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

Testing against a linear regression model using ideas from shape-restricted estimation