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.

翻译：众所周知，用于检验边界参数的似然比统计量的渐近分布为卡方混合分布。已有研究表明，混合权重对应于关联切锥的本征体积，从而统一了先前众多孤立的特例。尽管对于边界上任意数量的感兴趣参数，其权重已得到充分理解，但当多余参数同样受限于边界时——这种情况在应用中频繁出现——相关认知则少得多。我们首次对边界上感兴趣参数与多余参数数量均为任意时的似然比检验统计量的渐近分布进行了普适性刻画。我们分析了从具有K个边界感兴趣参数的问题转变为具有K-m个感兴趣参数与m个多余参数问题时锥几何结构的变化。在正交情形下，我们证明由此产生的卡方条权重的变化呈现出封闭式的差分模式，该模式将概率质量重新分配到相邻自由度上，且当多余向量为一维时，该模式在任意协方差结构下仍是权重转移的主导成分。对于一般数量的多余参数，我们引入了一种新的基于秩的本征体积聚合方法，可精确逼近混合权重。综合仿真实验支持了理论结果，并验证了所提逼近方法的准确性。