does, these distributions usually do not have simple
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M. F. M. Naser [email protected]
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Faculty of Engineering Technology, Al-Balqa Applied University, Amman, Jordan
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Department of Mathematics, Faculty of Arts and Sciences, University of Petra, Amman, Jordan
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O. M. Bdair et al.
expressions. The hazard function of the Weibull distribution can only be increasing, decreasing or constant. Thus, it cannot be used to model lifetime data with a bathtubshaped hazard function, such as human mortality and machine life cycles. A detail discussion of Weibull distribution has been provided by Johnson et al. [14]. The Weibull distribution has played an important role in analyzing skewed data, and it is an appropriate model in reliability and life testing problems such as: time to failure or life length of a component or a product. It is a quite useful in various fields ranging from engineering to medical scopes (Lawless [21]). Murthy et al. [26] discussed additional applications and gave a methodological review of the ‘Weibull area’. They also suggested further study of various Weibull-type distributions, their properties, re
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