Minimax Density Estimation Under Radial Symmetry

This study illustrates a dimensionality reduction effect of radial symmetry in nonparametric density estimation. To deal with the class of radially symmetric functions, we adopt a generalized translation operation that preserves the symmetry structure. Radial kernel density estimators based on directly or indirectly observed random samples are proposed. For the latter case, we analyze deconvolution problems with four distinct scenarios depending on the symmetry assumptions on the signal and noise. Minimax upper and lower bounds are established for each scheme to investigate the role of the radial symmetry in determining optimal rates of convergence. The results confirm that the radial symmetry reduces the dimension of the estimation problems so that the optimal rate of convergence coincides with the univariate convergence rate except at the origin where a singularity occurs. The results also imply that the proposed estimators are rate optimal in the minimax sense for the Sobolev class of densities.