  Regression splines are smooth, flexible, and parsimonious nonparametric
function estimators. They are known to be sensitive to knot number and
placement, but if assumptions such as monotonicity or convexity may be imposed
on the regression function, the shape-restricted regression splines are robust
to knot choices. Monotone regression splines were introduced by Ramsay
[Statist. Sci. 3 (1998) 425--461], but were limited to quadratic and lower
order. In this paper an algorithm for the cubic monotone case is proposed, and
the method is extended to convex constraints and variants such as
increasing-concave. The restricted versions have smaller squared error loss
than the unrestricted splines, although they have the same convergence rates.
The relatively small degrees of freedom of the model and the insensitivity of
the fits to the knot choices allow for practical inference methods; the
computational efficiency allows for back-fitting of additive models. Tests of
constant versus increasing and linear versus convex regression function, when
implemented with shape-restricted regression splines, have higher power than
the standard version using ordinary shape-restricted regression.
