Marshall Olkin Bivariate Weibull Distribution Research Articles

On competing risk model under step-stress stage life testing

In this article, we consider the inference of a dependent competing risk model when the data are collected from a simple step-stress stage life testing experiment. We assume that there are two causes of failure and they are dependent causes. The dependency structure is explained through Marshall–Olkin bivariate Weibull distribution. The distribution under two stress levels is connected through Khamis Higgins model. We obtain the maximum likelihood estimates and the Bayes estimates of the model parameters. We also provide the asymptotic confidence intervals, symmetric credible intervals and the highest posterior density credible intervals of the model parameters. An optimality study has been done under different optimality criteria. An extensive simulation study and a data analysis have been performed for illustrative purpose.

Objective Bayesian analysis of Marshall-Olkin bivariate Weibull distribution with partial information

ABSTRACT In competing risks problem, a subset of risks is needed more attention for inferential purposes. In the objective Bayesian paradigm, reference priors enable to achieve such inferential objectives. In this article, the Marshall-Olkin bivariate Weibull distribution is considered to model the competing risks data. In the availability of partial information for some of the parameters, the reference priors are derived as per the importance of the parameters. The Dirichlet prior is taken as a conditional subjective prior and the marginal reference prior has been derived. Also, the propriety of the resulting posterior density has been proved. The Bayesian estimates of the parameters are obtained under squared error and linear-exponential loss functions. Further, the derived reference prior is used for the computation of Bayes factors or posterior odds in testing the hypothesis that the competing risks are identical. The performance of established Bayesian estimators is illustrated using the Diabetic Retinopathy Study (DRS) and Prostate Cancer data sets. Finally, the model compatibility is done for the considered data sets under Bayesian Paradigm.

A Bayes Analysis of a Dependent Competing Risk Model Based on Marshall–Olkin Bivariate Weibull Distribution

This paper considers a competing risk model defined on the basis of minimum of two dependent failures where the two failures are assumed to jointly follow Marshall–Olkin bivariate Weibull distribution. This paper explores some important features of corresponding likelihood functions and performs a full Bayesian analysis of the model for data resulting from normal as well as accelerated life tests. The accelerated model is described by regressing the scale parameters of the model through inverse power-law relationship. Posterior-based inferences are drawn using the Gibbs sampler algorithm after specifying proper but vague priors for the model parameters. The numerical illustration is provided using real datasets. The performance of the model is assured by Bayesian tools of model compatibility and then the entertained model is compared with the competing risk model based on Marshall–Olkin bivariate exponential assumption.

Bayesian Inference of System Reliability for Multicomponent Stress-Strength Model under Marshall-Olkin Weibull Distribution

Industrial systems often have redundant structures for improving reliability and avoiding sudden failures, and a parallel system is one of the special redundant systems. In this paper, we consider the problem of reliability estimation for a parallel system when one stress variable is involved, which is called the multicomponent stress-strength model. The parallel system contains two components, and their joint lifetime follows a Marshall–Olkin bivariate Weibull distribution, while the stress variable is assumed to be the Weibull distribution. Due to the complicated form of the likelihood function, a data augmentation method is proposed, and then the Gibbs sampling algorithm is constructed to obtain the Bayesian estimation of the system reliability. The proposed method is evaluated by a simulated dataset and Monte Carlo simulation study. The simulation results show that the proposed method performs well in terms of relative bias, mean squared error and frequentist coverage probability.

Statistical inference of generalized progressive hybrid censored step-stress accelerated dependent competing risks model for Marshall-Olkin bivariate Weibull distribution

In this article, a step-stress accelerated life test based on the generalized Khamis-Higgins model in the presence of Marshall-Olkin bivariate Weibull distributed dependent competing risks is considered under generalized progressive hybrid censoring. It is assumed that the stress is changed when a pre-specified number of failures take place, and four different failure causes are discussed. The classical and Bayesian statistical methods are used to estimate the unknown parameters. In addition to the maximum likelihood estimation, the stochastic expectation-maximization algorithm is also applied. The Monte Carlo Markov Chain samples with importance sampling are used to obtain the Bayesian estimates under two loss functions, and the Bayesian highest posterior density credible intervals are constructed. Then the parameters and reliability function under normal use stress level are calculated. The Monte Carlo method is applied to compare the performance of the estimation methods. Finally, a real-life data set is analyzed for illustrative purposes.

