A test for linear versus convex regression function using shape-restricted regression

An unbiased test for the appropriateness of the simple linear regression model is presented. The null hypothesis is that the underlying regression function is indeed a line, and the alternative is that it is convex. The exact distribution for a likelihood ratio test statistic is that of a mixture of beta random variables, with the mixing distribution calculated from relative volumes of polyhedral convex cones determined by the convex shape restriction. Simulations show that the power of the test is favourable compared with the usual F-test against a quadratic model, for some nonquadratic choices of the underlying regression function.