Exact rates in density support estimation

Let f be an unknown multivariate probability density with compact support S"f. Given n independent observations X"1,...,X"n drawn from f, this paper is devoted to the study of the estimator [email protected]?"n of S"f defined as unions of balls centered at the X"i and of common radius r"n. To measure the proximity between [email protected]?"n and S"f, we employ a general criterion d"g, based on some function g, which encompasses many statistical situations of interest. Under mild assumptions on the sequence (r"n) and some analytic conditions on f and g, the exact rates of convergence of d"g([email protected]?"n,S"f) 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.