Ignacio Vidal

A new class of skew-symmetric distributions and related families

In this work, we investigate a new class of skew-symmetric distributions, which includes the distributions with the probability density function (pdf) given by g α(x)=2f(x) G(α x), introduced by Azzalini [A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]. We call this new class as the symmetric-skew-symmetric family and it has the pdf proportional to f(x) G β(α x), where G β(x) is the cumulative distribution function of g β(x). We give some basic properties for the symmetric-skew-symmetric family and study the particular case obtained from the normal distribution.

Comparison between a measurement error model and a linear model without measurement error

The regression of a response variable y on an explanatory variable @x from observations on (y,x), where x is a measurement of @x, is a special case of errors-in-variables model or measurement error model (MEM). In this work we attempt to answer the following question: given the data (y,x) under a MEM, is it possible to not consider the measurement error on the covariable @x in order to use a simpler model? To the best of our knowledge, this problem has not been treated in the Bayesian literature. To answer that question, we compute Bayes factors, the deviance information criterion and the posterior mean of the logarithmic discrepancy. We apply these Bayesian model comparison criteria to two real data sets obtaining interesting results. We conclude that, in order to simplify the MEM, model comparison criteria can be useful to compare structural MEM and a random effect model, but we would also need other statistic tools and take into account the final goal of the model.

Bayesian inference for dependent elliptical measurement error models

In this article we provide a Bayesian analysis for dependent elliptical measurement error models considering nondifferential and differential errors. In both cases we compute posterior distributions for structural parameters by using squared radial prior distributions for the precision parameters. The main result is that the posterior distribution of location parameters, for specific priors, is invariant with respect to changes in the generator function, in agreement with previous results obtained in the literature under different assumptions. Finally, although the results obtained are valid for any elliptical distribution for the error term, we illustrate those results by using the student-t distribution and a real data set.

Influential Observations in the Functional Measurement Error Model

Abstract In this work we propose Bayesian measures to quantify the influence of observations on the structural parameters of the simple measurement error model (MEM). Different influence measures, like those based on q-divergence between posterior distributions and Bayes risk, are studied to evaluate the influence. A strategy based on the perturbation function and MCMC samples is used to compute these measures. The samples from the posterior distributions are obtained by using the Metropolis–Hastings algorithm and assuming specific proper prior distributions. The results are illustrated with an application to a real example modeled with MEM in the literature.

Modified skew-slash distribution

Abstract In this work, we introduce a new skewed slash distribution. This modification of the skew-slash distribution is obtained by the quotient of two independent random variables. That quotient consists on a skew-normal distribution divided by a power of an exponential distribution with scale parameter equal to two. In this way, the new skew distribution has a heavier tail than that of the skew-slash distribution. We give the probability density function expressed by an integral, but we obtain some important properties useful for making inferences, such as moment estimators and maximum likelihood estimators. By way of illustration and by using real data, we provide maximum likelihood estimates for the parameters of the modified skew-slash and the skew-slash distributions. Finally, we introduce a multivariate version of this new distribution.

Bayesian inference for agreement measures

ABSTRACT The agreement of different measurement methods is an important issue in several disciplines like, for example, Medicine, Metrology, and Engineering. In this article, some agreement measures, common in the literature, were analyzed from a Bayesian point of view. Posterior inferences for such agreement measures were obtained based on well-known Bayesian inference procedures for the bivariate normal distribution. As a consequence, a general, simple, and effective method is presented, which does not require Markov Chain Monte Carlo methods and can be applied considering a great variety of prior distributions. Illustratively, the method was exemplified using five objective priors for the bivariate normal distribution. A tool for assessing the adequacy of the model is discussed. Results from a simulation study and an application to a real dataset are also reported.

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.