Bayesian regression analysis of data with censored initiating and terminating times: applications to aids

Data with censored initiating and terminating times arises quite frequently in acquired immunodeficiency syndrome (AIDS) epidemiologic studies. Analysis of such data involves a complicated bivariate likelihood, which is difficult to deal with computationally. Bayesian analysis, op the other hand, presents added complexities that have yet to be resolved. By exploiting the simple form of a complete data likelihood and utilizing the power of a Markov Chain Monte Carlo (MCMC) algorithm, this paper presents a methodology for fitting Bayesian regression models to such data. The proposed methods extend the work of Sinha (1997), who considered non-parametric Bayesian analysis of this type of data. The methodology is illustiated with an application to a cohort of HIV infected hemophiliac patients.     

Robust Bayesian analysis for parallel system with masked data under inverse Weibull lifetime distribution

Abstract

This paper considers the statistical analysis of masked data in a parallel system with inverse Weibull distributed components under type II censoring. Based on Gamma conjugate prior, the Bayesian estimation as well as the hierarchical Bayesian estimation for the parameters and the reliability function of system are obtained by using the Bayesian theory and the hierarchical Bayesian method. Finally, Monte Carlo simulations are provided to compare the performances of the estimates under different masking probabilities and effective sample sizes.     

Bayesian analysis of directed graphs data with applications to social networks

Summary.  A fully Bayesian analysis of directed graphs, with particular emphasis on applica- tions in social networks, is explored. The model is capable of incorporating the effects of covariates, within and between block ties and multiple responses. Inference is straightforward by using software that is based on Markov chain Monte Carlo methods. Examples are provided which highlight the variety of data sets that can be entertained and the ease with which they can be analysed.     

A Bayesian approach to estimating linear mixtures with unknown covariance structure

In this paper, we study a new Bayesian approach for the analysis of linearly mixed structures. In particular, we consider the case of hyperspectral images, which have to be decomposed into a collection of distinct spectra, called endmembers, and a set of associated proportions for every pixel in the scene. This problem, often referred to as spectral unmixing, is usually considered on the basis of the linear mixing model (LMM). In unsupervised approaches, the endmember signatures have to be calculated by an endmember extraction algorithm, which generally relies on the supposition that there are pure (unmixed) pixels contained in the image. In practice, this assumption may not hold for highly mixed data and consequently the extracted endmember spectra differ from the true ones. A way out of this dilemma is to consider the problem under the normal compositional model (NCM). Contrary to the LMM, the NCM treats the endmembers as random Gaussian vectors and not as deterministic quantities. Existing Bayesian approaches for estimating the proportions under the NCM are restricted to the case that the covariance matrix of the Gaussian endmembers is a multiple of the identity matrix. The self-evident conclusion is that this model is not suitable when the variance differs from one spectral channel to the other, which is a common phenomenon in practice. In this paper, we first propose a Bayesian strategy for the estimation of the mixing proportions under the assumption of varying variances in the spectral bands. Then we generalize this model to handle the case of a completely unknown covariance structure. For both algorithms, we present Gibbs sampling strategies and compare their performance with other, state of the art, unmixing routines on synthetic as well as on real hyperspectral fluorescence spectroscopy data.     

Bayesian inference for sinh-normal/independent nonlinear regression models

Sinh-normal/independent distributions are a class of symmetric heavy-tailed distributions that include the sinh-normal distribution as a special case, which has been used extensively in Birnbaum–Saunders regression models. Here, we explore the use of Markov Chain Monte Carlo methods to develop a Bayesian analysis in nonlinear regression models when Sinh-normal/independent distributions are assumed for the random errors term, and it provides a robust alternative to the sinh-normal nonlinear regression model. Bayesian mechanisms for parameter estimation, residual analysis and influence diagnostics are then developed, which extend the results of Farias and Lemonte [Bayesian inference for the Birnbaum-Saunders nonlinear regression model, Stat. Methods Appl. 20 (2011), pp. 423-438] who used the Sinh-normal/independent distributions with known scale parameter. Some special cases, based on the sinh-Student-t (sinh-St), sinh-slash (sinh-SL) and sinh-contaminated normal (sinh-CN) distributions are discussed in detail. Two real datasets are finally analyzed to illustrate the developed procedures.     

