Statistical Analysis And Reliability Evaluation For The Competing Risks Model Under Progressively Hybrid Censoring

Competing cause of failure is an important cause of failure of many industrial products.The reliability analysis of competing risks model has become one of the important topics in the field of reliability engineering.We study the problems of statistical analysis and reliability evaluation under progressively hybrid censoring(PHC)life tests as well as accelerated life tests(ALT)when the competing failure causes are dependent and independent,respectively.The main organizations and innovations of the thesis are as follows:(1)Under Type-I PHC life test,we study the statistical analysis of competing risks model from Weibull distributions.Firstly,we found the likelihood function based on the competing failure data,and present the properties of the profile log-likelihood function of the shape parameters.The maximum likelihood estimates(MLEs)and approximate maximum likelihood estimates(AMLEs)of parameters are derived.The method of percentile bootstrap of self-simulation is used for constructing the approximate confidence intervals for parameters.Secondly,choosing the noninformative priors of parameters,a Gibbs hybrid sampling algorithm and Monte Carlo Markov Chain(MCMC)method are presented to obtain Bayesian estimates and the highest posterior density(HPD)credible intervals of parameters,respectively.Finally,the simulation is given to illustrate the validity and effectiveness of the methods.(2)Under Type-I PHC life test with random removals,we study the Bayesian analysis of the competing risks model from Gompertz distributions.Assume the number of removals at each failure time follows a binomial distribution,and the likelihood function is found based on the competing failure data.Choosing the hybrid(discrete and continuous)priors of parameters,we obtain the Bayesian estimates of model parameters,removal probabilities,reliability functions and hazard rate functions under symmetric and asymmetric loss functions.In addition,the expected termination time under Type-I PHC life test is derived which is compared with the expected termination time under the whole sample test.The numerical simulation example is presented to analyze the effects of the scale parameter in loss functions on the Bayesian estimates,and the effects of different sample size,effective sample size and removal probability on the expected termination time are investigated.(3)Under Type-I PHC life test,we study the expected Bayesian(E-Bayesian)analysis of competing risks model from Gompertz distributions.Choosing the conjugate priors of the scale parameters,the explicit expressions of Bayesian estimates of the scale parameters and reliability functions are derived under symmetric and asymmetric loss functions.On the premise that the prior distributions are monotone decreasing functions,we choose three different priors of hyperparameters,and E-Bayesian estimates of scale parameters and reliability functions are derived under different loss functions.Some relations and properties among E-Bayesian estimates under different priors of hyperparameters are discussed in the end.(4)Under Type-I PHC ALT,we study the parametric estimation and reliability evaluation for competing risks model from Weibull distributions.Firstly,under Type-I PHC constant-stress ALT,we derive the MLEs and asymptotic confidence intervals of parameters.Based on the hybrid(discrete and noninformative)priors of parameters,the log-concave property of the conditional posterior densities of shape parameters is proved.A hybrid sampling algorithm combined Gibbs sampling with adaptive rejection sampling is considered for deriving Bayesian estimates and HPD credible intervals of parameters.The least squares approach is used to obtain the estimates of the parameters in the accelerated function,then the reliability under normal stress is evaluated.Secondly,under Type-I PHC step-stress ALT,we derive the MLEs of parameters by the cumulative exposure model and maximum likelihood theory,and the asymptotic confidence intervals and bias corrected percentile bootstrap confidence intervals are constructed for the parameters.The simulation is provided to evaluate the effectiveness of the methods.(5)Under Type-I PHC life test,we study the statistical analysis of dependent competing risks model from Gompertz distributions.The joint survival function and the joint probability density function of the dependent variables are derived,and the properties of the profile log-likelihood function of the shape parameters are studied.Then we obtain the MLEs of parameters.The approximate confidence intervals of parameters are constructed by large sample theory and percentile bootstrap method.A simulation is provided to evaluate the validity and effectiveness of the methods,and investigate the effects of different dependence structure on the estimates of parameters.(6)Under Type-I PHC constant-stress ALT,we study the reliability analysis of dependent competing risks model from Weibull distributions.Firstly,the Gumbel copula is used as the link function to build the reliability evaluation model of dependent competing failure products.Secondly,the classical and asymptotic statistical methods are considered to obtain the point estimates,asymptotic confidence intervals and percentile bootstrap confidence intervals of parameters.On the basis,the reliability under normal stress is evaluated.Finally,the simulation is provided to analyze the effects of different dependence structure on the estimates of parameters,and evaluate the effectiveness of the methods.