  We consider a compound testing problem within the Gaussian sequence model in
which the null and alternative are specified by a pair of closed, convex cones.
Such cone testing problem arise in various applications, including detection of
treatment effects, trend detection in econometrics, signal detection in radar
processing, and shape-constrained inference in non-parametric statistics. We
provide a sharp characterization of the GLRT testing radius up to a universal
multiplicative constant in terms of the geometric structure of the underlying
convex cones. When applied to concrete examples, this result reveals some
interesting phenomena that do not arise in the analogous problems of estimation
under convex constraints. In particular, in contrast to estimation error, the
testing error no longer depends purely on the problem complexity via a
volume-based measure (such as metric entropy or Gaussian complexity), other
geometric properties of the cones also play an important role. To address the
issue of optimality, we prove information-theoretic lower bounds for minimax
testing radius again in terms of geometric quantities. Our general theorems are
illustrated by examples including the cases of monotone and orthant cones, and
involve some results of independent interest.
