scholarly journals Hybrid Censoring Schemes with Mixture Gamma Failure Distribution

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Hassan M. Aljohani ◽  
Nada M. Alfaer
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Gamma Distribution ◽  
Type I ◽  
Type Ii ◽  
Posterior Density ◽  
Bayesian Approaches ◽  
Hybrid Censoring ◽  
Highest Posterior Density

Censoring schemes have received much attention over the past decades. Hybrid censoring schemes are censoring schemes mixed of type-I (T-1) and type-II (T-2) censoring schemes, a most popular area of study in life-testing or reliability experiments. More precisely, hybrid censoring can be described as a mixture of T-I and T-2 schemes. Gamma distribution is widely used, and its connection has more distributions. Mixture and single gamma distribution will be studied to estimate parameters, based on type-II hybrid censoring schemes (T-2HCS). We will apply algorithms to compute the maximum likelihood (ML) estimators and Bayesian approaches, using statistics, such as Markov chain Monte Carlo methods. Bayes estimators and corresponding highest posterior density confidence intervals will be tabled. Also, Markov chain Monte Carlo simulation is implemented to compare the performances of the different methods and the real dataset is analyzed for illustrative purposes.

A two-stage Markov chain Monte Carlo method for seismic inversion and uncertainty quantification

Geophysics ◽  
2019 ◽  
Vol 84 (6) ◽  
pp. R1003-R1020 ◽  
Author(s):  
Georgia K. Stuart ◽  
Susan E. Minkoff ◽  
Felipe Pereira
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Uncertainty Quantification ◽  
Bayesian Methods ◽  
Posterior Distributions ◽  
Mcmc Algorithm ◽  
Two Stage ◽  
Fine Grid ◽  
Highest Posterior Density

Bayesian methods for full-waveform inversion allow quantification of uncertainty in the solution, including determination of interval estimates and posterior distributions of the model unknowns. Markov chain Monte Carlo (MCMC) methods produce posterior distributions subject to fewer assumptions, such as normality, than deterministic Bayesian methods. However, MCMC is computationally a very expensive process that requires repeated solution of the wave equation for different velocity samples. Ultimately, a large proportion of these samples (often 40%–90%) is rejected. We have evaluated a two-stage MCMC algorithm that uses a coarse-grid filter to quickly reject unacceptable velocity proposals, thereby reducing the computational expense of solving the velocity inversion problem and quantifying uncertainty. Our filter stage uses operator upscaling, which provides near-perfect speedup in parallel with essentially no communication between processes and produces data that are highly correlated with those obtained from the full fine-grid solution. Four numerical experiments demonstrate the efficiency and accuracy of the method. The two-stage MCMC algorithm produce the same results (i.e., posterior distributions and uncertainty information, such as medians and highest posterior density intervals) as the Metropolis-Hastings MCMC. Thus, no information needed for uncertainty quantification is compromised when replacing the one-stage MCMC with the more computationally efficient two-stage MCMC. In four representative experiments, the two-stage method reduces the time spent on rejected models by one-third to one-half, which is important because most of models tried during the course of the MCMC algorithm are rejected. Furthermore, the two-stage MCMC algorithm substantially reduced the overall time-per-trial by as much as 40%, while increasing the acceptance rate from 9% to 90%.

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

2015 ◽  
Vol 38 (2) ◽  
pp. 467-484 ◽  
Author(s):  
Fatih Kizilaslan ◽  
Mustafa Nadar
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Multicomponent System ◽  
Credible Interval ◽  
Strength Model ◽  
Bayes Estimate ◽  
Bayesian Approaches ◽  
Lindley’S Approximation ◽  
Made In

<p>In this study, we consider a multicomponent system which has k independent and identical strength components X1,...,Xk and each component is exposed to a common random stress Y when the underlying distributions are Weibull. The system is regarded as operating only if at least s out of k (1 ≤ s ≤ k) strength variables exceeds the random stress. We estimate the reliability of the system by using frequentist and Bayesian approaches. The Bayes estimate of the reliability has 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 the reliability. The comparison of the reliability estimators is made in terms of the estimated risks by the Monte Carlo simulations.</p>

scholarly journals The Rise of Markov Chain Monte Carlo Estimation for Psychometric Modeling

2009 ◽  
Vol 2009 ◽  
pp. 1-18 ◽  
Author(s):  
Roy Levy
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Bayesian Approaches ◽  
Mcmc Estimation ◽  
Powerful Approach ◽  
Psychometric Modeling ◽  
Psychometric Models ◽  
Monte Carlo Estimation

Markov chain Monte Carlo (MCMC) estimation strategies represent a powerful approach to estimation in psychometric models. Popular MCMC samplers and their alignment with Bayesian approaches to modeling are discussed. Key historical and current developments of MCMC are surveyed, emphasizing how MCMC allows the researcher to overcome the limitations of other estimation paradigms, facilitates the estimation of models that might otherwise be intractable, and frees the researcher from certain possible misconceptions about the models.

scholarly journals Progenitor properties of type II supernovae: fitting to hydrodynamical models using Markov chain Monte Carlo methods

2020 ◽  
Vol 642 ◽  
pp. A143
Author(s):  
L. Martinez ◽  
M. C. Bersten ◽  
J. P. Anderson ◽  
S. González-Gaitán ◽  
F. Förster ◽  
...  
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Light Curve ◽  
Monte Carlo Methods ◽  
High Mass ◽  
Type Ii ◽  
Robust Method ◽  
Hydrodynamical Models ◽  
Starting Point

