Bayesian inferences given a type-2censored sample from a burr distribution

Bayesian statistics assumes that there are specific parameteric distributions for the unknown parameters. It fits the probability model of interest by incorporating prior information regarding the unknown parameters and the likelihood function of the observed data. Moreover, Bayesian statistics as well as non-Bayesian methods produce good asymptotic results. Using WinBUGS and R language, a binomial logistic regression model of voting choice in the 2002 Taipei mayoral election is developed. Adding the prior information from the first panel to the estimation of the second panel, the Bayesian model yields sharper estimates concerning the election outcome. Additionally, the replication of data provides a model check and the baseline of the new observations. The methodological contribution of this paper is the ability to fit a binary logistic regression model with the observed data using the Bayesian inference.

Bayesian approaches have been demonstrated as effective methods for reliability analysis of complex systems with small-amount data, which integrate prior information and sample data using Bayes’ theorem. However, there is an assumption that precise prior probability distributions are available for unknown parameters, yet these prior distributions are sometimes unavailable in practical engineering. A possible way to avoiding this assumption is to generalize Bayesian reliability analysis approach by using imprecise probability theory. In this paper, we adopt a set of imprecise Dirichlet distributions as priors to quantify uncertainty of unknown parameters and extend traditional Bayesian reliability analysis approach by introducing an imprecise Dirichlet model (IDM). When the prior information is rare, the result of imprecise Bayesian analysis method is too rough to support engineering decision-making, so we proposed an optimization model to reduce the imprecision of the new method. Spindles are crucial for machine tools and reliability data related to spindles of new-developed machine tools are often rare. We can then use the imprecise Bayesian reliability analysis method to assess its reliability. In this paper, we mainly investigate the reliability assessment of a motorized spindle to illustrate the effectiveness of the proposed method.

We tested whether it is beneficial for the accuracy of phylogenetic inference to sample characters that are evolving under different sets of parameters, using both Bayesian MCMC (Markov chain Monte Carlo) and parsimony approaches. We examined differential rates of evolution among characters, differential character-state frequencies and character-state space, and differential relative branch lengths among characters. We also compared the relative performance of parsimony and Bayesian analyses by progressively incorporating more of these heterogeneous parameters and progressively increasing the severity of this heterogeneity. Bayesian analyses performed better than parsimony when heterogeneous simulation parameters were incorporated into the substitution model. However, parsimony outperformed Bayesian MCMC when heterogeneous simulation parameters were not incorporated into the Bayesian substitution model. The higher the rate of evolution simulated, the better parsimony performed relative to Bayesian analyses. Bayesian and parsimony analyses converged in their performance as the number of simulated heterogeneous model parameters increased. Up to a point, rate heterogeneity among sites was generally advantageous for phylogenetic inference using both approaches. In contrast, branch-length heterogeneity was generally disadvantageous for phylogenetic inference using both parsimony and Bayesian approaches. Parsimony was found to be more conservative than Bayesian analyses, in that it resolved fewer incorrect clades.

Burr distribution is one of the most important types of distribution in Burr system and has gained special attention. It has an important role in various disciplines, such as reliability analysis, life testing, survival analysis, actuarial science, economics, forestry, hydrology and meteorology. Thus, the parameter estimation for Burr distribution becomes an important thing to do. The frequentist approach using the maximum likelihood method is the most commonly used way to estimate the parameters of a distribution. In this paper we considered using the Bayesian method to estimate the shape parameter k of Burr distribution using gamma prior which is a conjugate prior. The Bayes estimate for the shape parameter k is obtained under the squared-error loss function (SELF) which is one of the symmetric loss function and the precautionary loss function (PLF) which is one of the asymmetric loss function. Through a simulation study, the comparison was made on the performance of the Bayes estimate for the shape parameter k under these two loss functions with respect to the mean-squared error (MSE) and the posterior risk.

