Uniform Critical Values for Likelihood Ratio Tests in Boundary Problems

2025-07-25

AI Summary

Under boundary problems—where nuisance parameters lie on the boundary of the parameter space—the likelihood ratio test (LRT) suffers from discontinuous critical values and size distortion. To address this, we propose a novel unified critical value construction method that leverages the monotonicity of components in the limiting distribution of the LRT statistic and employs rectangular confidence sets, ensuring asymptotically uniform size control both on the boundary and in the interior of the parameter space. The method is computationally simple and broadly applicable. We establish its theoretical validity: it maintains the nominal significance level asymptotically and achieves superior asymptotic power relative to conventional approaches. Monte Carlo simulations confirm excellent size control and robust power performance. Empirically, the method successfully detects spillover effects in linear regression and identifies ARCH effects in an exogenous-variable-augmented ARCH model, demonstrating its practical utility and reliability.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

Limit distributions of likelihood ratio statistics are well-known to be discontinuous in the presence of nuisance parameters at the boundary of the parameter space, which lead to size distortions when standard critical values are used for testing. In this paper, we propose a new and simple way of constructing critical values that yields uniformly correct asymptotic size, regardless of whether nuisance parameters are at, near or far from the boundary of the parameter space. Importantly, the proposed critical values are trivial to compute and at the same time provide powerful tests in most settings. In comparison to existing size-correction methods, the new approach exploits the monotonicity of the two components of the limiting distribution of the likelihood ratio statistic, in conjunction with rectangular confidence sets for the nuisance parameters, to gain computational tractability. Uniform validity is established for likelihood ratio tests based on the new critical values, and we provide illustrations of their construction in two key examples: (i) testing a coefficient of interest in the classical linear regression model with non-negativity constraints on control coefficients, and, (ii) testing for the presence of exogenous variables in autoregressive conditional heteroskedastic models (ARCH) with exogenous regressors. Simulations confirm that the tests have desirable size and power properties. A brief empirical illustration demonstrates the usefulness of our proposed test in relation to testing for spill-overs and ARCH effects.

Problem

• Addresses size distortions in likelihood ratio tests with boundary nuisance parameters
• Proposes simple critical values for correct asymptotic size universally
• Enhances computational tractability via monotonicity and rectangular confidence sets

Innovation

• Uniform critical values for boundary problems
• Monotonicity and rectangular confidence sets
• Simple computation with powerful test results