Group decision analysis and its application to combining opinions

This paper is about the graphical depiction of the set of feasible gains-in-utilities accruing to three Bayesians involved in the joint estimation of a multivatiate normal mean vector. The basic theory for this problem is sketched. Then a suitable parametrization of surface contours is developed. These contours allow the surface to be mapped and graphically displayed. This is done with the LIG language for interactive graphics. As the opinions of the three Bayesians diverge, the illustrations contained in the paper show how the initially smooth balloon shaped structure develops and ‘clicks’ through a flat spot and eventually becomes highly irregular with a central indentation. The result provides insight into the nature of disensus where explicit mathematical analysis is extremely difficult.     

Bayesian analysis of skew-normal independent linear mixed models with heterogeneity in the random-effects population

We present a new class of models to fit longitudinal data, obtained with a suitable modification of the classical linear mixed-effects model. For each sample unit, the joint distribution of the random effect and the random error is a finite mixture of scale mixtures of multivariate skew-normal distributions. This extension allows us to model the data in a more flexible way, taking into account skewness, multimodality and discrepant observations at the same time. The scale mixtures of skew-normal form an attractive class of asymmetric heavy-tailed distributions that includes the skew-normal, skew-Student-t, skew-slash and the skew-contaminated normal distributions as special cases, being a flexible alternative to the use of the corresponding symmetric distributions in this type of models. A simple efficient MCMC Gibbs-type algorithm for posterior Bayesian inference is employed. In order to illustrate the usefulness of the proposed methodology, two artificial and two real data sets are analyzed.     

Bayesian growth curve models with the generalized error distribution

To deal with the longitudinal data with both leptokurtic and platykurtic errors, we extend growth curve models using the generalized error distribution (GED) model. The Metropolis–Hastings algorithm is used to estimate the GED model parameters in the Bayesian framework. The application of the GED model is illustrated through the analysis of mathematical development data. Results show that the GED model can correctly identify the deviation from normal of the error distributions.     

Objective Bayesian analysis based on upper record values from two-parameter Rayleigh distribution with partial information

In the life test, predicting higher failure times than the largest failure time of the observed is an important issue. Although the Rayleigh distribution is a suitable model for analyzing the lifetime of components that age rapidly over time because its failure rate function is an increasing linear function of time, the inference for a two-parameter Rayleigh distribution based on upper record values has not been addressed from the Bayesian perspective. This paper provides Bayesian analysis methods by proposing a noninformative prior distribution to analyze survival data, using a two-parameter Rayleigh distribution based on record values. In addition, we provide a pivotal quantity and an algorithm based on the pivotal quantity to predict the behavior of future survival records. We show that the proposed method is superior to the frequentist counterpart in terms of the mean-squared error and bias through Monte carlo simulations. For illustrative purposes, survival data on lung cancer patients are analyzed, and it is proved that the proposed model can be a good alternative when prior information is not given.     

Bayesian multivariate normal analysis with a wishart prior

This paper considers the Bayesian analysis of the multivariate normal distribution when its covariance matrix has a Wishart prior density under the assumption of a multivariate quadratic loss function. New flexible marginal posterior distributions of the mean μ and of the covariance matrix Σ are developed and univariate cases with graphical representations are given.     

Perfect aggregation of Bayesian analysis on compositional data

Sufficiency is a widely used concept for reducing the dimensionality of a data set. Collecting data for a sufficient statistic is generally much easier and less expensive than collecting all of the available data. When the posterior distributions of a quantity of interest given the aggregate and disaggregate data are identical, perfect aggregation is said to hold, and in this case the aggregate data is a sufficient statistic for the quantity of interest. In this paper, the conditions for perfect aggregation are shown to depend on the functional form of the prior distribution. When the quantity of interest is the sum of some parameters in a vector having either a generalized Dirichlet or a Liouville distribution for analyzing compositional data, necessary and sufficient conditions for perfect aggregation are also established.     

Bayesian expectile regression with asymmetric normal distribution

In this paper, we adopt the Bayesian approach to expectile regression employing a likelihood function that is based on an asymmetric normal distribution. We demonstrate that improper uniform priors for the unknown model parameters yield a proper joint posterior. Three simulated data sets were generated to evaluate the proposed method which show that Bayesian expectile regression performs well and has different characteristics comparing with Bayesian quantile regression. We also apply this approach into two real data analysis.     

