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.

中文翻译：

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约束似然比检验的自适应临界值

我们提出了一种针对所有替代方案测试有序假设的新方法，该方法在简单性和统计功效方面都优于经典方法。我们的新方法根据一个且仅一个卡方随机变量的分位数测试约束似然比统计量，该随机变量具有数据相关的自由度，而不是卡方的混合。与经典方法相比，我们的新测试被证明具有有效的有限样本显着性水平 $\alpha$，并且提供了更多的功效，尤其是对于稀疏替代方案（具有少量或中等数量的空约束违规的替代方案）。我们的方法也比需要计算或模拟一组复杂权重的经典方法更易于使用。更详细地考虑了两种特殊情况，即测试 orthants $\mu_1<0, \cdots, \mu_n<0$ 和针对所有替代品测试 $\mu_1<\mu_2<\mu_3$ 的等渗情况。这些示例显示了功率差异的轮廓，显示了我们新方法的兴趣。