The Flexible Beta Regression Model

A relevant problem in applied statistics concerns modelling rates, proportions or, more generally, continuous variables restricted to the interval (0,1). The aim of this contribution is to study the performances of a new regression model for continuous variables with bounded support that extends the well-known Beta regression model (Ferrari and Cribari-Neto, 2004, Journal of Applied Statistics). Under our new regression model (Migliorati, Di Brisco and Ongaro, submitted paper), the response variable is assumed to have a Flexible Beta (FB) distribution, a special mixture of two Beta distributions that can be interpreted as the univariate version of the Flexible Dirichlet distribution (Ongaro and Migliorati, 2013, Journal of Multivariate Analysis). In many respects, the FB can be considered as the counterpart on (0,1) to the well-established mixture of normal distributions sharing a common variance. The FB guarantees a greater flexibility than the Beta distribution for modelling bounded responses, especially in terms of bimodality, asymmetry and heavy tails. The peculiar mixture structure of the FB makes it identifiable in a strong sense and guarantees a likelihood a.s. bounded from above and a finite global maximum on the assumed parameter space. In the light of these many theoretical properties, the new model results to be very tractable from a computational perspective, in particular with respect to posterior computation. Therefore, we provide a Bayesian approach to inference and, in order to estimate its parameters, we propose a new mean-precision parametrization of the FB that guarantees a variation independent parametric space. Interestingly, the FB regression model can be understood itself as a mixture of regression models. Here we aim at showing the feasibility and strength of our new FB regression model by means of some simulation studies and applications on real datasets, with special attention to bimodal response variables and response variables characterized by the presence of outliers. To simulate values from the posterior distribution we shall implement the Gibbs sampling algorithm through the BUGS software.