  A formal likelihood ratio hypothesis test for the validity of a parametric
regression function is proposed, using a large-dimensional, nonparametric
double cone alternative. For example, the test against a constant function uses
the alternative of increasing or decreasing regression functions, and the test
against a linear function uses the convex or concave alternative. The proposed
test is exact, unbiased and the critical value is easily computed. The power of
the test increases to one as the sample size increases, under very mild
assumptions -- even when the alternative is mis-specified. That is, the power
of the test converges to one for any true regression function that deviates (in
a non-degenerate way) from the parametric null hypothesis. We also formulate
tests for the linear versus partial linear model, and consider the special case
of the additive model. Simulations show that our procedure behaves well
consistently when compared with other methods. Although the alternative fit is
non-parametric, no tuning parameters are involved.
