In nonstandard testing environments, researchers often derive ad hoc tests with correct (asymptotic) size, but their optimality properties are typically unknown a priori and difficult to assess. This paper develops a numerical framework for determining whether an ad hoc test is effectively optimal - approximately maximizing a weighted average power criterion for some weights over the alternative and attaining a power envelope generated by a single weighted average power-maximizing test. Our approach uses nested optimization algorithms to approximate the weight function that makes an ad hoc test's weighted average power as close as possible to that of a true weighted average power-maximizing test, and we show the surprising result that the rejection probabilities corresponding to the latter form an approximate power envelope for the former. We provide convergence guarantees, discuss practical implementation and apply the method to the weak instrument-robust conditional likelihood ratio test and a recently-proposed test for when a nuisance parameter may be on or near its boundary.

翻译：在非标准检验环境中，研究者常会推导出具有正确（渐近）尺寸的特设检验，但其最优性性质通常先验未知且难以评估。本文开发了一个数值框架，用于判定特设检验是否有效最优——即近似最大化备择假设上某个权重函数的加权平均功效准则，并达到由单一加权平均功效最大化检验生成的功效包络。我们的方法采用嵌套优化算法来逼近使特设检验加权平均功效尽可能接近真实加权平均功效最大化检验的权重函数，并证明了一个令人惊讶的结果：后者的拒绝概率构成了前者的近似功效包络。我们提供了收敛性保证，讨论了实际实施方案，并将该方法应用于弱工具变量稳健的条件似然比检验，以及针对干扰参数可能位于或接近其边界情况的最新检验方法。