Open Access
April 2019 The geometry of hypothesis testing over convex cones: Generalized likelihood ratio tests and minimax radii
Yuting Wei, Martin J. Wainwright, Adityanand Guntuboyina
Ann. Statist. 47(2): 994-1024 (April 2019). DOI: 10.1214/18-AOS1701


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.


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Yuting Wei. Martin J. Wainwright. Adityanand Guntuboyina. "The geometry of hypothesis testing over convex cones: Generalized likelihood ratio tests and minimax radii." Ann. Statist. 47 (2) 994 - 1024, April 2019.


Received: 1 April 2017; Revised: 1 March 2018; Published: April 2019
First available in Project Euclid: 11 January 2019

zbMATH: 07033159
MathSciNet: MR3909958
Digital Object Identifier: 10.1214/18-AOS1701

Primary: 62F03
Secondary: 52A05

Keywords: closed convex cone , Gaussian complexity , Hypothesis testing , likelihood ratio test , Minimax rate

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 2 • April 2019
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