Bayesian inference for random coefficient dynamic panel data models

We develop a hierarchical Bayesian approach for inference in random coefficient dynamic panel data models. Our approach allows for the initial values of each unit's process to be correlated with the unit-specific coefficients. We impose a stationarity assumption for each unit's process by assuming that the unit-specific autoregressive coefficient is drawn from a logitnormal distribution. Our method is shown to have favorable properties compared to the mean group estimator in a Monte Carlo study. We apply our approach to analyze energy and protein intakes among individuals from the Philippines.     

A hybrid Markov chain for the Bayesian analysis of the multinomial probit model

Bayesian inference for the multinomial probit model, using the Gibbs sampler with data augmentation, has been recently considered by some authors. The present paper introduces a modification of the sampling technique, by defining a hybrid Markov chain in which, after each Gibbs sampling cycle, a Metropolis step is carried out along a direction of constant likelihood. Examples with simulated data sets motivate and illustrate the new technique. A proof of the ergodicity of the hybrid Markov chain is also given.     

Bayesian analysis of two overdispersed poisson regression models

Shookri and Consul (1989) and Scollnik (1995) have previously considered the Bayesian analysis of an overdispersed generalized Poisson model. Scollnik (1995) also considered the Bayesian analysis of an ordinary Poisson and over-dispersed generalized Poisson mixture model. In this paper, we discuss the Bayesian analysis of these models when they are utilised in a regression context. Markov chain Monte Carlo methods are utilised, and an illustrative analysis is provided.     

Comparison of Two Methods for Calculating the Partition Functions of Various Spatial Statistical Models

Likelihood computation in spatial statistics requires accurate and efficient calculation of the normalizing constant (i.e. partition function) of the Gibbs distribution of the model. Two available methods to calculate the normalizing constant by Markov chain Monte Carlo methods are compared by simulation experiments for an Ising model, a Gaussian Markov field model and a pairwise interaction point field model.     

Bayesian estimation in the multidimensional three-parameter logistic model

The Gibbs sampler has a great potential to be an efficient and versatile estimation procedure in item response theory. In this article, based on a data augmentation scheme using the Gibbs sampler, we propose a Bayesian procedure to estimate the multidimensional three-parameter logistic model. With the introduction of the two latent variables, the full conditional distributions are tractable, and consequently the Gibbs sampling is easy to implement. Finally, the technique is illustrated by using simulated and real data, respectively.     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.     

Application of a predictive distribution formula to Bayesian computation for incomplete data models

We consider exact and approximate Bayesian computation in the presence of latent variables or missing data. Specifically we explore the application of a posterior predictive distribution formula derived in Sweeting And Kharroubi (2003), which is a particular form of Laplace approximation, both as an importance function and a proposal distribution. We show that this formula provides a stable importance function for use within poor man’s data augmentation schemes and that it can also be used as a proposal distribution within a Metropolis-Hastings algorithm for models that are not analytically tractable. We illustrate both uses in the case of a censored regression model and a normal hierarchical model, with both normal and Student t distributed random effects. Although the predictive distribution formula is motivated by regular asymptotic theory, it is not necessary that the likelihood has a closed form or that it possesses a local maximum.     

Bayesian semiparametric inference for the accelerated failure-time model

Bayesian semiparametric inference is considered for a loglinear model. This model consists of a parametric component for the regression coefficients and a nonparametric component for the unknown error distribution. Bayesian analysis is studied for the case of a parametric prior on the regression coefficients and a mixture-of-Dirichlet-processes prior on the unknown error distribution. A Markov-chain Monte Carlo (MCMC) method is developed to compute the features of the posterior distribution. A model selection method for obtaining a more parsimonious set of predictors is studied. The method adds indicator variables to the regression equation. The set of indicator variables represents all the possible subsets to be considered. A MCMC method is developed to search stochastically for the best subset. These procedures are applied to two examples, one with censored data.     

Bayesian estimation of the multidimensional graded response model with nonignorable missing data

A Bayesian approach is developed for analysing item response models with nonignorable missing data. The relevant model for the observed data is estimated concurrently in conjunction with the item response model for the missing-data process. Since the approach is fully Bayesian, it can be easily generalized to more complicated and realistic models, such as those models with covariates. Furthermore, the proposed approach is illustrated with item response data modelled as the multidimensional graded response models. Finally, a simulation study is conducted to assess the extent to which the bias caused by ignoring the missing-data mechanism can be reduced.     