A general family of distributions, namely Kumaraswamy generalized family of (Kw-G) distribution, is considered for estimation of the unknown parameters and reliability function based on record data from Kw-G distribution. The maximum likelihood estimators (MLEs) are derived for unknown parameters and reliability function, along with its confidence intervals. A Bayesian study is carried out under symmetric and asymmetric loss functions in order to find the Bayes estimators for unknown parameters and reliability function. Future record values are predicted using Bayesian approach and non Bayesian approach, based on numerical examples and a monte carlo simulation.

Summary A study is made regarding the precision of various types of samples in estimating the parameter of an exponential population. The following cases are considered : (i) a singly or doubly truncated sample of size N relative to a complete sample of the same size ; (ii) a singly or doubly censored sample of size N relative to a complete sample of size n, where n is the number of measured observations in the censored sample, It is observed that order case (i) right truncated and doubly truncated samples are inferior to the complete sample, whereas the left truncated sample is as good as the complete sample. Under case (ii) left censored and doubly censored samples are observed to be superior to the complete sample, while the right censored sample is as good as the complete sample.

Development and comparison in uncertainty assessment based Bayesian modularization method in hydrological modeling

Event Abstract Back to Event 2nd level modelling in fMRI analysis with a clinically depressed sample - Comparisons between classical and Bayesian methods Peter Goodin1, 2*, Joseph Ciorciari1, Susan Rossell1, 2, Matt Hughes1 and Richard Nibbs1 1 Swinburne University, BPsyC, Australia 2 Monash Alfred Psychiatric Research Centre, Australia The use of the Frequentist statistics in fMRI analysis has been the mainstay of the field for the last 25 years. Several other methods have been developed as alternatives, including those based on Bayesian statistics. Bayesian analysis methods have the benefit of allowing interrelation of data through effect size and confidence thresholds, can be more robust to outlier data and may negate the need for stringent multiple comparison correction. When examining functional differences in those with a mental illness , this can lead to detection of more subtle activation that may be of importance. Depression is a disorder that despite a common theme of symptoms is also characterised by functional heterogeneities between individuals, making inferences using Frequentist methods sometimes difficult. Utilising both Frequentist and Bayesian 2nd level analysis methods offered by SPM8 , we examined differences in activation between a healthy control sample and those with major depressive disorder during a novel emotional processing n-back task. Activation using classical methods showed small differences in more posterior visual regions however this was only detectable with liberal thresholding (p = .001 uncorrected, k = 0). In contrast Bayesian analysis (y = 0, 95% confidence) uncovered widespread differences (including overlap with those areas found with the Frequentist method) in multiple regions implicated with the disorder. This suggests that using Bayesian methods in fMRI analysis with a clinical population could be extremely useful in detecting responses that may otherwise go unnoticed or under-represented in more traditional methods, however determination of effect size importance is paramount. Keywords: Depression, fMRI, working memory, Bayesian, Frequentist Conference: XII International Conference on Cognitive Neuroscience (ICON-XII), Brisbane, Queensland, Australia, 27 Jul - 31 Jul, 2014. Presentation Type: Poster Topic: Methods Development Citation: Goodin P, Ciorciari J, Rossell S, Hughes M and Nibbs R (2015). 2nd level modelling in fMRI analysis with a clinically depressed sample - Comparisons between classical and Bayesian methods. Conference Abstract: XII International Conference on Cognitive Neuroscience (ICON-XII). doi: 10.3389/conf.fnhum.2015.217.00203 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 19 Feb 2015; Published Online: 24 Apr 2015. * Correspondence: Mr. Peter Goodin, Swinburne University, BPsyC, Hawthorn, Australia, peter@hitiq.com Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Peter Goodin Joseph Ciorciari Susan Rossell Matt Hughes Richard Nibbs Google Peter Goodin Joseph Ciorciari Susan Rossell Matt Hughes Richard Nibbs Google Scholar Peter Goodin Joseph Ciorciari Susan Rossell Matt Hughes Richard Nibbs PubMed Peter Goodin Joseph Ciorciari Susan Rossell Matt Hughes Richard Nibbs Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.

