Safe hypotheses testing with application to order restricted inference

TL;DR

The paper tackles hypothesis tests under order restrictions (ORI) and the risk of Type III errors when the imposed order is misspecified. It introduces safe tests that are asymptotically free of Type III errors by coupling a base distance-test against the constrained set with an auxiliary test against the unconstrained/alternative, and it provides a certificate of validity via a pre-test. The SAFE procedure guarantees α_SAFE ≤ α and yields practical procedures to compute critical values, including extensions to polyhedral cones with Monte Carlo-based weight estimation for face projections. Through simulations and illustrative examples, the method shows strong protection against Type III errors while maintaining power comparable to standard approaches in favorable regions. This framework supports adaptive ORI, enabling principled inference that imposes order constraints only when data support them, with broad applicability beyond the canonical problems studied.

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.

Figures (2)

• Figure 1: The acceptance rejection region of $T_n$ lies below the (red) curve $\mathfrak{L}$ whereas the acceptance region of $T_n^{'}$ lies above the (blue) curve $\mathfrak{L}^{'}$. For small values of $\gamma$ the curve $\mathfrak{L}^{'}$ lies at large distance from $\mathbb{R}_{+}^2$ whereas when $\gamma$ is large the curve closely hugs $\mathbb{R}_{+}^2$.
• Figure 2: Plots of the cones $(\mathbb{R}_{+}^{2})_{\mathop{\mathrm{\pmb{\Sigma}}}\nolimits}^{\circ}$, i.e., the regions in which the DT is not consistent, for $\rho\in\{+1/4,-1/4\}$. The cones, in blue, are bounded by their extreme rays which extend indefinitely.

Theorems & Definitions (15)

• Theorem 2.1
• Remark 2.1
• Example 2.1
• Theorem 2.2
• Theorem 2.3
• Example 2.2
• Definition 3.1
• Theorem 3.1
• Definition 3.2
• Theorem 3.2