| Zusammenfassung: | Kiefer and Wolfowitz ["Z. Wahrsch. Verw. Gebiete" 34 (1976) 73-85] 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 ${\Bbb F}_{n}$ , differs from the empirical distribution function in the uniform norm by no more than a constant times $(n^{-1}\text{log}n)^{2/3}$ 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 $\tilde{F}_{n}$ : if X₁,..., $X_{n}$ are sampled from a distribution function F with strictly convex density f, then the least squares estimator $\tilde{F}_{n}$ of F and the empirical distribution function ${\Bbb F}_{n}$ differ in the uniform norm by no more than a constant times $(n^{-1}\text{log}n)^{3/5}$ almost surely. The proofs rely on bounds on the interpolation error for complete spline interpolation due to Hall ["J. Approximation Theory" 1 (1968) 209-218], Hall and Meyer ["J. Approximation Theory 16" (1976) 105-122], building on earlier work by Birkhoff and de Boor ["J. Math. Mech." 13 (1964) 827-835]. These results, which are crucial for the developments here, are all nicely summarized and exposited in de Boor ["A Practical Guide to Splines" (2001) Springer, New York].
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