Testing uniformity on high-dimensional spheres
We consider the problem of testing uniformity on high-dimensional unit spheres. We are primarily interested in non-null issues and focus on spiked alternatives. We show that such alternatives lead to two Local Asymptotic Normality (LAN) structures. The first one is for a fixed spike direction theta and allows to derive locally asymptotically optimal tests under specified theta. The second one relates to the unspecified-theta problem and allows to identify locally asymptotically optimal invariant tests. Interestingly, symmetric and asymmetric spiked alternatives lead to very different optimal tests, based on sample averages and sample covariance matrices, respectively. Most of our results allow the dimension p to go to infinity in an arbitrary way as a function of the sample size n.
Reference : Cutting, Chr., Paindaveine, D., and Verdebout, Th. (Z017). Testing uniformity on high-dimensional spheres against monotone rotationally symmetric alternatives. Annals of Statistics, to appear.