Brazilian Journal of Probability and Statistics (2006), 20, pp. 179–190. c Associa¸ c˜ ao Brasileira de Estat´ ıstica A Bayesian inference for the extended skew-normal measurement error model Josemar Rodrigues Universidade Federal de S˜ ao Carlos Abstract: In this paper, we introduce the multivariate skew-normal model in the context of measurement error models in order to avoid data transformations or usual constraints on the parametric space. This distribu- tion was recently discussed by Capitanio, Azzalini and Stanghellini (2003) using graphical models. The motivation is to use the conditioning argument on the unobserved true value of the explanatory variable in the two-variable measurement error model in order to get the skewness parameters as a func- tion the original parameters of the proposed measurement error models. In- ferential problems are considered from the Bayesian point of view assuming proper noninformative priors via Winbugs. The usefulness of the proposed model with errors in variables is investigated with a simulation study and real data analysis. The main advantage of the Bayesian approach is the possibility to measure the degree of belief that the true value of the explanatory variable is greater than its mean value. This constraint implies a strong asymmetry on the distribution of the unobserved true value. We end the paper by con- cluding that the extended skew-normal measurement error model provides ﬂexibility in terms of skewness without making any additional assumptions to eliminate the usual identiﬁable problems in the measurement error models. Key words: Noninformative prior, posterior distribution, structural er- ror model, Winbugs. 1 Introduction An n-dimensional random vector Z is said to have an extended multivariate ran- dom distribution with parameters (μ, Ω, λ, τ ), denoted by ESN n (μ, Ω, λ, τ ), if it is continuous with density ϕ n (z, μ, Ω) Φ ( α o + λ t (z - μ) ) Φ(τ ) , (1.1) where • α o is a function of (Ω, λ, τ ) to be speciﬁed later. 179