Márcia D. Branco

A new class of multivariate skew distributions with applications to bayesian regression models

The authors develop a new class of distributions by introducing skewness in multivariate ellip- tically symmetric distributions. The class, which is obtained by using transformation and conditioning, contains many standard families including the multivariate skew-normal and distributions. The authors obtain analytical forms of the densities and study distributional properties. They give practical applica- tions in Bayesian regression models and results on the existence of the posterior distributions and moments under improper priors for the regression coefficients. They illustrate their methods using practical examples.

A skew item response model

We introduce a new skew-probit link for item response theory (IRT) by considering an accumulated skew-normal distribution. The model extends the symmetric probit-normal IRT model by considering a new item (or skewness) parameter for the item characteristic curve. A special interpretation is given for this parameter, and a latent linear structure is indicated for the model when an augmented likelihood is considered. Bayesian MCMC inference approach is developed and an efficiency study in the estimation of the model parameters is undertaken for a data set from Tanner (1996, p. 190) by using the notion of effective sample size (ESS) as defined in Kass et al. (1998) and the sample size per second (ESS/s) as considered in Sahu (2002). The methodology is illustrated using a data set corresponding to a Mathematical Test applied in Peruvian schools for which a sensitivity analysis of the chosen priors is conducted and also a comparison with seven parametric IRT models is conducted. The main conclusion is that the skew-probit item response model seems to provide the best fit.

A Framework for Skew-Probit Links in Binary Regression

We review several asymmetrical links for binary regression models and present a unified approach for two skew-probit links proposed in the literature. Moreover, under skew-probit link, conditions for the existence of the ML estimators and the posterior distribution under improper priors are established. The framework proposed here considers two sets of latent variables which are helpful to implement the Bayesian MCMC approach. A simulation study to criteria for models comparison is conducted and two applications are made. Using different Bayesian criteria we show that, for these data sets, the skew-probit links are better than alternative links proposed in the literature.

Bayesian analysis of the calibration problem under elliptical distributions

Abstract In this paper we discuss calibration problems under dependent and independent elliptical family of distributions. In the dependent case, it is shown that the posterior distribution of the quantity of interest is robust with respect to the distributions in the elliptical family. In particular, the results obtained by Hoadley (1970. J. Amer. Statist. 65, 356–369) showing that the inverse estimator is a Bayes estimator under normal models with a Student-t prior also holds under the dependent elliptical family of distributions. In the independent case, the use of the elliptical family allows the consideration of models which provide protection against possible outliers in the data. The multivariate calibration problem is also considered, where some results given in Brown (1993. Measurement, Regression and Calibration. Oxford University Press, Oxford) are extended. Finally, the results of the paper are applied to a real data problem, showing that the Student-t model can be a valid alternative to normality.

Bayesian analysis for nonlinear regression model under skewed errors, with application in growth curves

We have considered a Bayesian approach for the nonlinear regression model by replacing the normal distribution on the error term by some skewed distributions, which account for both skewness and heavy tails or skewness alone. The type of data considered in this paper concerns repeated measurements taken in time on a set of individuals. Such multiple observations on the same individual generally produce serially correlated outcomes. Thus, additionally, our model does allow for a correlation between observations made from the same individual. We have illustrated the procedure using a data set to study the growth curves of a clinic measurement of a group of pregnant women from an obstetrics clinic in Santiago, Chile. Parameter estimation and prediction were carried out using appropriate posterior simulation schemes based in Markov Chain Monte Carlo methods. Besides the deviance information criterion (DIC) and the conditional predictive ordinate (CPO), we suggest the use of proper scoring rules based on the posterior predictive distribution for comparing models. For our data set, all these criteria chose the skew-t model as the best model for the errors. These DIC and CPO criteria are also validated, for the model proposed here, through a simulation study. As a conclusion of this study, the DIC criterion is not trustful for this kind of complex model.

Bayesian robustness under a skew-normal class of prior distribution

Abstract We develop a global sensitivity analysis to measure the robustness of the Bayesian estimators with respect to a class of prior distributions. This class arises when we consider multiplicative contamination of a base prior distribution. A similar structure was presented by van der Linde [12] . Some particular specifications for this multiplicative contamination class coincide with well known families of skewed distributions. In this paper, we explore the skew-normal multiplicative contamination class for the prior distribution of the location parameter of a normal model. Results of a Bayesian conjugation and expressions for some measures of distance between posterior means and posterior variance are obtained. We also elaborate on the behavior of the posterior means and of the posterior variances through a simulation study.

Bayesian Sensitivity Analysis in Skew-elliptical Models

The main objective of this chapter is to investigate the influence of introducing skewness parameter in elliptical models. First, we review definitions and properties of skew distributions considered in the literature with emphasis on the so-called skew elliptical distributions. For univariate skew-normal models we study the influence of the skew parameter on the posterior distributions of the location and scale parameters. The influence is quantified by evaluating the L 1 distance between the posterior distributions obtained under the skew-normal model and normal model respectively. We then examine the problem of computing Bayes factors to test skewness in linear regression models, evaluating the performance trough simulations.