On the rate of convergence of an estimate of a functional of a probability density

Abstract

Abstract Bhattacharyya & Roussas (1970) have proposed that one estimates T(f) = ∫ ∞−∞f 2(u)du by estimates of the form T(fn ) where fn is a suitably chosen kernel estimator of the density f of the type studied by Rosenblatt (1956), Parzen (1962), and others. In this paper we first give the rate of strong uniform convergence of the distribution estimators Fn (x) = ∫x−∞ f (u)du to the population cdf F(x) = ∫x−∞ f(u)du. We then interpret T(f) as T(f, F) = ∫x−∞ f(u)dF(u) and estimate T(f, F) by T(fn , Fn ). Looking at the estimates in this fashion allows us to use both the properties of fn and Fn in establishing the rate of strong consistency of T(fn ) to T(f). We then consider the computationally simpler estimate where is the usual empirical distribution function. The rate of convergence remains the same.