Safe hypotheses testing with application to order restricted inference

TL;DR

This paper introduces safe hypothesis tests for order-restricted inference that prevent misleading conclusions due to misspecified constraints, ensuring valid inferences with controlled Type III errors.

Contribution

It proposes a novel pre-test approach that certifies the validity of hypotheses before testing, enhancing inference reliability in order-restricted problems.

Findings

Strong protection against Type III errors in simulations

Maintains power comparable to standard tests

Applicable beyond order-restricted inference

Abstract

Hypothesis tests under order restrictions arise in a wide range of scientific applications. By exploiting inequality constraints, such tests can achieve substantial gains in power and interpretability. However, these gains come at a cost: when the imposed constraints are misspecified, the resulting inferences may be misleading or even invalid, and Type III errors may occur, i.e., the null hypothesis may be rejected when neither the null nor the alternative is true. To address this problem, this paper introduces safe tests. Heuristically, a safe test is a testing procedure that is asymptotically free of Type III errors. The proposed test is accompanied by a certificate of validity, a pre--test that assesses whether the original hypotheses are consistent with the data, thereby ensuring that the null hypothesis is rejected only when warranted, enabling principled inference without risk of systematic error. Although the development in this paper focus on testing problems in order--restricted inference, the underlying ideas are more broadly applicable. The proposed methodology is evaluated through simulation studies and the analysis of well--known illustrative data examples, demonstrating strong protection against Type III errors while maintaining power comparable to standard procedures.