Classical and Bayesian Estimation of Reliability in Multicomponent Stress-Strength Model Based on Weibull Distribution

This paper deals with the classical and Bayesian estimation of multicomponent stress-strength reliability through assuming the exponentiated inverse Rayleigh distribution. Assuming that both stress and strength variates are follows to exponentiated inverse Rayleigh distribution with common and known shape parameter. The multicomponent stress-strength reliability of a system is obtained by the methods of maximum likelihood and Bayesian approach. The results are compared using Markov Chain Monte Carlo (MCMC) technique for both small and large samples. Finally, two data sets of coating weights of iron sheets are analyzed for illustrative purposes.

In this article, the reliability inference for a multicomponent stress-strength (MSS) model, when both stress and strength random variables follow inverse Topp-Leone distributions, was studied. The maximum likelihood and uniformly minimum variance unbiased estimates for the reliability of MSS model were obtained explicitly. The exact Bayes estimate of MSS reliability was derived the under squared error loss function. Also, the Bayes estimate was obtained using the Monte Carlo Markov Chain method for comparison with the aforementioned exact estimate. The asymptotic confidence interval was determined under the expected Fisher information matrix. Furthermore, the highest probability density credible interval was established through using Gibbs sampling method. Monte Carlo simulations were implemented to compare the different proposed methods. Finally, a real life example was presented in support of the suggested procedures.

ABSTRACTThe interplay between the Bayesian and Frequentist approaches: a general nesting spatial panel-data model. Spatial Economic Analysis. An econometric framework mixing the Frequentist and Bayesian approaches is proposed in order to estimate a general nesting spatial model. First, it avoids specific dependency structures between unobserved heterogeneity and regressors, which improves mixing properties of Markov chain Monte Carlo (MCMC) procedures in the presence of unobserved heterogeneity. Second, it allows model selection based on a strong statistical framework, characteristics that are not easily introduced using a Frequentist approach. We perform some simulation exercises, finding good performance of the properties of our approach, and apply the methodology to analyse the relation between productivity and public investment in the United States.

In the literature, there are well-developed estimation techniques that can be used to measure the reliability in multicomponent stress-strength models when the information about all experimental units are available. However, in real applications, only observations that exceed (or fall below) the current value may be recorded. In this article, assuming that the components of the system follow generalized Rayleigh distribution, we investigate classical and Bayesian estimation of the reliability of a multicomponent stress-strength model when the available data are reported in terms of record values. The maximum likelihood estimate of the reliability parameter and its asymptotic confidence interval are obtained. Moreover, two confidence intervals based on the parametric bootstrap methods are presented. The Bayes estimate of the reliability parameter is derived using different loss functions. Because there is no closed form for the Bayes estimate, we use the Markov chain Monte Carlo method to obtain an approximate Bayes estimate and the highest posterior density credible interval of the reliability. To evaluate the performances of different procedures, simulation studies are conducted, and an example of real data sets is provided.

Stress-strength models are of special importance in reliability literature and engineering applications. This paper deals with the estimation problem of a stress-strength model incorporating multi-component system. The system is regarded as alive only if at least S out of k ( S < k ) strength components exceed the stress. The reliability of such system is obtained when strength and stress variables have Weibull distributions. Maximum likelihood estimator of R S , k and asymptotic confidence intervals are obtained based on upper record values. Bayesian estimator under squared error and linear exponential loss functions using gamma prior distributions and the corresponding credible intervals are obtained. Due to the lack of explicit forms for the Bayes estimates, the Markov Chain Monte Carlo (MCMC) method is employed. A simulation study is implemented to assess the performance of estimates. A real-life example is presented to show how the proposed model may be utilized in breaking strength of jute fibre data.

