The Flexible Beta Regression Model

A relevant problem in applied statistics concerns modeling rates, proportions or, more generally, continuous variables restricted to the interval (0,1). 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). Under our new regression model, 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). 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 modeling 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 bounded likelihood and a finite global maximum on the assumed parameter space. In the light of these good 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 parameterization of the FB that guarantees a variation-independent parametric space. Interestingly, the FB regression (FBR) model can be understood itself as a mixture of regression models. The strength of our new FBR model is illustrated by means of application to a real dataset. To simulate values from the posterior distribution, we implement the Gibbs sampling algorithm through the BUGS software.