The Annals of Statistics

The geometry of hypothesis testing over convex cones: Generalized likelihood ratio tests and minimax radii

Yuting Wei, Martin J. Wainwright, and Adityanand Guntuboyina

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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.

Article information

Ann. Statist., Volume 47, Number 2 (2019), 994-1024.

Received: April 2017
Revised: March 2018
First available in Project Euclid: 11 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F03: Hypothesis testing
Secondary: 52A05: Convex sets without dimension restrictions

Hypothesis testing closed convex cone likelihood ratio test minimax rate Gaussian complexity


Wei, Yuting; Wainwright, Martin J.; Guntuboyina, Adityanand. The geometry of hypothesis testing over convex cones: Generalized likelihood ratio tests and minimax radii. Ann. Statist. 47 (2019), no. 2, 994--1024. doi:10.1214/18-AOS1701.

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Supplemental materials

  • Supplement to “The geometry of hypothesis testing over convex cones: Generalized likelihood ratio tests and minimax radii”. The supplementary material includes the explanation for the GLRT suboptimality and the proofs of more technical aspects.