On the impossibility of estimating densities in the extreme tail

We give a short proof of the following result. Let X1,...,Xn be independent and identically distributed observations drawn from a density f on the real line. Let fn be any estimate of the density gn of max(X1,...,Xn). We show that there exists a unimodal infinitely many times differentiable density f such that Thus, in the total variation sense, universally consistent density estimates do not exist. A similar result is derived concerning the supremum norm.