AI 中文总结
该研究推导了packing检验在高维FvML和Watson模型下的精确检测阈值，证实其在两种模型中检验对称性时严格次优，还分析了其极限分布及相变现象。
AI 中文摘要
采用泊松近似技术，我们推导了文献[Jiang13]中packing检验在高维Fisher-von Mises-Langevin（FvML）和Watson备择假设下检验球面对称性时的检测阈值。我们的结果严格证实了packing检验在这两种流行模型中检验对称性时严格次优的经验观察。在高维FvML模型中，其检测阈值恰好为κ=Θ(p^(3/4)/(log n)^(1/4))；在高维Watson模型中，其检测阈值为p-2κ=Θ(√(p log n))，等价于κ=p/2 - Θ(√(p log n))。我们还推导了packing检验在这两种模型下的非零极限分布，表明最大平方内积的极限尺度在Watson模型中经历不连续相变，而FvML模型中无类似现象。
英文摘要
Using Poisson approximation techniques, we derive the detection threshold of the packing test in \cite{Jiang13} when testing spherical uniformity under high-dimensional Fisher--von Mises--Langevin (FvML) and Watson alternatives. Our result rigorously confirms the empirical observation that the packing test is strictly suboptimal for testing uniformity in these two popular models. In the high-dimensional FvML model, its detection threshold is precisely \(κ=Θ\lb p^{3/4}/(\log n)^{1/4}\rb\). In the high-dimensional Watson model, its detection threshold is \(p-2κ=Θ(\sqrt{p\log n})\), or equivalently \(κ=p/2-Θ(\sqrt{p\log n})\). The non-null limiting distributions of the packing test under these two models are derived. We show that the limiting scalings of the largest squared inner product undergo a discontinuous phase transition in the Watson model, whereas no analogous phenomenon occurs in the FvML model.