In this paper, we propose an estimate of reliability in a multicomponent system. The system has $$k$$ components strengths are given by independently and identically distributed random variables $${X}_{1}$$ , $${X}_{2}$$ ,…, $${X}_{k}$$ and each component is exposed to random stress $${\rm Y}$$ . The reliability of such a system is obtained when strength and stress variables are given by inverse Weibull (IW) distribution with scale parameters $${\lambda }_{1}$$ , $${\lambda }_{2}$$ and common shape parameter $$\alpha$$ . The system reliability is estimated using maximum likelihood estimation (MLE) and the best two-observational percentile estimation (BTPE) methods in samples drawn from strength and stress distributions. Also, the asymptotic confidence interval for system reliability is obtained. The reliability estimators obtained from both methods are compared using average bias, mean squares error, and confidence interval length via Monte Carlo simulation. In the end, using two real data sets we illustrate the procedure.

In this study we compared the efficiency of frequentist approach with Bayesian approach by carrying out extreme value analysis of Annual Maximum Daily Rainfall (AMDR). For frequentist frequency analysis of AMDR, we used the data of one station i.e. Lahore in Punjab province, Pakistan while for Bayesian analysis we used the data of three other neighboring stations as prior information. During frequentist approach, Generalized Extreme Value (GEV) was found to be a best-fit distribution on the data. In frequentist method, the parameters of GEV distribution were estimated using Maximum Likelihood Estimation (MLE), while in the Bayesian framework the Markov Chain Monte Carlo (MCMC) simulation technique along with Metropolis-Hasting algorithm and Gibbs sampler were implemented. Findings of this study indicate that despite the asymptotic requirements of the MLE, its performance can be enhanced by adopting the Bayesian MCMC paradigm. In order to acquire the posterior densities of GEV parameters, non-informative and informative priors based on the historical data of surrounding weather stations were employed. The result of Bayesian MCMC might be affected by the choice of priors. In addition, the performance of the parameters estimation methods was verified by employing several robustness measures. Robustness measures results proved that the Bayesian MCMC method performed better than MLE in estimating GEV parameters and future return levels. Therefore, the findings of these analyses could be helpful in adopting proper flood protection measures and designing infrastructures of culverts, buildings, bridges and dams in the region.

As alluded to in the chapter “Linkage Analysis of Qualitative Traits”, neither the Elston–Steward algorithm nor the Lander–Green approach is amenable to genetic data from large complex pedigrees and a large number of markers. In such cases, Monte Carlo estimation methods provide a viable alternative to the exact solutions. Two types of Monte Carlo methods have been developed for linkage analysis, haplotype inference, and other kinds of genetic analysis. They are Markov chain Monte Carlo (MCMC) methods and Monte Carlo methods that are based on independent samples. Approaches based on Markov chain Monte Carlo methods are more widely applicable; there is practically no limit on the size or complexity of the pedigrees, nor on the number of markers to be considered simultaneously. In this chapter, we will review the basic principles of MCMC methods for multipoint linkage analysis with extended pedigrees. Both simulations and application to data from the Framingham study will be used to compare and contrast three MCMC software packages: LOKI, MORGAN, and SIMWALK.

In this paper, the authors outline the general principles behind an approach to Bayesian system identification and highlight the benefits of adopting a Bayesian framework when attempting to identify models of nonlinear dynamical systems in the presence of uncertainty. It is then described how, through a summary of some key algorithms, many of the potential difficulties associated with a Bayesian approach can be overcome through the use of Markov chain Monte Carlo (MCMC) methods. The paper concludes with a case study, where an MCMC algorithm is used to facilitate the Bayesian system identification of a nonlinear dynamical system from experimentally observed acceleration time histories.

Airport air traffic is one of the most important and hardest one among all the airport data forecasts. In this paper, the Markov Chain Monte Carlo (MCMC) method of applied theory of statistics has been introduced into the aviation sector, and the discussion on airport air traffic forecast has been conducted taking Shanghai airport as the application background. MCMC method is designed to solve the problem parameters expectations where some baysian inference can not be directly calculated. This paper conducts the iterative computation using WinBUGS, EViews is used to estimate the initial iteration parameter of the regression model, and it is on this basis that the regression equation under the MCMC method is obtained.