Assessing the reliability of two production lines, whether individually or simultaneously, is of significant importance for improving manufacturing processes and guaranteeing superior product quality. This paper examines the reliability assessment of two production lines utilizing joint progressively Type-II censored samples derived from the XLindley distribution. In addition to estimating the unknown parameters, the reliability functions for each production line, as well as for both production lines simultaneously, are analyzed. Both classical likelihood-based and Bayesian methodologies are employed for estimation purposes. The maximum likelihood method is applied to obtain point estimates for the unknown parameters and reliability functions, while Bayesian analysis is performed under the squared error loss function, employing the Markov Chain Monte Carlo technique to generate samples from the posterior distribution. The approximate confidence intervals, percentile bootstrap confidence intervals, and the highest posterior density credible intervals for the unknown parameters and various reliability functions are discussed. The problem of selecting the optimal censoring plan is also considered. A comprehensive simulation study is conducted to assess the performance of the proposed methods, comparing the accuracy and efficiency of the estimates across different censoring schemes. Furthermore, the applicability of the proposed methodology is demonstrated through the analysis of two real-world data sets, underscoring its practical utility within the field of reliability. Finally, three criteria, namely A-optimality, D-optimality, and F-optimality, are considered to determine the optimal censoring plan.

The Burr XII distribution is similar to the log-normal distribution in that it has a non-monotone hazard function which can accommodate many shapes of hazard. However, the Burr distribution has the advantages that: a) the Burr cumulative distribution function and reliability function can be written in closed form; thus, it simplifies the computation of the percentiles and the likelihood function for censored data; and b) the Burr XII distribution has algebraic tails which are effective for modeling failures that occur with lesser frequency than with corresponding models based on exponential tails. The Burr XII distribution gives the reliability practitioner another model for representing failure data. In this paper, the statistical and probabilistic properties of the Burr XII distribution are presented, and its relationship to other distributions used in reliability analyses is described. Its use as a model for failure data is outlined, methods for graphical estimation on probability paper are illustrated, and examples of its usage are presented.

This paper develops maximum likelihood estimators (MLEs) of unknown param-eters in an exponentiated half-logistic distribution based on a progressively type-IIcensored sample. We obtain approximate con dence intervals for the MLEs by usingasymptotic variance and covariance matrices. Using importance sampling, we obtainBayes estimators and corresponding credible intervals with the highest posterior densityand Bayes predictive intervals for unknown parameters based on progressively type-IIcensored data from an exponentiated half logistic distribution. For illustration pur-poses, we examine the validity of the proposed estimation method by using real andsimulated data.Keywords: Bayes predictive interval, exponentiated half-logistic distribution, HPD cred-ible interval, importance sampling, progressively type-II censored sample. 1. Introduction Many reliability and survival analysis studies have used half-logistic distributions, partic-ularly for censored data. Several studies have drawn inferences about half logistic distribu-tions. Balakrishnan and Puthenpura (1986) introduce the best linear unbiased estimator oflocation and scale parameters of half-logistic distributions by considering linear functionsof order statistics. Balakrishnan and Wong (1991) obtain approximate maximum likelihoodestimators (AMLEs) of location and scale parameters of half-logistic distributions by usinga type-II right-censored sample. Kang et al. (2008) derive AMLEs and MLEs of scale pa-rameters of half-logistic distributions by using progressively type-II censored samples. Kanget al. (2009) propose AMLEs of scale parameters of half-logistic distributions by consideringdouble-hybrid censored samples. Kang and Seo (2011) recently examine exponentiated half-logistic distributions and propose two types of AMLEs of scale parameters and reliabilityfunctions by using progressively type-II right-censored samples. Kim et al. (2011a) deriveBayesian estimators of shape parameters, reliability functions, and failure rate functions