An improper form of Weibull distribution for competing risks analysis with Bayesian approach

ABSTRACT

In survival analysis, individuals may fail due to multiple causes of failure called competing risks setting. Parametric models such as Weibull model are not improper that ignore the assumption of multiple failure times. In this study, a novel extension of Weibull distribution is proposed which is improper and then can incorporate to the competing risks framework. This model includes the original Weibull model before a pre-specified time point and an exponential form for the tail of the time axis. A Bayesian approach is used for parameter estimation. A simulation study is performed to evaluate the proposed model. The conducted simulation study showed identifiability and appropriate convergence of the proposed model. The proposed model and the 3-parameter Gompertz model, another improper parametric distribution, are fitted to the acute lymphoblastic leukemia dataset.     

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.     

Robust Bayesian analysis for exponential parameters under generalized Type-II progressive hybrid censoring

The Type-II progressive hybrid censoring scheme has received wide attention, but it has a disadvantage in that long time may be required to complete the life test. The generalized progressive Type-II hybrid censoring scheme has recently been proposed to solve this problem. Under the censoring scheme, the time on test does not exceed a predetermined time. In this paper, we propose a robust Bayesian approach based on a hierarchical structure when the generalized progressive Type-II hybrid censored sample has a two-parameter exponential distribution. For unknown parameters, marginal posterior distributions are provided in closed forms, and their statistical properties are discussed. To examine the robustness of the proposed method, Monte Carlo simulations are conducted and a real data set is analyzed. Further, the quality and adequacy of the proposed model are evaluated in an analysis based on the real data.     

On the measurement and analysis of asymmetry with applications to facial modelling

Summary.  Asymmetry is a feature of shape which is of particular interest in a variety of applications. With landmark data, the essential information on asymmetry is contained in the degree to which there is a mismatch between a landmark configuration and its relabelled and matched reflection. This idea is explored in the context of a study of facial shape in infants, where particular interest lies in identifying changes over time and in assessing residual deformity in children who have had corrective surgery for a cleft lip or cleft lip and palate. Interest lies not in whether the mean shape is asymmetric but in comparing the degrees of asymmetry in different populations. A decomposition of the asymmetry score into components that are attributable to particular features of the face is proposed. A further decomposition allows different sources of asymmetry due to position, orientation or intrinsic asymmetry to be identified for each feature. The methods are also extended to data representing anatomical curves across the face.     

Bayesian estimation of an AR(1) process with exponential white noise

The object of this paper is a Bayesian analysis of the autoregressive model X _t ?=?ρX _t?1?+?Y _t where 0?Y _t are independent random variables with an exponential distribution of parameter θ. Our study generalizes some results obtained by Turkmann (1990 Amaral Turkmann, M. A. (1990). Bayesian analysis of an autoregressive process with exponential white noise. Statistics, 4: 601–608.  [Google Scholar]). Our analysis is based on a more general non-informative prior which allows us to improve the estimators of ρ and θ.     

On the matrix-variate generalized hyperbolic distribution and its Bayesian applications

In the first part of the paper, we introduce the matrix-variate generalized hyperbolic distribution by mixing the matrix normal distribution with the matrix generalized inverse Gaussian density. The p-dimensional generalized hyperbolic distribution of [Barndorff-Nielsen, O. (1978). Hyperbolic distributions and distributions on hyperbolae. Scand. J. Stat., 5, 151–157], the matrix-T distribution and many well-known distributions are shown to be special cases of the new distribution. Some properties of the distribution are also studied. The second part of the paper deals with the application of the distribution in the Bayesian analysis of the normal multivariate linear model.     

Bayesian analysis of generalized elliptical semi-parametric models

In this paper, we study the statistical inference based on the Bayesian approach for regression models with the assumption that independent additive errors follow normal, Student-t, slash, contaminated normal, Laplace or symmetric hyperbolic distribution, where both location and dispersion parameters of the response variable distribution include nonparametric additive components approximated by B-splines. This class of models provides a rich set of symmetric distributions for the model error. Some of these distributions have heavier or lighter tails than the normal as well as different levels of kurtosis. In order to draw samples of the posterior distribution of the interest parameters, we propose an efficient Markov Chain Monte Carlo (MCMC) algorithm, which combines Gibbs sampler and Metropolis–Hastings algorithms. The performance of the proposed MCMC algorithm is assessed through simulation experiments. We apply the proposed methodology to a real data set. The proposed methodology is implemented in the R package BayesGESM using the function gesm().