Bivariate Weibull Distribution Research Articles

Estimation based on Ranked Set Sampling for Farlie--Gumbel--Morgenstern Bivariate Weibull Distribution Parameters with an application to medical data

Estimation based on Ranked Set Sampling for Farlie--Gumbel--Morgenstern Bivariate Weibull Distribution Parameters with an application to medical data

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.

UDTCWT-PHFMs Domain Statistical Image Watermarking Using Vector BW-Type R Distribution

For any image watermarking algorithm, how to achieve the trade-off among robustness, imperceptibility, and watermark capacity is a challenging problem because of their mutually constrained relationship. In this paper, we design a statistical-based image watermarking system to solve the trade-off problem. In embedding process, consider the imperceptibility and the robustness, we inventively combine the undecimated dual tree complex Wavelet transform (UDTCWT) and the polar harmonic Fourier moments (PHFMs) to obtain the UDTCWT-PHFMs magnitudes as the watermark carriers, and we embed watermark signals in multiplicative manner. In modeling phase, by analyzing the statistical property and the multiple correlations of the UDTCWT-PHFMs magnitudes, we bound the magnitudes of different directions into the vector groups, and then we model them with the vector Roy type bivariate Weibull (BW-Type R) distribution so that we can capture accurately the marginal characteristics, the inter-scale dependencies, and the inter-orientation dependencies at the same time. Moreover, we obtain the model parameters with the double looped maximum likelihood estimation (double-looped MLE). Benefit from the reliable modeling result, we finally derive a novel specific image watermark decoder to blindly extract the hidden watermarks. Extensive experiment results declare the designed statistical image watermarking system achieves the better balance among imperceptibility, robustness, and payload.

A hierarchical Bayesian analysis for bivariate Weibull distribution under left-censoring scheme

This paper presents a novel approach for analyzing bivariate positive data, taking into account a covariate vector and left-censored observations, by introducing a hierarchical Bayesian analysis. The proposed method assumes marginal Weibull distributions and employs either a usual Weibull likelihood or Weibull–Tobit likelihood approaches. A latent variable or frailty is included in the model to capture the possible correlation between the bivariate responses for the same sampling unit. The posterior summaries of interest are obtained through Markov Chain Monte Carlo methods. To demonstrate the effectiveness of the proposed methodology, we apply it to a bivariate data set from stellar astronomy that includes left-censored observations and covariates. Our results indicate that the new bivariate model approach, which incorporates the latent factor to capture the potential dependence between the two responses of interest, produces accurate inference results. We also compare the two models using the different likelihood approaches (Weibull or Weibull–Tobit likelihoods) in the application. Overall, our findings suggest that the proposed hierarchical Bayesian analysis is a promising approach for analyzing bivariate positive data with left-censored observations and covariate information.

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.

New efficient estimators for the Weibull distribution

For some distributions, there is no available closed-form maximum likelihood estimator (MLE). This could be a problem when estimating the parameters of state-space models or of real-time processing models, because of time consuming iterations required to obtain an MLE. A simple method of obtaining a closed-form and efficient estimators for the univariate Weibull distribution is presented. Furthermore, an alternative efficient estimator for bivariate Weibull distribution parameters other than the MLE is also suggested. For the bivariate Weibull distribution, one estimator is not in closed-form, but the others are all in closed-form. In general, if we have a closed-form n -consistent estimator for each parameter, then we have a closed-form solution set which is efficient and has asymptotic normality. This is quite striking since we can obtain such a closed-form and efficient estimator using a very simple strategy. Roughly speaking, our suggested estimator is approximately 11 times as fast as the MLE for univariate case and 85 times for bivariate case. A simulation study is done for univariate and bivariate Weibull distributions to find some small sample properties. A real data illustration is also provided.

New closed-form efficient estimators for a bivariate Weibull distribution

This study aimed to develop new closed-form and efficient estimators for the parameters of the bivariate Weibull distribution. New estimators can be produced using closed-form -consistent estimators for all parameters, except for the association parameter for which the estimator is not in closed form. This is carried out by utilizing a theorem that produces asymptotically efficient estimators. To achieve this, -consistent estimators are introduced. Fisher observed and expected information matrices are derived and used to develop new estimators. A simulation study and real data application are included to validate the new estimators. Given that the new estimators are as asymptotically efficient as maximum likelihood estimators and are in closed form, except for the association parameter, they can be used effectively in state-space modelling or real-time processing models. This is because of the shorter computing time associated with them than with maximum likelihood estimators.

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.

