(Received June 14, 1989; revised February 15, 1990)
Abstract. Let X1, X2,...., Xn be independent observations from an (unknown) absolutely continuous univariate distribution with density f and let ^f(x) = (nh)-1 \sum ni=1 K[(x-Xi)/h] be a kernel estimator of f(x) at the point x, - \infty < x < \infty, with h = hn (hn \to 0 and nhn \to \infty, as n \to \infty) the bandwidth and K a kernel function of order r. ``Optimal'' rates of convergence to zero for the bias and mean square error of such estimators have been studied and established by several authors under varying conditions on K and f. These conditions, however, have invariably included the assumption of existence of the r-th order derivative for f at the point x. It is shown in this paper that these rates of convergence remain valid without any differentiability assumptions on f at x. Instead some simple regularity conditions are imposed on the density f at the point of interest. Our methods are based on certain results in the theory of semi-groups of linear operators and the notions and relations of calculus of ``finite differences''.
Key words and phrases: Kernel density estimation, bias, mean square error, finite differences, semi-groups, linear operators.
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