For the standard linear regression estimator, the degrees of freedom of the error sum of squares is n-k, where k+1 is the dimension of the linear space onto which the data vector y is projected. For the shape restricted regression estimator, we project the data vector onto a convex polyhedral cone, which has dimension n. We want to find an estimator of the model variance of the form SSE/(n-c), where n-c is the surrogate degrees of freedom, and c is analogous to the "dimension" of the projection. We show that if D is the dimension of the face of the cone on which the projection falls, and we estimate the model variance with SSE/(n-cD), then $\sqrt{n}(\hat{\sigma}^2-\sigma^2)$ is asymptotically normal with mean zero and variance 2 sigma^4 if 1 <= c <= 2. For monotone regression, we show c is asymptotically equivalent to 1.5.
The degrees of freedom result can be used to test for parametric models against the shape-restricted alternative. The theory can also be used in the selection of a smoothing parameter in the smoothed version of the shape restricted regression estimator.