The geometry of hypothesis testing over convex cones: Generalized likelihood ratio tests and minimax radii
研究了高斯序列模型中凸锥上的复合检验问题，给出了广义似然比检验半径的精确刻画，并证明了极小化最大检验半径的信息论下界，揭示了检验误差不仅依赖于问题复杂度，还受锥的几何性质影响。
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 arises in various applications, including detection of treatment effects, trend detection in econometrics, signal detection in radar processing and shape-constrained inference in nonparametric 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. In order to address the issue of optimality, we prove information-theoretic lower bounds for the 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.