Marshall-Olkin Bivariate Exponential Distribution Research Articles

Reliability Estimation for Dependent Left-Truncated and Right-Censored Competing Risks Data with Illustrations

In this paper, a competing risks model with dependent causes of failure is considered under left-truncated and right-censoring scenario. When the dependent failure causes follow a Marshall–Olkin bivariate exponential distribution, estimation of model parameters and reliability indices are proposed from classic and Bayesian approaches, respectively. Maximum likelihood estimators and approximate confidence intervals are constructed, and conventional Bayesian point and interval estimations are discussed as well. In addition, E-Bayesian estimators are proposed and their asymptotic behaviors have been investigated. Further, another objective-Bayesian analysis is also proposed when a noninformative probability matching prior is used. Finally, extensive simulation studies are carried out to investigate the performance of different methods. Two real data examples are presented to illustrate the applicability.

Statistical inference of adaptive type II progressive hybrid censored data with dependent competing risks under bivariate exponential distribution

ABSTRACT Marshall–Olkin bivariate exponential distribution is used to statistically infer the adaptive type II progressive hybrid censored data under dependent competition risk model. For complex censored data with only partial failure reasons observed, maximum likelihood estimation and approximate confidence interval based on Fisher information are established. At the same time, Bayesian estimation is performed under the highly flexible Gamma–Dirichlet prior distribution and the highest posterior density interval using Gibbs sampling and Metropolis–Hastings algorithm is obtained. Then the performance of two methods is compared through several indexes. In addition, the Monte Carlo method is used for data simulation of multiple sets of variables to give experimental suggestions. Finally, a practical example is given to illustrate the operability and applicability of the proposed algorithm to efficiently carry out reliability test.

A new class of bivariate Lindley distributions based on stress and shock models and some of their reliability properties

Shock and stress models are used in reliability analysis to describe different kinds of applications. For instance, shock may refer to damage caused to vital organs by illness or environmental causes. On the other hand, stress can relate, for example, to two components that were maintained working independently, and had joint maintenance scheduled at a fixed time. In this context, the presented paper is aimed to introduce a new class of bivariate Lindley distribution of the Marshall–Olkin type to evaluate a 2-component series system’s reliability function. The proposed methodology is based on introducing a latent variable following an Exponential or a Lindley distribution. The obtained results generalize those achieved by using a Marshall–Olkin bivariate Exponential distribution, suggesting a useful way to search for appropriate bivariate lifetime distributions, especially under engineering reliability studies or in other areas of quantitative research.

A Note on the Time to Failure of a Two-Unit Parallel Redundant System with Deterioration on a Lattice

In this paper, we consider a two-unit parallel redundant system with deterioration on a lattice, where each unit has multi-stage deterioration levels, say, n levels. The transition from one deterioration level to the subsequent level occurs following the well-known Marshall-Olkin bivariate exponential distribution. We derive the closed form of the Laplace transform of the time to system failure in the two-unit parallel redundant system with deterioration on n×n lattice without repair and simultaneous failure, as well as the simple system on 3×3 lattice.

Statistical inference of Type-I progressively censored step-stress accelerated life test with dependent competing risks

This paper considers a step-stress accelerated dependent competing risks model under progressively Type-I censoring schemes. The dependence structure between competing risks is modeled by a general bivariate function, the cumulative exposure model is assumed and the accelerated model is described by the power rule model. The point and interval estimation of the model parameters and the reliability under normal usage level at mission time are obtained by using the maximum likelihood method and the asymptotic normal theory. We also consider the Bayesian estimators and the highest posterior density credible intervals based on conjugate priors, E-Bayesian, hierarchical Bayesian and empirical Bayesian methods. To illustrate the proposed methodology, the Marshall-Olkin bivariate exponential distribution is used to model the dependence structure between competing risks. A Monte Carlo simulation study and a real data analysis are presented to study the performance of different estimation methods.

A two-level Bayesian early fault detection for mechanical equipment subject to dependent failure modes

A two-level Bayesian control approach is presented to detect early fault for mechanical equipment subject to dependent degradation and catastrophic failures. The system degradation process is modeled using a continuous time stochastic process with three states. To model the dependence of two failure modes, we assume that the joint distribution of the time to catastrophic failure and sojourn time in the healthy state follows Marshall-Olkin bivariate exponential distribution. To avoid unnecessary sampling cost and to effectively detect impending failure, a two-level control policy, where longer sampling interval is applied for healthier state and shorter sampling interval is used in severe degradation state is proposed in Bayesian control chart framework for a multivariate observation process considering dependent failure modes. The optimization problem is formulated and solved in the semi-Markov decision process (SMDP) framework. A formula for the mean residual life (MRL) is also derived using the Bayesian approach. The validation of the proposed methodologies is carried out using real multivariate degradation data obtained from a milling machine. A comparison with the multivariate Bayesian control chart with a single sampling interval and a single control limit is given, which illustrates the effectiveness of the proposed approach.