Bayesian inference for the generalized exponential distribution

The two-parameter generalized exponential (GE) distribution was introduced by Gupta and Kundu [Gupta, R.D. and Kundu, D., 1999, Generalized exponential distribution. Australian and New Zealand Journal of Statistics, 41(2), 173–188.]. It was observed that the GE can be used in situations where a skewed distribution for a nonnegative random variable is needed. In this article, the Bayesian estimation and prediction for the GE distribution, using informative priors, have been considered. Importance sampling is used to estimate the parameters, as well as the reliability function, and the Gibbs and Metropolis samplers data sets are used to predict the behavior of further observations from the distribution. Two data sets are used to illustrate the Bayesian procedure.     

Modeling a mixture of ordinal and continuous repeated measures

We study the correlation structure for a mixture of ordinal and continuous repeated measures using a Bayesian approach. We assume a multivariate probit model for the ordinal variables and a normal linear regression for the continuous variables, where latent normal variables underlying the ordinal data are correlated with continuous variables in the model. Due to the probit model assumption, we are required to sample a covariance matrix with some of the diagonal elements equal to one. The key computational idea is to use parameter-extended data augmentation, which involves applying the Metropolis-Hastings algorithm to get a sample from the posterior distribution of the covariance matrix incorporating the relevant restrictions. The methodology is illustrated through a simulated example and through an application to data from the UCLA Brain Injury Research Center.     

Bayesian binary quantile regression for the analysis of Bachelor-to-Master transition

The multi-cycle organization of modern university systems stimulates the interest in studying the progression to higher level degree courses during the academic career. In particular, after the achievement of the first level qualification (Bachelor degree), students have to decide whether to continue their university studies, by enrolling in a second level (Master) programme, or to conclude their training experience. In this work we propose a binary quantile regression (BQR) approach to analyse the Bachelor-to-Master transition phenomenon with the adoption of the Bayesian inferential perspective. In addition to the traditional predictors of academic outcomes, such as the personal characteristics and the field of study, different aspects of student's performance are considered. Moreover, the role of a new contextual variable, representing the type of university regulations experienced during the academic path, is investigated. The utility of the Bayesian BQR to characterize the non-continuation decision after the first cycle studies is illustrated with an application to administrative data of Bachelor graduates at the School of Economics of Sapienza University of Rome. The method favourably compares with more conventional model specifications concerning the conditional mean of the binary response.     

Bayesian inference for the randomly censored Burr-type XII distribution

The article presents the Bayesian inference for the parameters of randomly censored Burr-type XII distribution with proportional hazards. The joint conjugate prior of the proposed model parameters does not exist; we consider two different systems of priors for Bayesian estimation. The explicit forms of the Bayes estimators are not possible; we use Lindley's method to obtain the Bayes estimates. However, it is not possible to obtain the Bayesian credible intervals with Lindley's method; we suggest the Gibbs sampling procedure for this purpose. Numerical experiments are performed to check the properties of the different estimators. The proposed methodology is applied to a real-life data for illustrative purposes. The Bayes estimators are compared with the Maximum likelihood estimators via numerical experiments and real data analysis. The model is validated using posterior predictive simulation in order to ascertain its appropriateness.     

Bayesian analysis of human movement curves

Summary.  We consider the Bayesian analysis of human movement data, where the subjects perform various reaching tasks. A set of markers is placed on each subject and a system of cameras records the three-dimensional Cartesian co-ordinates of the markers during the reaching movement. It is of interest to describe the mean and variability of the curves that are traced by the markers during one reaching movement, and to identify any differences due to covariates. We propose a methodology based on a hierarchical Bayesian model for the curves. An important part of the method is to obtain identifiable features of the movement so that different curves can be compared after temporal warping. We consider four landmarks and a set of equally spaced pseudolandmarks are located in between. We demonstrate that the algorithm works well in locating the landmarks, and shape analysis techniques are used to describe the posterior distribution of the mean curve. A feature of this type of data is that some parts of the movement data may be missing—the Bayesian methodology is easily adapted to cope with this situation.     

