Density Estimation With Confidence Sets Exemplified by Superclusters and Voids in the Galaxies

Abstract

A method is presented for forming both a point estimate and a confidence set of semiparametric densities. The final product is a three-dimensional figure that displays a selection of density estimates for a plausible range of smoothing parameters. The boundaries of the smoothing parameter are determined by a nonparametric goodness-of-fit test that is based on the sample spacings. For each value of the smoothing parameter our estimator is selected by choosing the normal mixture that maximizes a function of the sample spacings. A point estimate is selected from this confidence set by using the method of cross-validation. An algorithm to find the mixing distribution that maximizes the spacings functional is presented. These methods are illustrated with a data set from the astronomy literature. The measurements are velocities at which galaxies in the Corona Borealis region are moving away from our galaxy. If the galaxies are clustered, the velocity density will be multimodal, with clusters corresponding to modes. Natural candidates for examining the distribution of the data are finite normal mixtures and histograms. The shortcomings of these methods become apparent from the analysis of these data. By finding a confidence set of densities a set of estimates is obtained, ranging from smooth to rough; the number of modes ranges from three to seven. The confidence set of densities is further substantiated by performing nonparametric tests for the number of modes.