Consistency and power in tests with shape-restricted alternatives

Likelihood ratio tests of constant vs. monotone regression function, as well as linear vs. convex regression function and other tests with shape-restricted alternatives, are known to have null distributions equivalent to mixtures of beta random variates. The monotone and convex regression estimators are known to be inconsistent at the endpoints, where there is “spiking.” This spiking affects the critical values of the test statistic. Modified versions of the monotone and convex regression estimators are proposed that are consistent everywhere; when the modified versions are used in hypothesis testing, the null distribution is again an exact mixture of beta distributions, with different mixing parameters. Simulations show that the power of the test using the modified version is larger for the examples chosen.