Adaptive Critical Value for Constrained Likelihood Ratio Testing

We present a new way of testing ordered hypotheses against all alternatives which overpowers the classical approach both in simplicity and statistical power. Our new method tests the constrained likelihood ratio statistic against the quantile of one and only one chi-squared random variable with a data-dependent degrees of freedom instead of a mixture of chi-squares. Our new test is proved to have a valid finite-sample significance level $\alpha$ and provides more power especially for sparse alternatives (those with a few or moderate number of null constraints violations) in comparison to the classical approach. Our method is also easier to use than the classical approach which requires to calculate or simulate a set of complicated weights. Two special cases are considered with more details, namely the case of testing orthants $\mu_1<0, \cdots, \mu_n<0$ and the isotonic case of testing $\mu_1<\mu_2<\mu_3$ against all alternatives. Contours of the difference in power are shown for these examples showing the interest of our new approach.