Bayesian Estimation of the Parameters of Bivariate Exponential Distributions

In this article, we develop an empirical Bayesian approach for the Bayesian estimation of parameters in four bivariate exponential (BVE) distributions. We have opted for gamma distribution as a prior for the parameters of the model in which the hyper parameters have been estimated based on the method of moments and maximum likelihood estimates (MLEs). A simulation study was conducted to compute empirical Bayesian estimates of the parameters and their standard errors. We use moment estimators or MLEs to estimate the hyper parameters of the prior distributions. Furthermore, we compare the posterior mode of parameters obtained by different prior distributions and the Bayesian estimates based on gamma priors are very close to the true values as compared to improper priors. We use MCMC method to obtain the posterior mean and compared the same using the improper priors and the classical estimates, MLEs.     

Heteroscedastic Weibull-Normal Mixture Models: A Bayesian Approach

In this article, we propose Bayesian methodology to obtain parameter estimates of the mixture of distributions belonging to the normal and biparametric Weibull families, modeling the mean and the variance parameters. Simulated studies and applications show the performance of the proposed models.     

Bayesian Estimation of Bivariate Exponential Distributions Based on Linex and Quadratic Loss Functions: A Survival Approach with Censored Samples

In this article, four bivariate exponential (BVE) distributions with subject to right censoring samples are presented. Bayesian estimates of the parameters of BVE are obtained through Linex and quadratic loss functions. Gamma prior distribution has been suggested to reforming the posterior function. The estimations and standard errors of parameters have also been obtained through simulation method. Markov chain Monte Carlo (MCMC) method is employed for the case of Block-Buse bivariate distribution because there was no closed form for estimator criteria. Simulation studies have been conducted to show that the computation parts can be implemented easily and comparing the estimated values due to two methods and with the true values as well.     

Mixture of Bivariate Exponential Distributions

The exponential distribution is one of the most used type of distribution because of its importance in many lifetime applications and its properties. So is its bivariate form. Simply used, there can be limitations specially for the heterogeneous type population. Its mixture form adds a lot of characters and desirable properties. We propose a mixture of bivariate exponential distribution, study properties of the associated parameters and predict the elements of the mixture. We include the presence of covariate information through a linear relationship, capturing the now famous idea by Marshall and Olkin.     

Bayesian Approach to Multicentre Sparse Data

In a 2 × 2 contingency table, when the sample size is small, there may be a number of cells that contain few or no observations, usually referred to as sparse data. In such cases, a common recommendation in the conventional frequentist methods is adding a small constant to every cell of the observed table to find the estimates of the unknown parameters. However, this approach is based on asymptotic properties of the estimates and may work poorly for small samples. An alternative approach would be to use Bayesian methods in order to provide better insight into the problem of sparse data coupled with fewer centers, which would otherwise be difficult to carry out the analysis. In this article, an attempt has been made to use hierarchical Bayesian model to a multicenter data on the effect of a surgical treatment with standard foot care among leprosy patients with posterior tibial nerve damage which is summarized as seven 2 × 2 tables. Monte Carlo Markov Chain (MCMC) techniques are applied in estimating the parameters of interest under sparse data setup.     

A Simple Solution to Bayesian Mixture Labeling

The label-switching problem is one of the fundamental problems in Bayesian mixture analysis. Using all the Markov chain Monte Carlo samples as the initials for the expectation-maximization (EM) algorithm, we propose to label the samples based on the modes they converge to. Our method is based on the assumption that the samples converged to the same mode have the same labels. If a relative noninformative prior is used or the sample size is large, the posterior will be close to the likelihood and then the posterior modes can be located approximately by the EM algorithm for mixture likelihood, without assuming the availability of the closed form of the posterior. In order to speed up the computation of this labeling method, we also propose to first cluster the samples by K-means with a large number of clusters K. Then, by assuming that the samples within each cluster have the same labels, we only need to find one converged mode for each cluster. Using a Monte Carlo simulation study and a real dataset, we demonstrate the success of our new method in dealing with the label-switching problem.     

A New Bayesian Nonparametric Mixture Model

We propose a new mixture model for Bayesian nonparametric inference. Rather than considering extensions from current approaches, such as the mixture of Dirichlet process model, we end up shrinking it, by making the weights less complex. We demonstrate the model and discuss its performance.     

