A Likelihood Ratio Test regarding Two Nested but Oblique Order-Restricted Hypotheses

Warrack, Giles; Robertson, Tim

doi:10.1080/01621459.1984.10477106

Cited by 18 publications

(7 citation statements)

References 9 publications

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“…Our analysis also highlights some new settings in which the GLRT is non-optimal. Although past work [52,33,37] has exhibited non-optimality of the GLRT in certain settings, in the context of cones, all of these past examples involve oblique cones. In Section 3.1.4, we gave an example of suboptimality which, to the best of our knowledge, is the first for a non-oblique pair of cones-namely, the cone {0}, and a certain type of Cartesian product cone.…”

Section: Discussionmentioning

confidence: 99%

“…Thus, past work has studied conditions on the cone pair under which the null distribution has a simple characterization. One such condition is a certain nonobliqueness property that is common to much past work on the GLRT (e.g., [52,33,34,21]). The non-obliqueness condition, first introduced by Warrack et al [52], is also motivated by the fact that are many instances of oblique cone pairs for which the GLRT is known to dominated by other tests.…”

Section: Cone-based Glrts and Non-oblique Pairsmentioning

confidence: 99%

“…One such condition is a certain nonobliqueness property that is common to much past work on the GLRT (e.g., [52,33,34,21]). The non-obliqueness condition, first introduced by Warrack et al [52], is also motivated by the fact that are many instances of oblique cone pairs for which the GLRT is known to dominated by other tests. Menendez et al [32] provide an explanation for this dominance in a very general context; see also the papers [34,21] for further studies of non-oblique cone pairs.…”

Section: Cone-based Glrts and Non-oblique Pairsmentioning

confidence: 99%

“…For suitably regular problems, there is a great deal of asymptotic theory on the GLRT, and in particular when its distribution under the null is independent of nuisance parameters (e.g., [3,42,39]). On the other hand, there are some isolated cases in which the GLRT can be shown to dominated by other tests (e.g., [52,33,32,26]).…”

Section: Introductionmentioning

confidence: 99%

“…Based on observing y, our goal is to test whether a given parameter θ belongs to the smaller cone C 1 -corresponding to the null hypothesis-or belongs to the larger cone C 2 . Cone testing problems of this type arise in many different settings, and there is a fairly substantial literature on the behavior of the GLRT in application to such problems (e.g., see the papers and books [8,25,40,39,41,44,35,33,34,14,47,52], as well as references therein).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Wei¹,

Wainwright²,

Guntuboyina³

2019

Ann. Statist.

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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.

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Section: Discussionmentioning

confidence: 99%

Section: Cone-based Glrts and Non-oblique Pairsmentioning

confidence: 99%

Section: Cone-based Glrts and Non-oblique Pairsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Wei¹,

Wainwright²,

Guntuboyina³

2019

Ann. Statist.

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Application of the Limit of Truncated Isotonic Regression in Optimization Subject to Isotonic and Bounding Constraints

1999

Journal of Multivariate Analysis

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An isotonic regression truncated by confining its domain to a union of its level sets is the isotonic regression in the reduced function space. When some of the weights with which the inner product system is defined go to infinity, the truncated isotonic regression converges. This limit can be used in discribing the projection onto the set of vectors which satisfy an order restriction and have one or more of its coordinates bounded above andÂor below. Through this characterization, two inequalities associated with the projection are established and found useful in order and bound restricted statistical inference. The results obtained show that for an exponential distribution family the inequalities lead to the linkage of the order and bound restricted MLE with the projection of the unrestricted MLE, and the dominance of the order and bound restricted MLE over the unrestricted MLE with respect to two classes of loss functions and risks as well as Bayes risks. 1999Academic Press AMS 1991 subject classifications: 62F10, 62C99.

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Likelihood ratio tests for a class of non-oblique hypotheses

Wright

1994

Ann Inst Stat Math

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A Likelihood Ratio Test regarding Two Nested but Oblique Order-Restricted Hypotheses

Cited by 18 publications

References 9 publications

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

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

Application of the Limit of Truncated Isotonic Regression in Optimization Subject to Isotonic and Bounding Constraints

Likelihood ratio tests for a class of non-oblique hypotheses

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