Smoothing Spline Density Estimation: A Dimensionless Automatic Algorithm

Abstract

Abstract As a sequel to an earlier article by Gu and Qiu, this article describes and illustrates a dimensionless automatic algorithm for nonparametric probability density estimation using smoothing splines. The algorithm is designed to calculate an adaptive finite dimensional solution to the penalized likelihood problem, which was shown by Gu and Qiu to share the same asymptotic convergence rates as the nonadaptive infinite dimensional solution. The smoothing parameter is updated jointly with the estimate in a performance-oriented iteration via a cross-validation performance estimate, where the performance is measured by proxies of the symmetrized Kullback-Leibler distance between the true density and the estimate. Simulations of limited scale are conducted to examine the relative effectiveness of the automatic smoothing parameter selection procedure and to assess the practical statistical performance of the methodology in general. The method is also applied to some real data sets. The algorithm is implemented in a few Ratfor routines, which will be included in later versions of RKPACK for public access.