Inference for Weibull competing risks model with partially observed failure causes under generalized progressive hybrid censoring
Section snippets
Introduction and notation
Quite often reliability experiments are conducted under cost and time limitations. In such studies statistical inferences upon unknown quantities of interest are usually derived on the basis of censored data. Two commonly used censoring schemes (CSs) are the Type-I censoring and Type-II censoring, wherein the test terminated at a prefixed time and upon observing certain number of failures. To allow for more flexibility in removing surviving units from the test, progressive censoring schemes
Model description
Suppose identical units are put in test and their lifetimes are described by independent and identical distributed (i.i.d.) random variables . Consider there are two causes of failure, then the latent failure times are given by The PDF and CDF of are denoted by and , and the associated survival function by .
Under GPHC, suppose and are prefixed monitoring points, and are prefixed CS, then the experiment stops at the
Classical estimation
In this section, maximum likelihood estimators (MLEs) and approximate confidence intervals (ACIs) are established when there is no order information.
Bayesian estimation
Inference from Bayesian approach is provided in this subsection when there is no order restriction on parameters and .
From Eqs. (6), (7) and (9), the joint posterior PDF of and can be written as where and
Theorem 3 The marginal posterior density function in (14) is log-concave.
Proof See Appendix C. □
The Bayes estimator of some parametric
Inference under order restriction
In this section, classical and Bayesian estimates are conducted when the order restriction information is available.
Simulation studies
In this subsection, we perform Monte Carlo simulations to investigate the behavior of proposed methods. The performance of all the point estimators are evaluated in terms of their mean square error (MSE) and average bias (AB) values. Likewise, proposed confidence intervals are compared in terms of their coverage probabilities (CPs) and average width (AW). The following algorithm is used to generate GPHC competing risks samples from the Weibull distribution.
- Step 1
Generate Type-II PCS data from
Concluding remarks
Inference is studied for the Weibull competing risks model when the failure sample is generalized progressively hybrid censored. Under the situation that failure causes are partially observed and the scale parameters are restricted and unrestricted, point and interval estimates of unknown parameters are studied under classical and Bayesian framework with a flexible class of priors, respectively. Numerical illustration shows that both classical and Bayes approaches perform quit good, and that
Acknowledgments
The authors would like to thank the Editor and the referees for their insightful comments which have led to a substantial improvement to an earlier version of the paper. The authors also thank professor Li Yan from Université du Québec en Outaouais, Canada for his careful reading and useful comments in improving the English writing for this paper. This work of Liang Wang was supported by China Postdoctoral Science Foundation (No. 2019M650260), the National Natural Science Foundation of China
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