Inference for Weibull competing risks model with partially observed failure causes under generalized progressive hybrid censoring

https://doi.org/10.1016/j.cam.2019.112537Get rights and content

Highlights

  • Generalized progressive hybrid censoring is considered for Weibull competing risks model.

  • Partially causes of failure are observed for censored data.

  • Order restriction information for scale parameters is considered.

  • Maximum likelihood estimators are established.

  • Bayesian estimates are derived via MCMC sampling method.

  • Data example on electric appliances data is illustrated.

Abstract

In this paper, a competing risks model is studied when the latent failure times follow Weibull distribution. When the failure times are observed under generalized progressive hybrid censoring and the causes of failure are partially observed, the maximum likelihood estimators of the model parameters are established together with associated existence and uniqueness, and the approximate confidence intervals are constructed based on large sample theory. Bayes estimators and associated credible intervals are obtained under fairly general priors. Moreover, classical and Bayesian inferences are also discussed when there is an order restriction on the scale parameters of the Weibull distributions. Finally, a simulation study and a real data example are presented for illustration.

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 n identical units are put in test and their lifetimes are described by independent and identical distributed (i.i.d.) random variables X1,X2,,Xn. Consider there are two causes of failure, then the latent failure times are given by Xi=min{X1i,X2i},i=1,2,,n.The PDF and CDF of X1,X2,,Xn are denoted by f(x) and F(x), and the associated survival function by S(x).

Under GPHC, suppose T and k are prefixed monitoring points, and r1,r2,,rm 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 β1 and β2.

From Eqs. (6), (7) and (9), the joint posterior PDF of β1,β2 and α can be written as π(β1,β2,α|data)=π(β1,β2|α,data)π(α|data),where π(β1,β2|α,data)BG(a1+d,b0+W(α),c1+e1,c2+e2),and π(α|data)αa1+d1[b0+W(α)][a0+d]exp[b1i=1dlnxi:m:n]α.

Theorem 3

The marginal posterior density function π(α|data) 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 β1<β2 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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