🤖 AI Summary
In nonstandard testing scenarios, evaluating the optimality of heuristic test statistics is challenging. Method: This paper proposes a nested numerical optimization framework to assess whether a given heuristic test is approximately optimal—i.e., whether its power curve approximates the power envelope generated by a weighted average power (WAP)-maximizing test. Contribution/Results: We introduce, for the first time, a data-driven approach to approximate the optimal weighting function. Crucially, we establish theoretically that the rejection probability of the WAP-optimal test itself constitutes a tight upper bound on the power of any heuristic test—a result both theoretically unexpected and practically valuable. The method is provably convergent and is successfully applied to two canonical problems: robust conditional likelihood ratio (CLR) testing under weak instruments and testing at nuisance parameter boundaries. Empirical results confirm the near-optimality of the heuristic tests in these settings.
📝 Abstract
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.