Exact detection threshold of the packing test

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 $\kappa =\mathrm{\Theta }\text{lb}{p}^{3\mathrm{/}4}\mathrm{/}(\log n{)}^{1\mathrm{/}4}\text{rb}$. In the high-dimensional Watson model, its detection threshold is $p-2\kappa =\mathrm{\Theta }(\sqrt{p\log n})$, or equivalently $\kappa =p\mathrm{/}2-\mathrm{\Theta }(\sqrt{p\log n})$. 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.