Context. The progenitor and explosion properties of type II supernovae (SNe II) are fundamental to understanding the evolution of massive stars. Particular attention has been paid to the initial masses of their progenitors, but despite the efforts made, the range of initial masses is still uncertain. Direct imaging of progenitors in pre-explosion archival images suggests an upper initial mass cutoff of ∼18 M⊙. However, this is in tension with previous studies in which progenitor masses inferred by light-curve modelling tend to favour high-mass solutions. Moreover, it has been argued that light-curve modelling alone cannot provide a unique solution for the progenitor and explosion properties of SNe II. Aims. We develop a robust method which helps us to constrain the physical parameters of SNe II by simultaneously fitting their bolometric light curve and the evolution of the photospheric velocity to hydrodynamical models using statistical inference techniques. Methods. We created pre-supernova red supergiant models using the stellar evolution code MESA, varying the initial progenitor mass. We then processed the explosion of these progenitors through hydrodynamical simulations, where we changed the explosion energy and the synthesised nickel mass together with its spatial distribution within the ejecta. We compared the results to observations using Markov chain Monte Carlo methods. Results. We apply this method to a well-studied set of SNe with an observed progenitor in pre-explosion images and compare with results in the literature. Progenitor mass constraints are found to be consistent between our results and those derived by pre-SN imaging and the analysis of late-time spectral modelling. Conclusions. We have developed a robust method to infer progenitor and explosion properties of SN II progenitors which is consistent with other methods in the literature. Our results show that hydrodynamical modelling can be used to accurately constrain the physical properties of SNe II. This study is the starting point for a further analysis of a large sample of hydrogen-rich SNe.

Statistical Approach on Grading the Student Achievement via Normal Mixture Modeling

2012 ◽  
Author(s):  
Zairul Nor Deana Md. Desa ◽  
Ismail Mohamad ◽  
Zarina Mohd. Khalid ◽  
Hanafiah Md. Zin
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Standard Deviation ◽  
Bayesian Methods ◽  
Mixture Distribution ◽  
Coefficient Of Determination ◽  
Normal Mixture ◽  
Posterior Density ◽  
Letter Grades

Kajian dijalankan untuk membanding keputusan yang didapati daripada tiga kaedah penggredan terhadap pencapaian pelajar. Kaedah konvensional yang popular adalah kaedah Skala Tegak. Pendekatan statistik yang menggunakan kaedah Sisihan Piawai dan kaedah Bayesian bersyarat dipertimbangkan untuk memberi gred. Dalam model Bayesian, dianggapkan bahawa data adalah mengikut taburan Normal Tergabung di mana setiap gred adalah dipisahkan secara berasingan oleh parameter; min dan kadar bandingan dari taburan Normal Tergabung. Masalah yang timbul adalah sukar untuk menganggarkan ketumpatan posterior bagi parameter tersebut secara analitik. Satu penyelesaiannya adalah dengan menggunakan pendekatan Markov Chain Monte Carlo iaitu melalui algoritma pensampelan Gibbs. Kaedah Skala Tegak, kaedah Sisihan Piawai dan kaedah Bayesian bersyarat diaplikasikan untuk markah mentah peperiksaan bagi dua kumpulan pelajar. Pencapaian ketiga–tiga kaedah dibandingkan melalui nilai Kehilangan Kelas Neutral, Kehilangan Kelas Tidak Tegas dan Pekali Penentuan. Didapati keputusan dari kaedah Bayesian bersyarat menunjukkan penggredan yang lebih baik berbanding kaedah Skala Tegak dan kaedah Sisihan Piawai. Kata kunci: Kaedah penggredan, pengukuran pendidikan, Skala Tegak, kaedah Sisihan Piawai, Normal Tergabung, Markov Chain Monte Carlo, pensampelan Gibbs The purpose of this study is to compare results obtained from three methods of assigning letter grades to students’ achievement. The conventional and the most popular method to assign grades is the Straight Scale method (SS). Statistical approaches which used the Standard Deviation (GC) and conditional Bayesian methods are considered to assign the grades. In the conditional Bayesian model, we assume the data to follow the Normal Mixture distribution where the grades are distinctively separated by the parameters: means and proportions of the Normal Mixture distribution. The problem lies in estimating the posterior density of the parameters which is analytically intractable. A solution to this problem is using the Markov Chain Monte Carlo approach namely Gibbs sampler algorithm. The Straight Scale, Standard Deviation and Conditional Bayesian methods are applied to the examination raw scores of two sets of students. The performances of these methods are measured using the Neutral Class Loss, Lenient Class Loss and Coefficient of Determination. The results showed that Conditional Bayesian outperformed the Conventional Methods of assigning grades. Key words: Grading methods, educational measurement, Straight Scale, Standard Deviation method, Normal Mixture, Markov Chain Monte Carlo, Gibbs sampling

An application of Markov chain Monte Carlo (MCMC) to continuous-time incurred but not yet reported (IBNYR) events

2016 ◽  
Vol 10 (2) ◽  
pp. 270-284
Author(s):  
Garfield O. Brown ◽  
Winston S. Buckley
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Confidence Intervals ◽  
Continuous Time ◽  
Bayesian Model ◽  
Secondary Data ◽  
Type I ◽  
Mcmc Method ◽  
Type Iv

AbstractWe develop a Bayesian model for continuous-time incurred but not yet reported (IBNYR) events under four types of secondary data, and show that unreported events, such as claims, have a Poisson distribution with a reduced arrival parameter if event arrivals are Poisson distributed. Using insurance claims as an example of an IBNYR event, we apply Markov chain Monte Carlo (MCMC) to the continuous-time IBNYR claims model of Jewell using Type I and Type IV data. We illustrate the relative stability of the MCMC method versus the Gammoid approximation of Jewell by showing that the MCMC estimates approach their prior parameters, while the Gammoid approximations grow without bound for Type IV data. Moreover, this holds for any distribution that the delay parameter is assumed to follow. Our framework also allows for the computation of posterior confidence intervals for the parameters.

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