A weighted version of Lindley Distribution has been presented in this paper, offering a new single-parameter lifetime distribution. The expressions for the moment generating function (mgf), cumulant generating function (cgf), characteristic function, moments, reliability and hazard rate functions of the new weighted Lindley distribution have also been obtained. Tierney and Kadane approximation method has been used to derive Bayes estimators for its parameter (θ), reliability function R(t), and hazard rate function h(t) under three priors namely uniform, exponential and gamma. The findings have been illustrated using various randomly produced data sets from the proposed model using simulation technique, with each sample replicated 10,000 times. The Bayes Risks have been estimated under Squared Error Loss Function (SELF). Two real life data sets have further been used to illustrate its utility. It is finally concluded that gamma prior outperforms uniform and exponential priors for computing the Bayes estimates of the parameter θ, reliability function R(t) and hazard rate function h(t) of the proposed weighted Lindley distribution.

This study aims to construct the model for the length of hospital stay for patients with COVID-19 using quantile regression and Bayesian quantile approaches. The quantile regression models the relationship at any point of the conditional distribution of the dependent variable on several independent variables. The Bayesian quantile regression combines the concept of quantile analysis into the Bayesian approach. In the Bayesian approach, the Asymmetric Laplace Distribution (ALD) distribution is used to form the likelihood function as the basis for formulating the posterior distribution. All 688 patients with COVID-19 treated in M. Djamil Hospital and Universitas Andalas Hospital in Padang City between March-July 2020 were used in this study. This study found that the Bayesian quantile regression method results in a smaller 95% confidence interval and higher value than the quantile regression method. It is concluded that the Bayesian quantile regression method tends to yield a better model than the quantile method. Based on the Bayesian quantile regression method, it investigates that the length of hospital stay for patients with COVID-19 in West Sumatra is significantly influenced by Age, Diagnoses status, and Discharge status.

The Intra-Voxel Incoherent Motion (IVIM) model is largely adopted to estimate slow and fast diffusion coefficients of water molecules in biological tissues, which are used in cancer applications. The most reported fitting approach is a voxel-wise segmented non-linear least square, whereas Bayesian approaches with a direct fit, also considering spatial regularization, were proposed too. In this work a novel segmented Bayesian method was proposed, also in combination with a spatial regularization through a Conditional Autoregressive (CAR) prior specification. The two segmented Bayesian approaches, with and without CAR specification, were compared with two standard least-square and a direct Bayesian fitting methods. All approaches were tested on simulated images and real data of patients with head-and-neck and rectal cancer. Estimation accuracy and maps noisiness were quantified on simulated images, whereas the coefficient of variation and the goodness of fit were evaluated for real data. Both versions of the segmented Bayesian approach outperformed the standard methods on simulated images for pseudo-diffusion (D∗ ) and perfusion fraction (f), whilst the segmented least-square fitting remained the less biased for the diffusion coefficient (D). On real data, Bayesian approaches provided the less noisy maps, and the two Bayesian methods without CAR generally estimated lower values for f and D∗ coefficients with respect to the other approaches. The proposed segmented Bayesian approaches were superior, in terms of estimation accuracy and maps quality, to the direct Bayesian model and the least-square fittings. The CAR method improved the estimation accuracy, especially for D∗ .

For the purpose of improving the statistical efficiency of estimators in life-testing experiments, generalized Type-I hybrid censoring has lately been implemented by guaranteeing that experiments only terminate after a certain number of failures appear. With the wide applications of bathtub-shaped distribution in engineering areas and the recently introduced generalized Type-I hybrid censoring scheme, considering that there is no work coalescing this certain type of censoring model with a bathtub-shaped distribution, we consider the parameter inference under generalized Type-I hybrid censoring. First, estimations of the unknown scale parameter and the reliability function are obtained under the Bayesian method based on LINEX and squared error loss functions with a conjugate gamma prior. The comparison of estimations under the E-Bayesian method for different prior distributions and loss functions is analyzed. Additionally, Bayesian and E-Bayesian estimations with two unknown parameters are introduced. Furthermore, to verify the robustness of the estimations above, the Monte Carlo method is introduced for the simulation study. Finally, the application of the discussed inference in practice is illustrated by analyzing a real data set.