Marshall–Olkin frailty survival models for bivariate right-censored failure time data

ABSTRACT The aim of this paper is to explore multivariate survival techniques for the analysis of bivariate right-censoring failure time data. In particular, a new family of parametric bivariate frailty models is investigated. To take into account the correlation between two survival times, the Marshall–Olkin Bivariate Exponential Distribution (MOBVE) is exploited to model the joint distribution of two frailties. The reason is twofold: on the one hand, it allows one to model shocks that affect individual-specific frailties; on the other hand, the parameter underlying the Poisson process characterizing the common shock is used to capture the dependence between two lifetimes. The proposed methodology is applied to the investigation of association in death on different-sex couples followed within the Cache County Study on Memory Health and Aging (CCSMHA). A cure rate extension of the model is also described.

On bivariate inverse Weibull distribution

Inverse Weibull distribution has been used quite successfully to analyze lifetime data which has non-monotone hazard function. The main aim of this paper is to introduce bivariate inverse Weibull distribution along the same line as the Marshall–Olkin bivariate exponential distribution, so that the marginals have inverse Weibull distributions. The proposed bivariate inverse Weibull distribution has four parameters and it has a singular component. Therefore, it can be used quite successfully if there are ties in the data. The joint probability density function, the joint cumulative distribution function and the joint survival function are all in closed forms. Several properties of this distribution have been discussed. It is observed that the proposed distribution can be obtained from the Marshall–Olkin copula. The maximum likelihood estimators of the unknown parameters cannot be obtained in closed form, and we propose to use EM algorithm to compute the maximum likelihood estimators. We propose to use parametric bootstrap method for construction of confidence intervals of the different parameters. We present some simulation experiments results to show the performances of the EM algorithm and they are quite satisfactory. We provide the Bayesian analysis of the unknown parameters based on very flexible priors. We analyze one bivariate American Football League data set for illustrative purposes, and it is observed that this model provides a slightly better fit than some of the existing models. Finally, we present some generalization to the multivariate case.

Fuzzification of Linear Combination Based on Marshall-Olkin Bivariate Exponential Distribution

The exponential distribution is applied to a very wide range of life analysis models, therefore, the research on it is of great significance in the practice of life. Based on the basic theory of general reliability and fuzzy reliability, on account of the Marshall-Olkin binary exponential distribution model, by establishing the fuzzy probability density function of the linear combination of two - dimensional random variables, this paper gives the fuzzy reliability function of the linear combination of the most widely used two - dimensional exponential distribution in reliability engineering.

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.

Fisher Information in Censored Samples from the Marshall-Olkin Bivariate Exponential Distribution

We obtain explicit expressions for the elements of the Fisher information matrix (FIM) for a single pair of order statistic and its concomitant, and Type II right, left, and doubly censored samples from the Marshall-Olkin bivariate exponential distribution. We also obtain the limiting form of the FIM for the censored samples. We evaluate the FIM for selected parameter values and sample size of 10, and determine simple, close approximations based on the limiting form. We discuss implications of our findings to inference based on small and large samples and for ranked-set samples from this distribution.

Correlation structure of the Marshall–Olkin bivariate exponential distribution

We first review the basic properties of Marshall–Olkin bivariate exponential distribution (BVE) and then investigate its correlation structure. We provide the correct reasonings for deriving some properties of the Marshall–Olkin BVE and show that the correlation of the BVE is always smaller than that of its copula regardless of the parameters. The latter implies that the BVE does not have Lancaster’s phenomenon (any nonlinear transformation of variables decreases the correlation in absolute value). The dependence structure of the BVE is also investigated.

Statistical Inference on Middle-Censored Data in a Dependent Setup

In this article, we deal with a dependent middle censoring with a censoring interval of fixed length, where the lifetime and lower bound of censoring interval are variables with a Marshall-Olkin bivariate exponential distribution. In this setup, we derive maximum likelihood estimates of the unknown parameters, using some iterative method. We also propose the Bayes estimates of the parameters under gamma priors and the squared error loss function. Finally, a Monte Carlo simulation is carried out to compare these estimators.

