Bayesian inference for semiparametric regression using a Fourier representation

This paper presents the Bayesian analysis of a semiparametric regression model that consists of parametric and nonparametric components. The nonparametric component is represented with a Fourier series where the Fourier coefficients are assumed a priori to have zero means and to decay to 0 in probability at either algebraic or geometric rates. The rate of decay controls the smoothness of the response function. The posterior analysis automatically selects the amount of smoothing that is coherent with the model and data. Posterior probabilities of the parametric and semiparametric models provide a method for testing the parametric model against a non-specific alternative. The Bayes estimator's mean integrated squared error compares favourably with the theoretically optimal estimator for kernel regression.