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

Summary

A formal likelihood ratio hypothesis test for the validity of a parametric regression function is proposed, using a large dimensional, non‐parametric 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 test proposed is exact and unbiased and the critical value is easily computed. The power of the test increases to 1 as the sample size increases, under very mild assumptions—even when the alternative is misspecified, i.e. the power of the test converges to 1 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. Supplementary materials with proofs and technical details are available on line.