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 S ˆ 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 S ˆ 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 ( S ˆ 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.