Sharp minimaxity and spherical deconvolution for super-smooth error distributions

Abstract

The spherical deconvolution problem was first proposed by Rooij and Ruymgaart (in: G. Roussas (Ed.), Nonparametric Functional Estimation and Related Topics, Kluwer Academic Publishers, Dordrecht, 1991, pp. 679–690) and subsequently solved in Healy et al. (J. Multivariate Anal. 67 (1998) 1). Kim and Koo (J. Multivariate Anal. 80 (2002) 21) established minimaxity in the L 2-rate of convergence. In this paper, we improve upon the latter and establish sharp minimaxity under a super-smooth condition on the error distribution.