Bayesian Interpretation and Credible Intervals for Regularized Linear Wavelet Estimator

The regularised linear wavelet estimator has been recently proposed as an alternative to the spline smoothing estimator, one of the most used linear estimator for the standard nonparametric regression problem. It has been demonstrated that the regularised linear wavelet estimator attains the optimal rate of convergence in the mean integrated squared error sense and compares favorably with the smoothing spline estimator in finite-sample situations, especially for less smooth response functions. We investigate further this estimator, extending Bayesian aspects of smoothing splines considered earlier in the literature. 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. We then use the posterior distribution to construct pointwise Bayesian credible intervals for the resulting regularised linear wavelet function estimate. Simulation results show that the wavelet-based pointwise Bayesian credible intervals have good empirical coverage rates for standard nominal coverage probabilities and compare favorably with the corresponding intervals obtained by smoothing splines, especially for less smooth response functions. Moreover, their construction algorithm is of order O(n) and it is easily implemented.