Bivariate Weibull Research Articles

A Bayesian Inference Approach for Bivariate Weibull Distributions Derived from Roy and Morgenstern Methods

Bivariate lifetime distributions are of great importance in studies related to interdependent components, especially in engineering applications. In this paper, we introduce two bivariate lifetime assuming three- parameter Weibull marginal distributions. Some characteristics of the proposed distributions as the joint survival function, hazard rate function, cross factorial moment and stress-strength parameter are also derived. The inferences for the parameters or even functions of the parameters of the models are obtained under a Bayesian approach. An extensive numerical application using simulated data is carried out to evaluate the accuracy of the obtained estimators to illustrate the usefulness of the proposed methodology. To illustrate the usefulness of the proposed model, we also include an example with real data from which it is possible to see that the proposed model leads to good fits to the data.

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.

An introduction to new robust linear and monotonic correlation coefficients

BackgroundThe most common measure of association between two continuous variables is the Pearson correlation (Maronna et al. in Safari an OMC. Robust statistics, 2019. https://login.proxy.bib.uottawa.ca/login?url=https://learning.oreilly.com/library/view/-/9781119214687/?ar&orpq&email=^u). When outliers are present, Pearson does not accurately measure association and robust measures are needed. This article introduces three new robust measures of correlation: Taba (T), TabWil (TW), and TabWil rank (TWR). The correlation estimators T and TW measure a linear association between two continuous or ordinal variables; whereas TWR measures a monotonic association. The robustness of these proposed measures in comparison with Pearson (P), Spearman (S), Quadrant (Q), Median (M), and Minimum Covariance Determinant (MCD) are examined through simulation. Taba distance is used to analyze genes, and statistical tests were used to identify those genes most significantly associated with Williams Syndrome (WS).ResultsBased on the root mean square error (RMSE) and bias, the three proposed correlation measures are highly competitive when compared to classical measures such as P and S as well as robust measures such as Q, M, and MCD. Our findings indicate TBL2 was the most significant gene among patients diagnosed with WS and had the most significant reduction in gene expression level when compared with control (P value = 6.37E-05).ConclusionsOverall, when the distribution is bivariate Log-Normal or bivariate Weibull, TWR performs best in terms of bias and T performs best with respect to RMSE. Under the Normal distribution, MCD performs well with respect to bias and RMSE; but TW, TWR, T, S, and P correlations were in close proximity. The identification of TBL2 may serve as a diagnostic tool for WS patients. A Taba R package has been developed and is available for use to perform all necessary computations for the proposed methods.

Local Dependence for Bivariate Weibull Distributions Created by Archimedean Copula

In multivariate survival analysis, estimating the multivariate distribution functions and then measuring the association between survival times are of great interest. Copula functions, such as Archimedean Copulas, are commonly used to estimate the unknown bivariate distributions based on known marginal functions. In this paper the feasibility of using the idea of local dependence to identify the most efficient copula model, which is used to construct a bivariate Weibull distribution for bivariate Survival times, among some Archimedean copulas is explored. Furthermore, to evaluate the efficiency of the proposed procedure, a simulation study is implemented. It is shown that this approach is useful for practical situations and applicable for real datasets. Moreover, when the proposed procedure implemented on Diabetic Retinopathy Study (DRS) data, it is found that treated eyes have greater chance for non-blindness compared to untreated eyes.

Beta Linear Failure Rate Geometric Distribution with Applications

In this paper we introduce bivariate Weibull distributions derived from copula functions in presence of cure fraction, censored data and covariates. Two copla functions are explored: the FGM (Farlie - Gumbel Morgenstern) copula and the Gumbel copula. Inferences for the proposed models are obtained under the Bayesian approach, using standard MCMC (Markov Chain Monte Carlo) methods. An illustration of the proposed methodology is given considering a medical data set.

Bivariate Weibull Distributions Derived From Copula Functions In The Presence Of Cure Fraction And Censored Data

In this paper, for evaluating and comparing the heterogeneous balance-variation order pair of any two decision-making trial and evaluation laboratory (DEMATEL) theories, in which one has larger balance and smaller variation, and on the contrary, the other one has smaller balance and larger variation, the first author proposed a useful integrated validity index to evaluate any DEMATEL theory by combining Liu's balanced coefficient and Liu's variation coefficient .Using this new validity index, three kinds of DEMATELs with a same direct relational matrix, including the traditional DEMATEL, shrinkage DEMATEL and balance DEMATEL, are compared, a simple validity experiment is conducted, the results show that the balance DEMATEL has the best performance, the performance of the shrinkage coefficient is better than that of the traditional DEMATEL.

