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