An approach for evaluating the stochastic behaviour of wave energy converters

Due to their random nature, obtaining reliable models that can describe the behaviour of waves is far from simple. This paper presents an approach for forecasting the capabilities of wave energy converters (WECs) for two points, one of them located offshore and the other nearshore. Bivariate Weibull distributions were fitted from spectral significant wave height and mean peak period data. Then, models relating the parameters of these distributions to the day of the year were obtained using mixture density networks, which give the distribution of the predicted variables instead of their expected value. Energy conversion capabilities were forecasted by generating a set of random values for the bivariate Weibull coefficients from the modelled distributions for the period in question. Predicted cumulative distributions for spectral significant wave heights and mean peak periods were then combined with the matrix of the converter in question, allowing the corresponding energy conversion capability to be computed. The proposed method was validated by considering data from the last three years, which were not used to train the models. The resulting predictions were consistent not only with the expected seasonal behaviour, but also with the expected differences between the offshore and nearshore points. It should be also noted that all the validation energy values fall into the forecasted 95% confidence intervals, showing the effectiveness of the approach.

Bivariate Generalized Half-Logistic Distribution: Properties and Its Application in Household Financial Affordability in KSA

The generalized half-logistic distribution is ideal to fit the lifetime of some products, such as ball bearings and electrical insulation. In this paper, we aim to extend this scope by creating a motivated bivariate version. We thus introduce the bivariate generalized half-logistic distribution using the Farlie Gumbel Morgenstern (FGM) copula, which is called the FGM bivariate generalized half-logistic distribution (FGMBGHLD for short). In particular, the FGMBGHLD finds application in describing bivariate lifetime datasets that have weak correlations between variables. Some statistical properties and functions of our new distribution, such as the product moments, moment generating function, reliability function, and hazard rate function, are derived. We discuss the maximum likelihood estimation method of the FGMBGHLD parameters. As an application of the FGMBGHLD in reliability, we consider the stress–strength model when the stress and strength random variables are dependent. We also derive the point and interval estimates of the stress–strength coefficient. Finally, we use the data from the household income and expenditure survey of KSA 2018 for Saudi households by administrative region to demonstrate the practicability of the proposed model. A comparison with a modern bivariate Weibull distribution is performed.

Progressive Type-II Censored Samples for Bivariate Weibull Distribution with Economic and Medical Applications

Progressive Type-II Censored Samples for Bivariate Weibull Distribution with Economic and Medical Applications

Some bivariate and multivariate families of distributions: Theory, inference and application

<abstract> <p>The bivariate and multivariate probability distributions are useful in joint modeling of several random variables. The development of bivariate and multivariate distributions is relatively tedious as compared with the development of univariate distributions. In this paper we have proposed a new method of developing bivariate and multivariate families of distributions from the univariate marginals. The properties of the proposed families of distributions have been studies. These properties include marginal and conditional distributions; product, ratio and conditional moments; joint reliability function and dependence measures. Statistical inference about the proposed families of distributions has also been done. The proposed bivariate family of distributions has been studied for Weibull baseline distribution giving rise to a new bivariate Weibull distribution. The properties of the proposed bivariate Weibull distribution have been studied alongside maximum likelihood estimation of the unknown parameters. The proposed bivariate Weibull distribution has been used for modeling of real bivariate data sets and we have found that the proposed bivariate Weibull distribution has been a suitable choice for the modeling of data used.</p> </abstract>

Methodology on Establishing Multivariate Failure Thresholds for Improved Remaining Useful Life Prediction in PHM

Prognostics and Health Management (PHM) methodologies and techniques have been much widely studied in the academia and practiced by the industry in recent years. Prognostic approaches commonly try to establish the relationship between Remaining Useful Life (RUL) and a single variable or health indicator (HI) which can be obtained from multi-sensor fusion or data-driven models. However, simply relying on a single variable could reduce RUL prediction robustness when it is less representative of the system health conditions. Taking multiple variables into consideration for RUL prediction, quantifying operating risks and determining multivariate failure threshold is essential yet rarely studied. Generally, there are three major challenges that limit the practicality of this topic. 1) How to determine the multivariate failure threshold? 2) How to quantify operation risks based on multiple variables? 3) How to make reliable extrapolations of future conditions? To address these questions, this paper proposes 1) a novel copula model to determine multivariate failure threshold, and 2) a Maximum Likelihood Estimation enhanced similarity-based Particle Filter (MLE-SMPF) to predict future system conditions. In the proposed methodology, the health assessment is firstly performed to obtain HI trajectory. The copula risk quantification model is then trained by two variables HI and life. The proposed copula model can easily include multiple variables compared with our previously published approach using bivariate Weibull Distribution[1]. Afterward, MLE-SMPF is used to extrapolate future HI for testing data. The prediction capability is further improved compared with [2] by introducing MLE for Particle Filter transition function parameter initialization. Finally, the system RUL is determined from the failure threshold which is obtained according to the quantified operation risk. The proposed methodology is validated on the C-MAPSS data from the PHM data competition 2008 hosted by PHM society. The result outperforms most of the benchmarks from recent publications. The proposed methodology is easy to transfer to other potential machine prognostic applications.

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.

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.

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.