Unbiased tests for the constant versus monotone regression function, as well as the linear versus convex regression function, are known to have null distributions equal to those of mixtures of beta random variables. Both monotone and convex regression estimators exhibit "spiking" at the endpoints of the data range, where the estimator is inconsistent. Consistent estimators for both shape-restricted alternatives are proposed, for which the test statistic using the consistent estimator has again the form of a mixture of beta densities. The power of the test is shown through simulations to improve with the consistent estimators in both examples, for many true underlying regression functions.
Improving the Power of Tests with Shape-Restricted Alternatives via Projections Onto Subcones