On Arbitrarily Slow Rates of Global Convergence in Density Estimation

Assume that one has to estimate a density f on R e from X1, ..., Xn, a sequence of independent random vectors with common density f A density estimate is a sequence (f,) of Borel measurable mappings: R a ( n + ~ R ; for fixed n, f (x) is estimated by f,~(x)=f,(x, X 1 . . . . , X,). In this note, we take a look at the rate of convergence of E(~ ]f , (x)f (x) lPdx) ( p > l ) for all density estimates. We could for instance inquire about the uniform rate of convergence over a suitable class of densities E, i.e. our criterion is