Statistical analysis of adaptive type-II progressively censored competing risks for Weibull models

In this article, the estimation of the unknown parameters and the survival and hazard functions of Weibull distribution is studied with the adaptive Type-II progressively censored competing risks data, where the lifetime random variables of the individual failure causes are independent and follow Weibull distribution with different scale and shape parameters. For frequentist estimation, the maximum likelihood estimators are obtained utilizing Newton-Raphson method, whose existence and uniqueness are also proved. Making use of the asymptotic normality of maximum likelihood estimators and delta method, we construct the respective confidence intervals of the parameters and the survival and hazard functions. For Bayesian estimation, we take advantage of Monte Carlo Markov Chain technique and importance sampling method to derive the Bayesian estimators and the credible intervals. Extensive Monte Carlo simulation experiments and a real-life mortality dataset analysis are implemented to evaluate the performance of the developed methods. And then, we discuss the issue of expected experimentation time. In the end, the model is extended to the case of dependent failure modes. Marshall-Olkin bivariate Weibull distribution is considered and the theoretical derivations are presented.

Bayesian inference of a dependent competing risk data

Recently, Feizjavadian and Hashemi (Analysis of dependent competing risks in presence of progressive hybrid censoring using Marshall–Olkin bivariate Weibull distribution. Comput Stat Data Anal. 2015;82:19–34) provided a classical inference of a competing risks data set using Marshall–Olkin bivariate Weibull distribution when the failure of an unit at a particular time point can happen due to more than one cause. The aim of this paper is to provide the Bayesian analysis of the same model based on a very flexible Gamma–Dirichlet (GD) prior on the scale parameters. The Bayesian inference has certain advantages over the classical inference in this case. We provide the Bayes estimates of the unknown parameters and the associated highest posterior density credible intervals based on Gibbs sampling technique. We further consider the Bayesian inference of the model parameters assuming partially ordered GD prior on the scale parameters when one cause is more severe than the other cause. We have extended the results for different censoring schemes also.

Inference of accelerated dependent competing risks model for Marshall–Olkin bivariate Weibull distribution with nonconstant parameters

In this paper, we discuss the statistical inference of constant-stress accelerated dependent competing risks model under Type-II hybrid censoring schemes. The dependency structure is modeled by a Marshall–Olkin bivariate Weibull distribution. Both the shape and the scale parameters in the model are assumed to be dependent on the stress levels through a log-linear relationship. The maximum likelihood estimates (MLEs) of the model parameters are derived. Confidence intervals (CIs) of the model parameters are constructed based on the asymptotic normality of MLEs and the bias-corrected percentile bootstrap method. Bayes estimates with the squared error loss function and the highest posterior density (HPD) credible intervals (CIs) are obtained by using an importance sampling method. In addition, the estimates for the accelerated coefficients and the reliability under normal use stress level at mission time are derived. A Monte Carlo simulation study is used to evaluate the performance of the proposed statistical inference methods. A real data example is used to illustrate the methodologies proposed in this paper and to compare the proposed model with the copula model.

On bivariate discrete Weibull distribution

Recently, Lee and Cha proposed two general classes of discrete bivariate distributions. They have discussed some general properties and some specific cases of their proposed distributions. In this paper we have considered one model, namely bivariate discrete Weibull distribution, which has not been considered in the literature yet. The proposed bivariate discrete Weibull distribution is a discrete analogue of the Marshall–Olkin bivariate Weibull distribution. We study various properties of the proposed distribution and discuss its interesting physical interpretations. The proposed model has four parameters, and because of that it is a very flexible distribution. The maximum likelihood estimators of the parameters cannot be obtained in closed forms, and we have proposed a very efficient nested EM algorithm which works quite well for discrete data. We have also proposed augmented Gibbs sampling procedure to compute Bayes estimates of the unknown parameters based on a very flexible set of priors. Two data sets have been analyzed to show how the proposed model and the method work in practice. We will see that the performances are quite satisfactory. Finally, we conclude the paper.