Concomitants of order statistics for bivariate exponentiated inverted Weibull distribution

Concomitants of order statistics for bivariate exponentiated inverted Weibull distribution

A Bi-Level Weibull Model with Applications to Two Ordered Events

In this paper, we propose and study a new bivariate Weibull model, called Bi-levelWeibullModel, which arises when one failure occurs after the other. Under some specific regularity conditions, the reliability function of the second event can be above the reliability function of the first event, and is always above the reliability function of the transformed first event, which is a univariate Weibull random variable. This model is motivated by a common physical feature that arises fromseveral real applications. The two marginal distributions are a Weibull distribution and a generalized three-parameter Weibull mixture distribution. Some useful properties of the model are derived, and we also present the maximum likelihood estimation method. A real example is provided to illustrate the application of the model.

Multivariate Inverted Kumaraswamy Distribution: Derivation and Estimation

Industrial revolution leads to the manufacturing of multicomponent products; to guarantee the sufficiency of the product and consumer satisfaction, the producer has to study the lifetime of the products. This leads to the use of bivariate and multivariate lifetime distributions in reliability engineering. The most popular and applicable is Marshall–Olkin family of distributions. In this paper, a new bivariate lifetime distribution which is the bivariate inverted Kumaraswamy (BIK) distribution is found and its properties are illustrated. Estimation using both maximum likelihood and Bayesian approaches is accomplished. Using different selection criteria, it is found that BIK provides the best performance compared with other bivariate distributions like bivariate exponential and bivariate inverse Weibull distributions. As a generalization, the multivariate inverted Kumaraswamy (MIK) distribution is derived. Few studies have been conducted on the multivariate Marshall–Olkin lifetime distributions. To the best of our knowledge, none of them handle estimation process. In this paper, we developed an algorithm to show how to estimate the unknown parameters of MIK using both maximum likelihood and Bayesian approaches. This algorithm could be applied in estimating other Marshall–Olkin multivariate lifetime distributions.

Ageing Intensity Function for Conditionally Specified Models

Jiang et al. have introduced a quantitative measure known as the ageing intensity function for evaluating the ageing properties of a component/system. The present study extends this ageing intensity function to the conditionally specified and conditional survival models. In a two component system, these two conditional ageing intensity functions provide the ageing patterns of one component when the other component has either failed or survived a specified period of time. The proposed conditionally ageing intensity function models provide simple expressions for some commonly used bivariate lifetime models. Characterization relationships are established for bivariate exponential, bivariate Weibull, conditional proportional hazards and bivariate weighted models.

The Mean Value Theorem for Integrals Method for Estimating Two-Dimensional Renewal Functions

An important aspect in the provision of a two-dimensional warranty is the expected number of failures of a component during the two-dimensional warranty period. The purpose of this paper is to present a new method to obtain the expected number of failures of a nonrepairable compo­nent from the two-dimensional renewal functions as the so­lution of two-dimensional renewal integral equations through the Mean Value Theorem for Integrals (MeVTI) method. The two-dimensional renewal integral equation involves Lu-Bhattacharyya’s bivariate Weibull model as a two-dimensional failure model. It turns out that the estimation of the expected number of failures using the MeVTI method is close to that of the other method, Riemann-Stieljies method. The bivariate data behaviour of the failures of an automobile component is also studied in this paper.

Bivariate Software Reliability Growth Models under Budget Constraint for Development Management

Software reliability growth is observed by investing not only the testing-time but also the testing-effort in the testing-phase of software development process. If the testing-time (testing-effort) is reduced to some extent, it is possible to observe the software reliability growth by investing the amount of testing-effort (testing-time) which can compensate the insufficiency of the testing-time (testing-effort). However, most of the existing software reliability growth models (SRGMs) are constructed as univariate models and the substitutability between the testing-time and testing-effort is not considered. Additionally, it is necessary to remove many faults efficiently within the budget. In this paper, we develop bivariate Weibull type SRGMs under budget constraint based on the Cobb-Douglas type and CES (constant elasticity of substitution) type testing-time functions. Simultaneously, we evaluate the substitutability between the testing-time and testing-effort factors which are software reliability growth factors. Finally, we conduct the sensitivity analysis and show numerical examples by using actual data sets.

Posterior and predictive inferences for Marshall Olkin bivariate Weibull distribution via Markov chain Monte Carlo methods

This paper deals with the well known bi-variate Weibull distribution developed by Marshall and Olkin. In the light of prior information, this paper derives the posterior distribution and performs Markov chain Monte Carlo methods to obtain posterior based inferences. This paper also checks the sensitivity of posterior estimates by changing the prior variances followed by Bayesian prediction using sample-based approaches. Numerical illustrations are provided for real as well as simulated data sets.

