Estimation of Densities and Derivatives of Densities with Directional Data

Estimating the density function of a random vector taking values on the d-dimensional unit sphere is considered. Also the estimation of the Laplacian of the density and estimation of other types of derivatives is considered. Fast convergence rate theory is developed for pointwise, L1, and L2 error, extending some results of Hall, Watson and Cabrera (1987). It is also proved that asymptotically the plug-in method is as good as using the asymptotically optimal deterministic smoothing parameter sequence.