Xuchao Bai

Inference for Constant-Stress Accelerated Life Tests With Dependent Competing Risks From Bivariate Birnbaum–Saunders Distribution Based on Adaptive Progressively Hybrid Censoring

In life testing, the competing risks model is usually discussed under the assumption of independence. In this paper, we consider a dependent competing risks model using bivariate Birnbaum–Saunders distribution in constant-stress accelerated life testing. To observe expected failure times and terminate the life tests around a predetermined time, the adaptive progressively hybrid censoring scheme is adopted. Based on the accelerated competing risks model with the adaptive progressively hybrid censoring scheme, we obtain the maximum-likelihood estimators, approximate confidence intervals, and bootstrap confidence intervals of unknown parameters. To test the independence between the bivariate competing risks and find the relationship of shape and scale parameters, we discuss the likelihood ratio tests for hypotheses of interest. In addition, we compute the maximum-likelihood predictors of unobserved competing risks times in the constant-stress accelerated life tests. Finally, a simulation study and an illustrative example are provided to support the proposed model and methods, and to examine the performance of estimators and testing.

Reliability estimation of multicomponent stress–strength model based on copula function under progressively hybrid censoring

Abstract In reliability analysis of the stress–strength models, the stress and strength variables are typically assumed as independent. However, such an assumption may be unrealistic in some applications. It is a meaningful issue to estimate the reliability of the stress–strength model for dependent stress and strength variables. In this paper, we estimate the reliability of multicomponent stress–strength model by assuming the dependent Weibull stress variables and exponential strength variables based on Gumbel copula under Type-I progressively hybrid censoring scheme. The estimators of the unknown parameters and reliability are obtained by using the maximum likelihood estimation method. Also, the asymptotic confidence intervals and Bootstrap percentile confidence intervals of the unknown parameters and reliability of stress–strength model are derived. Monte Carlo simulations are used to evaluate the performance of the maximum likelihood estimators, asymptotic confidence intervals and Bootstrap percentile confidence intervals. Finally, real data are analyzed to demonstrate the practicability of the stress–strength model in this article.

Reliability estimation of stress–strength model using finite mixture distributions under progressively interval censoring

Abstract This paper considers the reliability estimation of the stress–strength model based on progressively Type-I interval censored data, where the random stress variable follows a Lindley distribution and the random strength variable follows a finite mixture of exponential distributions. The maximum likelihood estimation and 95% confidence interval estimation of the stress–strength reliability are deduced by using EM algorithm and Bootstrap sampling, respectively. The Bayesian estimation and 95% highest posterior density credible interval of the stress–strength reliability under squared error loss function are obtained by using the Metropolis–Hastings within Gibbs algorithm. To test the homogeneity of the finite mixture distributions, the D-test statistic is introduced. Then, we use the D-test statistic to test the homogeneity of a real data, compare the finite mixture exponential distributions with a single exponential distribution by using the Akaike information criterion (AIC) values, and analyze this data using the proposed methodology. Finally, Monte Carlo simulations are performed for illustrative purpose.

Statistical analysis of dependent competing risks model in constant stress accelerated life testing with progressive censoring based on copula function

ABSTRACT In this paper, we consider the statistical analysis for the dependent competing risks model in the constant stress accelerated life testing with Type-II progressive censoring. It is focused on two competing risks from Lomax distribution. The maximum likelihood estimators of the unknown parameters, the acceleration coefficients and the reliability of unit are obtained by using the Bivariate Pareto Copula function and the measure of dependence known as Kendalls tau. In addition, the 95% confidence intervals as well as the coverage percentages are obtained by using Bootstrap-p and Bootstrap-t method. Then, a simulation study is carried out by the Monte Carlo method for different measures of Kendalls tau and different testing schemes. Finally, a real competing risks data is analysed for illustrative purposes. The results indicate that using copula function to deal with the dependent competing risks problems is effective and feasible.

Dynamic stress–strength reliability estimation of system with survival signature

ABSTRACT In this paper, we proposed a dynamic stress–strength model for coherent system. It is supposed that the system consists of n components with initial random strength and each component is subjected to random stresses. The stresses, applied repeatedly at random cycle times, will cause the degradation of strength. In addition, the number of cycles in an interval is assumed to follow a Poisson distribution. In the case of the strength and stress random variables following exponential distributions, the expression for the reliability of the proposed dynamic stress–strength model is derived based on survival signature. The reliability is estimated by using the best linear unbiased estimation (BLUE). Considering the Type-II censored failure times, the best linear unbiased predictors (BLUP) for the unobserved coherent system failure times are developed based on the observed failure times. Monte Carlo simulations are performed to compare the BLUE of parameters with different values and compute the BLUP. A real data set is also analysed for an illustration of the findings.