A Bayesian analysis for the Block and Basu bivariate exponential distribution in the presence of covariates and censored data

In this paper, we introduce a Bayesian Analysis for the Block and Basu bivariate exponential distribution using Markov Chain Monte Carlo (MCMC) methods and considering lifetimes in presence of covariates and censored data. Posterior summaries of interest are obtained using the popular WinBUGS software. Numerical illustrations are introduced considering a medical data set related to the recurrence times of infection for kidney patients and a medical data set related to bone marrow transplantation for leukemia.     

Bayesian multivariate Poisson mixtures with an unknown number of components

In this paper we present Bayesian analysis of finite mixtures of multivariate Poisson distributions with an unknown number of components. The multivariate Poisson distribution can be regarded as the discrete counterpart of the multivariate normal distribution, which is suitable for modelling multivariate count data. Mixtures of multivariate Poisson distributions allow for overdispersion and for negative correlations between variables. To perform Bayesian analysis of these models we adopt a reversible jump Markov chain Monte Carlo (MCMC) algorithm with birth and death moves for updating the number of components. We present results obtained from applying our modelling approach to simulated and real data. Furthermore, we apply our approach to a problem in multivariate disease mapping, namely joint modelling of diseases with correlated counts.     

Bayesian Estimation Using Warner's Randomized Response Model through Simple and Mixture Prior Distributions

The linear discriminant function (LDF) is known to be optimal in the sense of achieving an optimal error rate when sampling from multivariate normal populations with equal covariance matrices. Use of the LDF in nonnormal situations is known to lead to some strange results. This paper will focus on an evaluation of misclassification probabilities when the power transformation could have been used to achieve at least approximate normality and equal covariance matrices in the sampled populations for the distribution of the observed random variables. Attention is restricted to the two-population case with bivariate distributions.     

A comparison of nonparametric priors in hierarchical mixture modelling for AFT regression

We will pursue a Bayesian nonparametric approach in the hierarchical mixture modelling of lifetime data in two situations: density estimation, when the distribution is a mixture of parametric densities with a nonparametric mixing measure, and accelerated failure time (AFT) regression modelling, when the same type of mixture is used for the distribution of the error term. The Dirichlet process is a popular choice for the mixing measure, yielding a Dirichlet process mixture model for the error; as an alternative, we also allow the mixing measure to be equal to a normalized inverse-Gaussian prior, built from normalized inverse-Gaussian finite dimensional distributions, as recently proposed in the literature. Markov chain Monte Carlo techniques will be used to estimate the predictive distribution of the survival time, along with the posterior distribution of the regression parameters. A comparison between the two models will be carried out on the grounds of their predictive power and their ability to identify the number of components in a given mixture density.     

A package of four edited books on structural equations modeling

We consider two analytical and a bootstrap bias correction scheme existing in the literature for maximum likelihood estimators (MLEs) in the special case of a particular biparametric exponential family, the estimators being obtained from i.i.d. samples. We assess the performances of the estimators through numerical simulations for three particular cases of the family explored here. We observe that the two analytical proposals display very similar behavior for these distributions and that all proposed estimators are effective in reducing bias and mean square error of the MLEs.     

基于MCMC模拟的贝叶斯分层信用风险评估模型

缺少违约数据与债务人异质性是度量信用风险时面临的重要问题。贝叶斯模型中分层先验信息和马尔可夫链蒙特卡罗（MCMC）模拟方法的应用可以有效缓解数据缺失和测量误差问题,并能对债务人异质性进行评价和比较,从而避免低估风险。针对银行数据的模型拟合与模型诊断均展现了分层估计的适应性和灵活性,相关方法简洁清晰,利于国内风险分析人员采用。同时,涵盖宏观经济协变量的贝叶斯分层模型可以用于更加复杂的风险分析。     

On Unimodal Regression in the Exponential Family

In this article, we discuss statistical methods for curve-estimation under the assumption of unimodality for variables with distributions belonging to the two-parameter exponential family with known or constant dispersion parameter. An important special case is a one-parameter distribution. We suggest a nonparametric method based on monotonicity properties. The method is applied to Swedish data on laboratory verified diagnoses of influenza and data on inflation from an episode of hyperinflation in Bulgaria.     

