Locally Adaptive Density Estimation on the Unit Sphere Using Needlets

Abstract: The problem of estimating a probability density function f on the (d-1)-dimensional unit sphere S^{d-1} from directional data using the needlet frame is considered. It is shown that the decay of needlet coefficients supported near a point of a function f depends only on local H\"{o}lder continuity properties of f at x. This is then used to show that the thresholded needlet estimator introduced in Baldi, Kerkyacharian, Marinucci and Picard adapts to the local regularity properties of f. Moreover an adaptive confidence interval for f based on the thresholded needlet estimator is proposed, which is asymptotically honest over suitable classes of locally H\"{o}lderian densities.