Stress–strength reliability analysis of multi-state system based on generalized survival signature

Abstract The stress–strength model has been widely used in reliability design of system. In the traditional stress–strength reliability theory, the system and each component are assumed to be only in one of two possible states being either working or failed, and the notion of stress–strength reliability is the probability that the strength is larger than the stress. In this paper, we study the stress–strength reliability of multi-state system based on generalized survival signature. It is supposed that the state of multi-state system is defined by using the ratio between strength and stress random variables. The definitions of generalized survival signature for a certain class of multi state systems with multi-state components in both discrete and continuous cases are given. In addition, the expressions of stress–strength reliability in both discrete and continuous situations are derived. In the case of continuous multi-state system, it is assumed that the random strength and stress are both from the Weibull distributions with different scale parameters, and the two different continuous kernel functions are Pareto and generalized half logistic distribution functions, respectively. Based on the assumptions, the stress–strength reliability is estimated by using both classical and Bayesian statistical theories. The uniformly minimum variance unbiased estimator and maximum likelihood estimator for the stress–strength reliability of the continuous multi-state system are derived. Under the squared error loss function, the exact expression of Bayes estimator for the stress–strength reliability of the continuous multi-state system is developed by using Gauss hypergeometric function. Finally, the Monte Carlo simulations are performed to compare the performances of the proposed stress–strength reliability estimators, and a real data set is also analyzed for an illustration of the findings.

Stress–strength reliability analysis of system with multiple types of components using survival signature

Abstract In this paper, we study the estimation for stress–strength reliability of the system with multiple types of components based on survival signature. In the situation that different types of components are subjected to different types of random stresses, the maximum likelihood estimator, maximum spacing estimator, bootstrap-p confidence interval, two point estimators and generalized confidence interval using generalized pivotal quantity for system stress–strength reliability are derived under the assumption that the stresses and strengths variables follow the Gompertz distributions with common or unequal scale parameters. Additionally, when the stresses and strengths variables follow the Gompertz distributions with unequal scale parameters, a modified generalized confidence interval for the system stress–strength reliability based on the Fisher Z transformation is also proposed. In the situation that the system is subjected to the common stress, the above point estimators and confidence intervals for the system stress–strength reliability are also developed. Monte Carlo simulations are performed to compare the performance of these point estimators and confidence intervals. A real data analysis is presented for an illustration of the findings.

Statistical analysis of competing risks model from Marshall–Olkin extended Chen distribution under adaptive progressively interval censoring with random removals

Abstract In this paper, a new censoring scheme named by adaptive progressively interval censoring scheme is introduced. The competing risks data come from Marshall–Olkin extended Chen distribution under the new censoring scheme with random removals. We obtain the maximum likelihood estimators of the unknown parameters and the reliability function by using the EM algorithm based on the failure data. In addition, the bootstrap percentile confidence intervals and bootstrap-t confidence intervals of the unknown parameters are obtained. To test the equality of the competing risks model, the likelihood ratio tests are performed. Then, Monte Carlo simulation is conducted to evaluate the performance of the estimators under different sample sizes and removal schemes. Finally, a real data set is analyzed for illustration purpose.

Statistical inference of Marshall-Olkin bivariate Weibull distribution with three shocks based on progressive interval censored data

ABSTRACT There are several failure modes may cause system failed in reliability and survival analysis. It is usually assumed that the causes of failure modes are independent each other, though this assumption does not always hold. Dependent competing risks modes from Marshall-Olkin bivariate Weibull distribution under Type-I progressive interval censoring scheme are considered in this paper. We derive the maximum likelihood function, the maximum likelihood estimates, the 95% Bootstrap confidence intervals and the 95% coverage percentages of the parameters when shape parameter is known, and EM algorithm is applied when shape parameter is unknown. The Monte-Carlo simulation is given to illustrate the theoretical analysis and the effects of parameters estimates under different sample sizes. Finally, a data set has been analyzed for illustrative purposes.

Statistical analysis of masked data in a hybrid system based on copula theory under progressive hybrid censoring

ABSTRACT This article considers the reliability analysis of a hybrid system with dependent components, which are linked by a copula function. Based on Type I progressive hybrid censored and masked system lifetime data, we drive some probability results for the hybrid system and then the maximum likelihood estimates as well as the asymptotic confidence intervals and bootstrap confidence intervals of the unknown parameters are obtained. The effects of different dependence structures on the estimates of the parameter and the reliability function are investigated. Finally, Monte Carlo simulations are implemented to compare the performances of the estimates when the components are dependent with those when the components are independent.