Bayesian inference for an extended simple regression measurement error model using skewed priors

Abstract

In this paper, we introduce a Bayesian extended regression model with two-stage priors when the covariate is positive and measured with error. Connections are made with some results in Arellano-Valle and Azzalini (2006), related to the multivariate skew-normal distributions. The usefulness of the proposed model with errors in variables, via the two-stage priors formulated by O'Hagan and Leonard (1976), is illustrated with an example abstracted from Fuller (1987, pg. 18). The main advantage of this extended Bayesian approach is the use of skewed priors, typically rare in most Bayesian applications, and to treat the true value of the explanatory variable as positive, consideration that is sometimes ignored in measurement error models. Such consideration makes naturally the model identifiable, a problem that significantly has troubled users of other approaches listed in the literature. This constraint implies also a strong asymmetry in the distribution of the response variable. Strong connections are shown with results in Li (1997) on non-random samples and with Berkson models, which are important in practical applications. Extensions of Copas and Li's results for models with vector explanatory variables are presented.