Exact rates in density support estimation

Let ff be an unknown multivariate probability density with compact support SfSf. Given nn independent observations X1,…,XnX1,…,Xn drawn from ff, this paper is devoted to the study of the estimator Sˆn of SfSf defined as unions of balls centered at the XiXi and of common radius rnrn. To measure the proximity between Sˆn and SfSf, we employ a general criterion dgdg, based on some function gg, which encompasses many statistical situations of interest. Under mild assumptions on the sequence (rn)(rn) and some analytic conditions on ff and gg, the exact rates of convergence of dg(Sˆn,Sf) 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.