Bayesian interpretation and credible intervals for regularised linear wavelet estimators

We first consider a Bayesian formalism in the wavelet domain that gives rise to the regularised linear wavelet estimator obtained in the standard nonparametric regression setting when the unknown response function belongs to a Sobolev space with non-integer regularity s > 1/2. We then use the posterior distribution of the wavelets coefficients to construct pointwise Bayesian credible intervals for the resulting regularised linear wavelet function estimate. These results extend Bayesian aspects of smoothing splines considered earlier in the literature for response functions belonging to Sobolev spaces with integer regularity s > 1. Simulation results show that the waveletbased pointwise Bayesian credible intervals have good empirical coverage rates for standard nominal coverage probabilities and compare favourably with the corresponding pointwise Bayesian credible intervals obtained by smoothing splines, especially for less smooth response functions. Moreover, their construction algorithm does not suffer from instability and, compared with smoothing splines, it is much easier and can be applied to any noninteger s > 1/2.