ON A CLASS OF GENERALIZED MARSHALL–OLKIN BIVARIATE DISTRIBUTIONS AND SOME RELIABILITY CHARACTERISTICS

We consider here a general class of bivariate distributions from reliability point of view, and refer to it as generalized Marshall–Olkin bivariate distributions. This class includes as special cases the Marshall–Olkin bivariate exponential distribution and the class of bivariate distributions studied recently by Sarhan and Balakrishnan [25]. For this class, the reliability, survival, hazard, and mean residual life functions are all derived, and their monotonicity is discussed for the marginal as well as the conditional distributions. These functions are also studied for the series and parallel systems based on this bivariate distribution. Finally, the Clayton association measure for this bivariate model is derived in terms of the hazard gradient.

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.

A method of moments to estimate bivariate survival functions: the copula approach

In this paper we discuss the problem on parametric and non parametric estimation of the distributions generated by the Marshall-Olkin copula. This copula comes from the Marshall-Olkin bivariate exponential distribution used in reliability analysis. We generalize this model by the copula and different marginal distributions to construct several bivariate survival functions. The cumulative distribution functions are not absolutely continuous and they unknown parameters are often not be obtained in explicit form. In order to estimate the parameters we propose an easy procedure based on the moments. This method consist in two steps: in the first step we estimate only the parameters of marginal distributions and in the second step we estimate only the copula parameter. This procedure can be used to estimate the parameters of complex survival functions in which it is difficult to find an explicit expression of the mixed moments. Moreover it is preferred to the maximum likelihood one for its simplex mathematic form; in particular for distributions whose maximum likelihood parameters estimators can not be obtained in explicit form.

Generalized Marshall–Olkin distributions and related bivariate aging properties

A class of generalized bivariate Marshall–Olkin distributions, which includes as special cases the Marshall–Olkin bivariate exponential distribution and the Marshall–Olkin type distribution due to Muliere and Scarsini (1987) [19] are examined in this paper. Stochastic comparison results are derived, and bivariate aging properties, together with properties related to evolution of dependence along time, are investigated for this class of distributions. Extensions of results previously presented in the literature are provided as well.

The bivariate generalized linear failure rate distribution and its multivariate extension

The two-parameter linear failure rate distribution has been used quite successfully to analyze lifetime data. Recently, a new three-parameter distribution, known as the generalized linear failure rate distribution, has been introduced by exponentiating the linear failure rate distribution. The generalized linear failure rate distribution is a very flexible lifetime distribution, and the probability density function of the generalized linear failure rate distribution can take different shapes. Its hazard function also can be increasing, decreasing and bathtub shaped. The main aim of this paper is to introduce a bivariate generalized linear failure rate distribution, whose marginals are generalized linear failure rate distributions. It is obtained using the same approach as was adopted to obtain the Marshall–Olkin bivariate exponential distribution. Different properties of this new distribution are established. The bivariate generalized linear failure rate distribution has five parameters and the maximum likelihood estimators are obtained using the EM algorithm. A data set is analyzed for illustrative purposes. Finally, some generalizations to the multivariate case are proposed.

Bayesian Inference in Dependent Right Censoring

In this article, we consider dependent right censoring when the lifetime and censoring variables have a Marshall–Olkin bivariate exponential distribution and obtain MLEs, MMEs and UMVUEs of the unknown parameters. The Bayes estimators as well as the Posterior Regret Gamma Minimax (PRGM) estimators of the parameters of interest under the SEL function are also obtained and a Monte Carlo simulation study is carried out to compare these estimators.

Modified Sarhan–Balakrishnan singular bivariate distribution

Recently Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527] introduced a new bivariate distribution using generalized exponential and exponential distributions. They discussed several interesting properties of this new distribution. Unfortunately, they did not discuss any estimation procedure of the unknown parameters. In this paper using the similar idea as of Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527], we have proposed a singular bivariate distribution, which has an extra shape parameter. It is observed that the marginal distributions of the proposed bivariate distribution are more flexible than the corresponding marginal distributions of the Marshall–Olkin bivariate exponential distribution, Sarhan–Balakrishnan's bivariate distribution or the bivariate generalized exponential distribution. Different properties of this new distribution have been discussed. We provide the maximum likelihood estimators of the unknown parameters using EM algorithm. We reported some simulation results and performed two data analysis for illustrative purposes. Finally we propose some generalizations of this bivariate model.