On the impossibility of estimating densities in the extreme tail

We give a short proof of the following result. Let X-1,...,X-n be independent and identically distributed observations drawn from a density f on the real line. Let f(n) be any estimate of the density g(n) of max (X-1,...,X-n). We show that there exists a unimodal infinitely many times differentiable density f such that inf(n)E{ integral\f(n)(x)-g(n)(x)\ dx} greater than or equal to 1/49. Thus, in the total variation sense, universally consistent density estimates do not exist. A similar result is derived concerning the supremum norm. (C) 1999 Elsevier Science B.V. All rights reserved. MSG: primary: 62G07; secondary: 62G05; 62F12; 60F25.