On the dependent competing risks using Marshall–Olkin bivariate Weibull model: Parameter estimation with different methods

ABSTRACTCompeting risks models are of great importance in reliability and survival analysis. They are often assumed to have independent causes of failure in literature, which may be unreasonable. In this article, dependent causes of failure are considered by using the Marshall–Olkin bivariate Weibull distribution. After deriving some useful results for the model, we use ML, fiducial inference, and Bayesian methods to estimate the unknown model parameters with a parameter transformation. Simulation studies are carried out to assess the performances of the three methods. Compared with the maximum likelihood method, the fiducial and Bayesian methods could provide better parameter estimation.

Bayesian analysis of series system with dependent causes of failure

ABSTRACTMost studies of series system assume the causes of failure are independent, which may not hold in practice. In this paper, dependent causes of failure are considered by using a Marshall–Olkin bivariate Weibull distribution. We derived four reference priors based on several grouping orders. Gibbs sampling combined with the rejection sampling algorithm and Metropolis–Hastings algorithm is developed to obtain the estimates of the unknown parameters. The proposed approach is compared with the maximum-likelihood method via simulation. We find that the root-mean-squared errors of the Bayesian estimates are much smaller for the case of small sample size, and that the coverage probabilities of the Bayesian estimates are much closer to the nominal levels. Finally, a real data-set is analysed for illustration.

Analysis of incomplete data in the presence of dependent competing risks from Marshall–Olkin bivariate Weibull distribution under progressive hybrid censoring

ABSTRACTThis article considers the statistical analysis of dependent competing risks model with incomplete data under Type-I progressive hybrid censored condition using a Marshall–Olkin bivariate Weibull distribution. Based on the expectation maximum algorithm, maximum likelihood estimators for the unknown parameters are obtained, and the missing information principle is used to obtain the observed information matrix. As the maximum likelihood approach may fail when the available information is insufficient, Bayesian approach incorporated with auxiliary variables is developed for estimating the parameters of the model, and Monte Carlo method is employed to construct the highest posterior density credible intervals. The proposed method is illustrated through a numerical example under different progressive censoring schemes and masking probabilities. Finally, a real data set is analyzed for illustrative purposes.

A simulation study on the correlation structure of Marshall–Olkin bivariate Weibull distribution

ABSTRACTThe correlation structure of Marshall–Olkin bivariate exponential distribution (BVE) is well known. However, we are unable to compute the correlation of Marshall–Olkin bivariate Weibull distribution analytically. Fortunately, bivariate observations from this family can be obtained easily through extensive simulations. As expected, the key factors that determine the size of the correlation are the common shape parameter α and the scale parameters and of the marginal variates. In particular, the common scale parameter is found to be the most influential factor that affects the correlation. Plots from the simulations illustrate how each parameter impacts on the correlation while holding other parameters fixed. We also discuss how the correlation behaves when certain nonlinear transformations are applied to marginal variates of BVE.

Estimation of Reliability in a Multicomponent Stress-Strength Model Based on a Marshall-Olkin Bivariate Weibull Distribution

In this paper, we consider a system which has $k$ s-independent and identically distributed strength components, and each component is constructed by a pair of s-dependent elements. These elements $(X_{1},Y_{1}),(X_{2},Y_{2}),\ldots ,(X_{k},Y_{k})$ follow a Marshall-Olkin bivariate Weibull distribution, and each element is exposed to a common random stress $T$ which follows a Weibull distribution. The system is regarded as operating only if at least $s$ out of $k (1\leq s\leq k)$ strength variables exceed the random stress. The multicomponent reliability of the system is given by $R_{s,k}=P$ (at least $s$ of the $(Z_{1},\ldots ,Z_{k})$ exceed $T$ ) where $Z_{i}=\min (X_{i},Y_{i})$ , $i=1,\ldots ,k$ . We estimate $R_{s,k}$ by using frequentist and Bayesian approaches. The Bayes estimates of $R_{s,k}$ have been developed by using Lindley's approximation, and the Markov Chain Monte Carlo methods, due to the lack of explicit forms. The asymptotic confidence interval, and the highest probability density credible interval are constructed for $R_{s,k}$ . The reliability estimators are compared by using the estimated risks through Monte Carlo simulations.

