A Kiefer-Wolfowitz theorem for convex densities

Abstract

Kiefer and Wolfowitz showed that if $F$ is a strictly curved concave distribution function (corresponding to a strictly monotone density $f$), then the Maximum Likelihood Estimator $\widehat{F}_n$, which is, in fact, the least concave majorant of the empirical distribution function $\FF_n$, differs from the empirical distribution function in the uniform norm by no more than a constant times $(n^{-1} \log n)^{2/3}$ 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 $\widehat{F}_n$ of $F$ replaced by the least squares estimator $\widetilde{F}_n$: if $X_1 , \ldots , X_n$ are sampled from a distribution function $F$ with strictly convex density $f$, then the least squares estimator $\widetilde{F}_n$ of $F$ and the empirical distribution function $\FF_n$ differ in the uniform norm by no more than a constant times $(n^{-1} \log n )^{3/5}$ almost surely. The proofs rely on bounds on the interpolation error for complete spline interpolation due to Hall, Hall and Meyer. These results, which are crucial for the developments here, are all nicely summarized and exposited in de Boor.