  Consider a nonparametric regression model with one-sided errors and
regression function in a general H\"older class. We estimate the regression
function via minimization of the local integral of a polynomial approximation.
We show uniform rates of convergence for the simple regression estimator as
well as for a smooth version. These rates carry over to mean regression models
with a symmetric and bounded error distribution. In such a setting, one obtains
faster rates for irregular error distributions concentrating sufficient mass
near the endpoints than for the usual regular distributions. The results are
applied to prove asymptotic $\sqrt{n}$-equivalence of a residual-based
(sequential) empirical distribution function to the (sequential) empirical
distribution function of unobserved errors in the case of irregular error
distributions. This result is remarkably different from corresponding results
in mean regression with regular errors. It can readily be applied to develop
goodness-of-fit tests for the error distribution. We present some examples and
investigate the small sample performance in a simulation study. We further
discuss asymptotically distribution-free hypotheses tests for independence of
the error distribution from the points of measurement and for monotonicity of
the boundary function as well.
