On arbitrarily slow rates of global convergence in density estimation

Abstract

Let a density f on Rd be estimated by fn(x, X1, ..., Xn) where x∈Rd, fn is a Borel measurable function of its arguments, and X1, ..., Xn are independent random vectors with common density f. Let p≧1 be a constant. One of the main results of this note is that for every sequence fn, and for every positive number sequence an satisfying lim an=0, there exists an f such that {Mathematical expression} infinitely often. Here it suffices to look at all the f that are bounded by 2 and vanish outside [0, 1]d. For p=1, f can always be restricted to the class of infinitely many times continuously differentiable densities with all derivatives absolutely bounded and absolutely integrable. © 1983 Sringer-Verlag.