A Kiefer - Wolfowitz Theorem for Convex Densities

Kiefer and Wolfowitz [14] showed that if F is a strictly curved concave distribution function (corresponding to a strictly monotone density f), then the Maximum Likelihood Estimator F̂n, which is, in fact, the least concave majorant of the empirical distribution function Fn, differs from the empirical distribution function in the uniform norm by no more than a constant times (n logn)2/3 almost surely. We review their result and give an updated version of their proof. We prove a comparable theorem for the class of distribution functions F with convex decreasing densities f , but with the maximum likelihood estimator F̂n of F replaced by the least squares estimator F̃n: if X1, . . . , Xn are sampled from a distribution function F with strictly convex density f , then the least squares estimator F̃n of F and the empirical distribution function Fn differ in the uniform norm by no more than a constant times (n logn)3/5 almost surely. The proofs rely on bounds on the interpolation error for complete spline interpolation due to Hall [12], Hall and Meyer [13], building on earlier work by Birkhoff and de Boor [4]. These results, which are crucial for the developments here, are all nicely summarized and exposited in de Boor [5].