Spline density estimation and inference with model-based penalties

In this paper we propose model-based penalties for smoothing spline density estimation and inference. These model-based penalties incorporate indefinite prior knowledge that the density is close to, but not necessarily in a family of distributions. We will use the Pearson and generalisation of the generalised inverse Gaussian families to illustrate the derivation of penalties and reproducing kernels. We also propose new inference procedures to test the hypothesis that the density belongs to a specific family of distributions. We conduct extensive simulations to show that the model-based penalties can substantially reduce both bias and variance in the decomposition of the Kullback-Leibler distance, and the new inference procedures are more powerful than some existing ones. We further demonstrate the empirical performance of the proposed method with a real world data set.