A bivariate inverse Weibull distribution and its application in complementary risks model

ABSTRACTIn reliability and survival analysis the inverse Weibull distribution has been used quite extensively as a heavy tailed distribution with a non-monotone hazard function. Recently a bivariate inverse Weibull (BIW) distribution has been introduced in the literature, where the marginals have inverse Weibull distributions and it has a singular component. Due to this reason this model cannot be used when there are no ties in the data. In this paper we have introduced an absolutely continuous bivariate inverse Weibull (ACBIW) distribution omitting the singular component from the BIW distribution. A natural application of this model can be seen in the analysis of dependent complementary risks data. We discuss different properties of this model and also address the inferential issues both from the classical and Bayesian approaches. In the classical approach, the maximum likelihood estimators cannot be obtained explicitly and we propose to use the expectation maximization algorithm based on the missing value principle. In the Bayesian analysis, we use a very flexible prior on the unknown model parameters and obtain the Bayes estimates and the associated credible intervals using importance sampling technique. Simulation experiments are performed to see the effectiveness of the proposed methods and two data sets have been analyzed to see how the proposed methods and the model work in practice.

Inference about the bivariate new extended Weibull distribution based on complete and censored data

In this article, we have discussed the problem of point estimation of the three unknown parameters of a bivariate new extended Weibull distribution under complete and randomly right-censored samples. The expectation-maximization algorithm is used to estimate the unknown parameters. Simulation experiments are performed to see the effectiveness of the estimators for complete and censored data. One dataset has been considered to illustrate the practical utility of the article.

A Nifty mean chart method based on median ranked set sampling design

In the case of contamination for skewed distributions, the modified Shewhart, modified weighted variance, and modified skewness correction methods are newly introduced by Karagoz (Hacet J Math Stat 47(1):223–242, 2018). In this study, we propose to modify these methods by considering simple random sampling (SRS), ranked set sampling (RSS) and median ranked set sampling (MRSS) designs under the contaminated type I Marshall–Olkin bivariate Weibull and lognormal distributions. These bivariate distributions are chosen since they can represent a wide variety of shapes from nearly symmetric to highly skewed. We evaluate the performance of proposed modified methods based on different ranked set sampling designs by using Monte Carlo Simulation. The type I risks of $${\bar{X}}$$ charts for existing and newly proposed modified methods by using SRS, RSS and MRSS designs in the case of contamination for these distributions are obtained via simulation study. The proposed modified methods using RSS and MRSS designs for the $${\bar{X}}$$ chart can be a favorable substitute in process monitoring when the distribution is highly skewed and contaminated.

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.

Discrete Bivariate Distributions Generated By Copulas: DBEEW Distribution

Copulas are the most powerful tools in constructing continuous multivariate distributions given marginals and certain dependence structures. In this paper, we shall propose a general and novel method of generating discrete bivariate distributions using copulas. The advantage of our method is that, contrary to the standard methods, we do not need to have the joint distribution of the base variables, but we only need the marginal distributions. In particular, we shall concentrate on generating a new family of discrete bivariate exponentiated extended Weibull (DBEEW) distribution by a Cuadras–Auge copula. We shall study several mathematical properties of a DBEEW distribution. Three special submodels of our new proposed distribution are studied in more detail. Finally, estimation of parameters for these three submodels is investigated and their potential and flexibility are examined using a real data set.

Bivariate Weibull Distribution: Properties and Different Methods of Estimation

The bivariate Weibull distribution is an important lifetime distribution in survival analysis. In this paper, Farlie–Gumbel–Morgenstern (FGM) copula and Weibull marginal distribution are used for creating bivariate distribution which is called FGM bivariate Weibull (FGMBW) distribution. FGMBW distribution is used for describing bivariate data that have weak correlation between variables in lifetime data. It is a good alternative to bivariate several lifetime distributions for modeling real-valued data in application. Some properties of the FGMBW distribution are obtained such as product moment, skewness, kurtosis, moment generation function, reliability function and hazard function. Three different estimation methods for parameters estimation are discussed for FGMBW distribution namely; maximum likelihood estimation, inference function for margins method and semi-parametric method. To evaluate the performance of the estimators, a Monte Carlo simulations study is conducted to compare the preferences between estimation methods. Also, a real data set is introduced, analyzed to investigate the model and useful results are obtained for illustrative purposes.