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.
Hypothesis testing has been studied extensively within the framework of order-restricted inference (ORI); see the monographs of Barlow et al. (1972), Robertson et al. (1988) Sen (2005). Silvapulle and Sen (2005) classified a large subset of the testing problems arising in ORI into Type A or Type B Problems. Type A Problems are formulated as
where L is a linear subspace and C is closed convex cone with L ⊂ C. A classic example of such testing problems is H 0 : θ = 0 versus H 1 : θ ∈ R m + {0} where R m + is the positive orthant. Type A problems are common in applications and often referred to as testing for an order. Type B Problems are formulated as
A canonical Type B Problem is
This class of tests is referred to as testing against an order.
It is well known that accounting for constraints, as specified in (1) or (2), improves the power of the resulting tests (e.g.,Praestgaard 2012) as well as the accuracy of the associated estimators (e.g., Hwang and Peddada 1994, Silvapulle and Sen 2005, Rosen and Davidov 2017). These improvements are often substantial (cf., Singh et al. 2021). A case in point is ANOVA type problems where the superior performance of ORI has been well known for over fifty years, cf., Barlow et al. (1972). Singh and Davidov (2019) recently showed that striking gains are possible when experiments are both designed and analyzed using methods that properly account for the underlying constraints. However, to date, few scientific studies have capitalized on these findings, and the methods of ORI remain vastly underutilized. In our view, barriers to the broad adoption of the methods of ORI are both practical and principled. Practically, ORI requires constrained estimation and nonstandard asymptotic theory, making it more complex to understand and implement. Moreover, standard tools such as the bootstrap may fail when parameters lie on the boundary of the parameter space (Andrews, 2000), and user-friendly software remains limited. Principled objections concern the behavior of tests and estimators when the assumed constraints, e.g., θ ∈ C are misspecified. For example, one may ask how a test for (1) behaves when θ / ∈ C. Additionally, several authors, including Silvapulle (1997) and Cohen and Sackrowitz (2004), have discussed methodological concerns and potential deficiencies of the likelihood ratio test (LRT) in ORI. See also Perlman and Wu (1999) and the references therein. Such concerns have motivated the development of alternative procedures, including cone-order monotone tests as advocated by Cohen and Sackrowitz (1998). This communication addresses the aforementioned principled concerns, thereby resolving many of the issues raised in the literature.
Despite the well-known possibility of misspecifying ordered restrictions and the widely recognized risk of Type III errors, there is a clear gap in the literature concerning their formal treatment. Addressing this gap, the paper introduces and studies a novel, easy-to-apply safe test, a testing procedure that is asymptotically free of Type III errors. Safe tests constitute a first step toward adaptive ORI, methodologies for estimation, prediction, and related tasks, in which order constraints are imposed only when supported by the data.
The paper is organized in the following way. In Section 2 the geometry of the distance test is studied. Section 3 introduces and studies a novel safe test. Simulation results and illustrative examples, including the reanalysis of some well known case studies from the literature, are provided in Section 4. We conclude in Section 5 with a brief summary and a discussion. All proofs are collected in Appendix A.
Suppose that there exists a statistic S S S n which estimates a parameter θ θ θ ∈ Θ ⊆ R m and satisfies
as n → ∞ where ⇒ denotes convergence in distribution. We further assume that Σ Σ Σ n , a consistent estimator for Σ Σ Σ, exists. Numerous tests for ( 1) and ( 2) assuming (3) have been proposed in the literature (Silvapulle and Sen, 2005). The most common in both applications as well in theoretical studies is the distance test (DT) which is of the form
where (Θ 0 , Θ 1 ) = (L, C) for Type A Problems and (Θ 0 , Θ 1 ) = (C, R m ) for Type B Problems. Here Π Σ Σ Σn (S S S n |Θ i ) is the Σ Σ Σ n -projection of S S S n onto Θ i where i ∈ {0, 1} and ∥ • ∥ Σ Σ Σn is the corresponding norm. The DT is the large sample version of the LRT under normality, i.e., if (3) holds exactly and Σ Σ Σ is known up to a constant multiple, then (4) is the LRT. The null is rejected in favor of the alternative at the level α if T n ≥ c α the α level critical value. We say that the DT is consistent at
Understanding the geometry of the DT requires additional notation. First, for any cone C let C • Σ Σ Σ denote its polar cone with respect to the inner product ⟨u,
Σ Σ Σ whenever no ambiguity arises. Next, note that the continuity of projections onto convex sets and the continuous mapping theorem imply that
so T n → ∞ if and o
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