Exact rates in density support estimation

Let f be an unknown multivariate probability density with compact support Sf. Given n independent observations X1,...,Xn drawn from f, this paper is devoted to the study of the estimator of Sf defined as unions of balls centered at the Xi and of common radius rn. To measure the proximity between and Sf, we employ a general criterion dg, based on some function g, which encompasses many statistical situations of interest. Under mild assumptions on the sequence (rn) and some analytic conditions on f and g, the exact rates of convergence of are obtained using tools from Riemannian geometry. The conditions on the radius sequence are found to be sharp and consequences of the results are discussed from a statistical perspective.