  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
$\hat{F}_n$, which is, in fact, the least concave majorant of the empirical
distribution function $\mathbb {F}_n$, 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 $\hat{F}_n$ of $F$ replaced by the least squares estimator
$\widetilde{F}_n$: if $X_1,..., X_n$ 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 $\mathbb
{F}_n$ 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 [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].