This paper presents an overview of Markov Chain Monte Carlo (MCMC) methods for statistical inference and applications. The article begins by describing ordinary Monte Carlo methods, which in principle has the same goals as the MCMC but can hardly be implemented in practice. Following that basic Markov Chain Monte Carlo is discussed, which is founded on the Hastings algorithm and includes Metropolis method and the Gibbs sampler as special cases. Finally, various special applications of Markov Chain Monte Carlo methods are briefly mentioned and some recent development of MCMC are covered in final remarks section.KeywordsReal EstateMarkov Chain Monte CarloGibbs SamplerDetailed BalanceMarkov Chain Monte Carlo MethodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The conventional Markov chain Monte Carlo (MCMC) method is limited to the selected shape and size of proposal distribution and is not easy to start when the initial proposal distribution is far away from the target distribution. To overcome these drawbacks of the conventional MCMC method, two useful improvements in MCMC method, adaptive Metropolis (AM) algorithm and delayed rejection (DR) algorithm, are attempted to be combined. The AM algorithm aims at adapting the proposal distribution by using the generated estimators, and the DR algorithm aims at enhancing the efficiency of the improved MCMC method. Based on the improved MCMC method, a Bayesian amplitude versus offset (AVO) inversion method on the basis of the exact Zoeppritz equation has been developed, with which the P- and S-wave velocities and the density can be obtained directly, and the uncertainty of AVO inversion results has been estimated as well. The study based on the logging data and the seismic data demonstrates the feasibility and robustness of the method and shows that all three parameters are well retrieved. So the exact Zoeppritz-based nonlinear inversion method by using the improved MCMC is not only suitable for reservoirs with strong-contrast interfaces and long-offset ranges but also it is more stable, accurate and anti-noise.

The maximum likelihood and Bayesian approaches for parameter estimations and prediction of future record values have been considered for the two-parameter Burr Type XII distribution based on record values with the number of trials following the record values (inter-record times). Firstly, the Bayes estimates are obtained based on a joint bivariate prior for the shape parameters. In this case, the Bayes estimates of the parameters have been developed by using Lindley's approximation and the Markov Chain Monte Carlo (MCMC) method due to the lack of explicit forms under the squared error and the linear-exponential loss functions. The MCMC method has been also used to construct the highest posterior density credible intervals. Secondly, the Bayes estimates are obtained with respect to a discrete prior for the first shape parameter and a conjugate prior for other shape parameter. The Bayes and the maximum likelihood estimates are compared in terms of the estimated risk by the Monte Carlo simulations. We further consider the non-Bayesian and Bayesian prediction for future lower record arising from the Burr Type XII distribution based on record data. The comparison of the derived predictors is carried out by using Monte Carlo simulations. A real data are analysed for illustration purposes.

This paper takes into account the estimation of system reliability of a multi-component stress–strength model of a s-out-of-k system, where both strength and stress are non-identical components and follow Weibull and Burr-III distributions, respectively. The reliability of such a system is obtained by the methods of maximum likelihood and Bayesian approach. We consider Bayes estimation under squared error loss function using conjugate priors for the parameters involved in the models. We propose Markov Chain Monte Carlo (MCMC) techniques to generate samples from the posterior distributions and in turn computing the Bayes estimator of the system reliability. Simulation study shows that Bayes estimator works better as compared to maximum likelihood estimator.

Markov chain Monte Carlo (MCMC) methods offer a rapid parametric approach that can test for linkage throughout the entire genome. It has an advantage similar to nonparametric methods in that the model does not have to be completely specified a priori. However, unlike nonparametric methods, there are no limitations on pedigree size and MCMC methods can also handle relatively complex pedigree structures. In addition MCMC methods can be used to carry segregation analysis in order to answer questions on the genetic components of a disease phenotype. Segregation analysis gave evidence for between two and eight alcoholism susceptibility loci, each having a modest effect on the phenotype. MCMC methods were used to map alcoholism loci using the phenotypes ALDX1 (DSM-III-R and Feighner criteria) and ALDX2 (World Health Organization diagnosis ICD-10 criteria). There was mild evidence for quantitative trait loci on chromosomes 2, 10, and 11.