Analysis of dependent competing risks in the presence of progressive hybrid censoring using Marshall–Olkin bivariate Weibull distribution

The lifetime of subjects in reliability and survival analysis in the presence of several causes of failure (i.e., competing risks) has attracted attention in the literature. Most studies have simplified the computations by assuming that the causes are independent, though this does not hold. Dependent competing risks under progressively hybrid censoring condition using a Marshall–Olkin bivariate Weibull distribution is investigated. Maximum likelihood and approximated maximum likelihood estimators are developed for estimating the unknown parameters. Asymptotic distributions of the maximum likelihood estimators are used to construct approximate confidence intervals using the observed Fisher information matrix. Based on a simulation and real applications, it is illustrated that when a parametric distributional assumption is nearly true, a close approximation could be achieved by deliberately censoring the number of subjects and the study duration using Type-II progressively hybrid censoring, which might help to save time and money in research studies.

Bayes estimation for the Marshall–Olkin bivariate Weibull distribution

In this paper, we consider the Bayesian analysis of the Marshall–Olkin bivariate Weibull distribution. It is a singular distribution whose marginals are Weibull distributions. This is a generalization of the Marshall–Olkin bivariate exponential distribution. It is well known that the maximum likelihood estimators of the unknown parameters do not always exist. The Bayes estimators are obtained with respect to the squared error loss function and the prior distributions allow for prior dependence among the components of the parameter vector. If the shape parameter is known, the Bayes estimators of the unknown parameters can be obtained in explicit forms under the assumptions of independent priors. If the shape parameter is unknown, the Bayes estimators cannot be obtained in explicit forms. We propose to use the importance sampling method to compute the Bayes estimators and also to construct associated credible intervals of the unknown parameters. The analysis of one data set is performed for illustrative purposes. Finally we indicate the analysis of data sets obtained from series and parallel systems.

An EM algorithm for estimating the parameters of bivariate Weibull distribution under random censoring

We consider the problem of estimation of the parameters of the Marshall-Olkin Bivariate Weibull distribution in the presence of random censoring. Since the maximum likelihood estimators of the parameters cannot be expressed in a closed form, we suggest an EM algorithm to compute the same. Extensive simulations are carried out to conclude that the estimators perform efficiently under random censoring.

Marshall–Olkin bivariate Weibull distributions and processes

In this paper we introduce a new probability model known as type 2 Marshall–Olkin bivariate Weibull distribution as an extension of type 1 Marshall–Olkin bivariate Weibull distribution of Marshall–Olkin (J Am Stat Assoc 62:30–44, 1967). Various properties of the new distribution are considered. Bivariate minification processes with the two types of Weibull distributions as marginals are constructed and their properties are considered. It is shown that the processes are strictly stationary. The unknown parameters of the type 1 process are estimated and their properties are discussed. Some numerical results of the estimates are also given.

Estimating the parameters of the Marshall–Olkin bivariate Weibull distribution by EM algorithm

In this paper we consider the Marshall–Olkin bivariate Weibull distribution. The Marshall–Olkin bivariate Weibull distribution is a singular distribution, whose both the marginals are univariate Weibull distributions. This is a generalization of the Marshall–Olkin bivariate exponential distribution. The cumulative joint distribution of the Marshall–Olkin bivariate Weibull distribution is a mixture of an absolute continuous distribution function and a singular distribution function. This distribution has four unknown parameters and it is observed that the maximum likelihood estimators of the unknown parameters cannot be obtained in explicit forms. In this paper we discuss about the computation of the maximum likelihood estimators of the unknown parameters using EM algorithm. We perform some simulations to see the performances of the EM algorithm and re-analyze one data set for illustrative purpose.