Bayesian P-Splines Applied to Semiparametric Models with Errors Following a Scale Mixture of Normals

This work is about semiparametric models assuming that the error follows a scale mixture of gaussian distributions and such that the functional relation between the response and explanatory variables is unknown. This class of distributions is particularly interesting since it includes heavy tailed distributions such as the Student's t, symmetric stable distributions and double exponential. They are specially useful for modelling data with high incidence of extreme values. Exploring the nature of these distributions and using the concept of P-splines we obtain the complete posterior conditional distributions of all the parameters involved in the model and apply the Gibbs sampler. In this way, we show how to combine P-splines and mixture of normals under a Bayesian perspective in order to estimate such curves. We conduct some simulations in order to illustrate the proposed methodology and also analyze the case of partially linear models.