Estimation of a two-dimensional distribution function under association

Lower bounds for the asymptotic minimax risk are given when estimating the density function with spherical data. Both the pointwise risk and the integrated risk are studied. Results are derived by constructing a sequence of experiments that converge weakly to a Gaussian shift experiment. Lower bounds derived in this article complement some previously derived upper bounds. We give one upper bound for the pointwise risk which is only a factor of 1.18 larger than the corresponding lower bound. The lower bounds are formulated in a general setting so that also the Euclidean results follow as a special case.