Asymptotic distribution of the likelihood ratio test statistic with inequality-constrained nuisance parameters

TL;DR

This paper characterizes the asymptotic distribution of the likelihood ratio test when both parameters of interest and nuisance parameters are constrained to the boundary, extending previous results and providing practical approximation methods.

Contribution

It provides the first general characterization of the asymptotic distribution for cases with boundary-constrained nuisance parameters, including closed-form weight patterns and approximation techniques.

Findings

Closed-form difference pattern for chi-bar weights in orthogonal case

New rank-based aggregation method for mixture weights

Simulations confirm the accuracy of the proposed approximation

Abstract

The asymptotic distribution of the likelihood-ratio statistic for testing parameters on the boundary is well known to be a chi-squared mixture. The mixture weights have been shown to correspond to the intrinsic volumes of an associated tangent cone, unifying a wide range of previously isolated special cases. While the weights are fully understood for an arbitrary number of parameters of interest on the boundary, much less is known when nuisance parameters are also constrained to the boundary, a situation that frequently arises in applications. We provide the first general characterization of the asymptotic distribution of the likelihood-ratio test statistic when both the number of parameters of interest and the number of nuisance parameters on the boundary are arbitrary. We analyze how the cone geometry changes when moving from a problem with K parameters of interest on the boundary to one with K-m parameters of interest and m nuisances. In the orthogonal case we show that the resulting change in the chi-bar weights admits a closed-form difference pattern that redistributes probability mass across adjacent degrees of freedom, and that this pattern remains the dominant component of the weight shift under arbitrary covariance structures when the nuisance vector is one-dimensional. For a generic number of nuisance parameters, we introduce a new rank-based aggregation of intrinsic volumes that yields an accurate approximation of the mixture weights. Comprehensive simulations support the theory and demonstrate the accuracy of the proposed approximation.