AI 中文总结
研究半正定矩阵秩至多为规定值的似然比检验，通过对正则干扰参数轮廓分析，推导出简化高斯实验，得出顶层定律最不利等结论，还证明了相关转变优势，获得极限分布及临界值的条件形状导数。
AI 中文摘要
我们研究关于半正定矩阵的秩至多为规定值这一假设的似然比检验。原假设是分层的：最大允许秩的点位于正则边界层，而低秩点是奇异的。因此，顶层的通常卡方校准本身并不能描述整个复合原假设，特别是沿着秩在局部\(n^{-1/2}\)尺度变化的序列。在对正则干扰参数进行轮廓分析后，我们为每个固定的原假设秩和所有可允许的局部秩转变推导出一个共同的简化高斯实验。在顶层，恢复了经典的卡方定律。在较低秩时，极限通常涉及投影到非凸的秩约束半定集上。我们的主要校准结果表明，在各向同性下，顶层定律在所有固定的原假设层和所有局部原假设秩转变中是最不利的。当有效余秩为一时，我们还证明了在任意各向异性下的相应转变优势。最后，在顶层，我们获得了极限分布及其临界值的条件形状导数。高斯协方差模型和有限样本实验说明了干扰轮廓分析、秩转变、各向异性和方向敏感性。
英文摘要
We study likelihood-ratio tests for the hypothesis that a positive-semidefinite matrix has rank at most a prescribed value. The null hypothesis is stratified: points of maximal allowed rank lie on a regular boundary stratum, whereas lower-rank points are singular. Consequently, the usual chi-bar-square calibration on the top stratum does not by itself describe the whole composite null, especially along sequences whose rank changes at the local $n^{-1/2}$ scale. After profiling regular nuisance parameters, we derive a common reduced Gaussian experiment for every fixed null rank and for all admissible local rank transitions. On the top stratum, the classical chi-bar-square law is recovered. At lower ranks, the limit generally involves projection onto a nonconvex rank-constrained semidefinite set. Our main calibration result shows that, under isotropy, the top-stratum law is least favourable over all fixed null strata and all local null rank transitions. We also prove the corresponding transition dominance under arbitrary anisotropy when the active corank is one. Finally, on the top stratum, we obtain a conditional shape derivative for the limiting distribution and its critical value. Gaussian covariance models and finite-sample experiments illustrate nuisance profiling, rank transitions, anisotropy, and orientation sensitivity.
Comments24 pages, 23 pages supplement, 1 R code
