  Almost sure bounds are established on the uniform error of smoothing spline
estimators in nonparametric regression with random designs. Some results of
Einmahl and Mason (2005) are used to derive uniform error bounds for the
approximation of the spline smoother by an ``equivalent'' reproducing kernel
regression estimator, as well as for proving uniform error bounds on the
reproducing kernel regression estimator itself, uniformly in the smoothing
parameter over a wide range. This admits data-driven choices of the smoothing
parameter.