指数族分布中参数的经验贝叶斯估计

指数族分布是一类应用广泛的分布类,包括了泊松分布、Gamma分布、Beta分布、二项分布等常见分布.在非寿险中,索赔额或索赔次数过程常常被假定服从指数族分布,由于风险的非齐次性,指数族分布中的参数θ也为随机变量,假定服从指数族共轭先验分布.此时风险参数的估计落入了Bayes框架,风险参数θ的Bayes估计被表达“信度”形式.然而,在实际运用中,由于先验分布与样本分布中仍然含有结构参数,根据样本的边际分布的似然函数估计结构参数,从而获得风险参数的经验Bayes估计,最后证明了该经验Bayes估计是渐近最优的.     

Bayesian item response model: a generalized approach for the abilities' distribution using mixtures

Traditional Item Response Theory models assume the distribution of the abilities of the population in study to be Gaussian. However, this may not always be a reasonable assumption, which motivates the development of more general models. This paper presents a generalized approach for the distribution of the abilities in dichotomous 3-parameter Item Response models. A mixture of normal distributions is considered, allowing for features like skewness, multimodality and heavy tails. A solution is proposed to deal with model identifiability issues without compromising the flexibility and practical interpretation of the model. Inference is carried out under the Bayesian Paradigm through a novel MCMC algorithm. The algorithm is designed in a way to favour good mixing and convergence properties and is also suitable for inference in traditional IRT models. The efficiency and applicability of our methodology is illustrated in simulated and real examples.     

Censored Exponential Data: Large Deviations for MLEs and Posterior Distributions

In this article, we consider several statistical models for censored exponential data. We prove a large deviation result for the maximum likelihood estimators (MLEs) of each model, and a unique result for the posterior distributions which works well for all the cases. Finally, comparing the large deviation rate functions for MLEs and posterior distributions, we show that a typical feature fails for one model; moreover, we illustrate the relation between this fact and a well-known result for curved exponential models.     

A Multi-Stage Two-Machines Replacement Strategy Using Mixture Models,Bayesian Inference,and Stochastic Dynamic Programming

If at least one out of two serial machines that produce a specific product in manufacturing environments malfunctions, there will be non conforming items produced. Determining the optimal time of the machines' maintenance is the one of major concerns. While a convenient common practice for this kind of problem is to fit a single probability distribution to the combined defect data, it does not adequately capture the fact that there are two different underlying causes of failures. A better approach is to view the defects as arising from a mixture population: one due to the first machine failures and the other due to the second one. In this article, a mixture model along with both Bayesian inference and stochastic dynamic programming approaches are used to find the multi-stage optimal replacement strategy. Using the posterior probability of the machines to be in state λ₁, λ₂ (the failure rates of defective items produced by machine 1 and 2, respectively), we first formulate the problem as a stochastic dynamic programming model. Then, we derive some properties for the optimal value of the objective function and propose a solution algorithm. At the end, the application of the proposed methodology is demonstrated by a numerical example and an error analysis is performed to evaluate the performances of the proposed procedure. The results of this analysis show that the proposed method performs satisfactorily when a different number of observations on the times between productions of defective products is available.     

Estimating Mixing Densities in Exponential Family Models for Discrete Variables

This paper is concerned with estimating a mixing density g using a random sample from the mixture distribution f(x)=∫f x | θ)g(θ)dθ where f(· | θ) is a known discrete exponen tial family of density functions. Recently two techniques for estimating g have been proposed. The first uses Fourier analysis and the method of kernels and the second uses orthogonal polynomials. It is known that the first technique is capable of yielding estimators that achieve (or almost achieve) the minimax convergence rate. We show that this is true for the technique based on orthogonal polynomials as well. The practical implementation of these estimators is also addressed. Computer experiments indicate that the kernel estimators give somewhat disappoint ing finite sample results. However, the orthogonal polynomial estimators appear to do much better. To improve on the finite sample performance of the orthogonal polynomial estimators, a way of estimating the optimal truncation parameter is proposed. The resultant estimators retain the convergence rates of the previous estimators and a Monte Carlo finite sample study reveals that they perform well relative to the ones based on the optimal truncation parameter.     

Bayesian Model Choice in Exponential Survival Models

ABSTRACT

Log-linear models for the distribution on a contingency table are represented as the intersection of only two kinds of log-linear models. One assuming that a certain group of the variables, if conditioned on all other variables, has a jointly independent distribution and another one assuming that a certain group of the variables, if conditioned on all other variables, has no highest order interaction. The subsets entering into these models are uniquely determined by the original log-linear model. This canonical representation suggests considering joint conditional independence and conditional no highest order association as the elementary building blocks of log-linear models.