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.

翻译：有序约束下的假设检验在众多科学领域有着广泛应用。通过利用不等式约束，此类检验能够在统计功效与可解释性方面获得显著提升。然而，这些优势伴随着相应代价：当施加的约束存在误设时，所得推断可能产生误导甚至无效，并可能出现第三类错误（即当原假设与备择假设均不成立时错误拒绝原假设）。为解决该问题，本文提出安全检验方法。直观而言，安全检验是一种渐近消除第三类错误的检验程序。所提出的检验方法附带有有效性证明——通过预检验评估原始假设与数据的相容性，从而确保仅在具备充分依据时拒绝原假设，实现无系统性风险的原则性推断。尽管本文主要针对有序约束推断中的检验问题展开研究，但其核心思想具有更广泛的适用性。通过模拟研究与经典示例数据分析，所提方法在保持与标准方法相当统计功效的同时，展现出对第三类错误的强效防护能力。