The Trace Restriction: An Alternative Identification Strategy for the Bayesian Multinomial Probit Model

Previous authors have made Bayesian multinomial probit models identifiable by fixing a parameter on the main diagonal of the covariance matrix. The choice of which element one fixes can influence posterior predictions. Thus, we propose restricting the trace of the covariance matrix, which we achieve without computational penalty. This permits a prior that is symmetric to permutations of the nonbase outcome categories. We find in real and simulated consumer choice datasets that the trace-restricted model is less prone to making extreme predictions. Further, the trace restriction can provide stronger identification, yielding marginal posterior distributions that are more easily interpreted.     

Assessing The Impact of Managed-Care on the Distribution of Length-of-Stay Using Bayesian Hierarchical Models

Hierarchical models provide a useful framework for the complexities encountered in policy-relevant research in which the impact of social programs is being assessed. Such complexities include multi-site data, censored data and over-dispersion. In this paper, Bayesian inference through Markov Chain Monte Carlo methods is used for the analysis of a complex hierarchical log-normal model that shows the impact of a managed care strategy aimed at limiting length of hospital stays. Parameters in this model allow for variability in baseline length-of-stay as well as the program effect across hospitals. The authors demonstrate elicitation and sensitivity analysis with respect to prior distributions. All calculations for the posterior and predictive distributions were obtained using the software BUGS.     

Bayesian multiple change-point estimation for exponential distribution with truncated and censored data

This paper considers the multiple change-point estimation for exponential distribution with truncated and censored data by Gibbs sampling. After all the missing data of interest is filled in by some sampling methods such as rejection sampling method, the complete-data likelihood function is obtained. The full conditional distributions of all parameters are discussed. The means of Gibbs samples are taken as Bayesian estimations of the parameters. The implementation steps of Gibbs sampling are introduced in detail. Finally random simulation test is developed, and the results show that Bayesian estimations are fairly accurate.     

基于贝叶斯方法的比例数据分位数推断及其应用

为了尝试使用贝叶斯方法研究比例数据的分位数回归统计推断问题,首先基于Tobit模型给出了分位数回归建模方法,然后通过选取合适的先验分布得到了贝叶斯层次模型,进而给出了各参数的后验分布并用于Gibbs抽样。数值模拟分析验证了所提出的贝叶斯推断方法对于比例数据分析的有效性。最后,将贝叶斯方法应用于美国加州海洛因吸毒数据,在不同的分位数水平下揭示了吸毒频率的影响因素。     

Bayesian estimation of reliability using time-to-failure distribution of parametric degradation models

This paper focusses on computing the Bayesian reliability of components whose performance characteristics (degradation – fatigue and cracks) are observed during a specified period of time. Depending upon the nature of degradation data collected, we fit a monotone increasing or decreasing function for the data. Since the components are supposed to have different lifetimes, the rate of degradation is assumed to be a random variable. At a critical level of degradation, the time to failure distribution is obtained. The exponential and power degradation models are studied and exponential density function is assumed for the random variable representing the rate of degradation. The maximum likelihood estimator and Bayesian estimator of the parameter of exponential density function, predictive distribution, hierarchical Bayes approach and robustness of the posterior mean are presented. The Gibbs sampling algorithm is used to obtain the Bayesian estimates of the parameter. Illustrations are provided for the train wheel degradation data.     

Bayesian Methods for Missing Covariates in Cure Rate Models

We propose methods for Bayesian inference for missing covariate data with a novel class of semi-parametric survival models with a cure fraction. We allow the missing covariates to be either categorical or continuous and specify a parametric distribution for the covariates that is written as a sequence of one dimensional conditional distributions. We assume that the missing covariates are missing at random (MAR) throughout. We propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. The proposed class of priors are shown to be useful in recovering information on the missing covariates especially in situations where the missing data fraction is large. Properties of the proposed prior and resulting posterior distributions are examined. Also, model checking techniques are proposed for sensitivity analyses and for checking the goodness of fit of a particular model. Specifically, we extend the Conditional Predictive Ordinate (CPO) statistic to assess goodness of fit in the presence of missing covariate data. Computational techniques using the Gibbs sampler are implemented. A real data set involving a melanoma cancer clinical trial is examined to demonstrate the methodology.