Improved confidence sets for the mean of a multivariate normal distribution

Shinozaki, Nobuo

doi:10.1007/bf00049400

Cited by 9 publications

(5 citation statements)

References 11 publications

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“…Confidence sets for the multivariate normal mean with other shapes have been proposed by Faith (1976), Berger (1980), Shinozaki (1989), Tseng and Brown (1997) and Efron (2006). Reviews of the literature on confidence sets for the multivariate normal mean are provided by Efron (2006) and Casella and Hwang (2012).…”

Section: Resultsmentioning

confidence: 99%

Optimized recentered confidence spheres for the multivariate normal mean

Abeysekera¹,

Kabaila²

2017

Electron. J. Statist.

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Casella and Hwang, 1983, JASA, introduced a broad class of recentered confidence spheres for the mean θ of a multivariate normal distribution with covariance matrix σ 2 I, for σ 2 known. Both the center and radius functions of these confidence spheres are flexible functions of the data. For the particular case of confidence spheres centered on the positive-part James-Stein estimator and with radius determined by empirical Bayes considerations, they show numerically that these confidence spheres have the desired minimum coverage probability 1 − α and dominate the usual confidence sphere in terms of scaled volume. We shift the focus from the scaled volume to the scaled expected volume of the recentered confidence sphere. Since both the coverage probability and the scaled expected volume are functions of the Euclidean norm of θ, it is feasible to optimize the performance of the recentered confidence sphere by numerically computing both the center and radius functions so as to optimize some clearly specified criterion. We suppose that we have uncertain prior information that θ = 0. This motivates us to determine the center and radius functions of the confidence sphere by numerical minimization of the scaled expected volume of the confidence sphere at θ = 0, subject to the constraints that (a) the coverage probability never falls below 1 − α and (b) the radius never exceeds the radius of the standard 1 − α confidence sphere. Our results show that, by focusing on this clearly specified criterion, significant gains in performance (in terms of this criterion) can be achieved. We also present analogous results for the much more difficult case that σ 2 is unknown.

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Section: Resultsmentioning

confidence: 99%

Optimized recentered confidence spheres for the multivariate normal mean

Abeysekera¹,

Kabaila²

2017

Electron. J. Statist.

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“…Smaller confidence ellipsoids with the same coverage can be obtained by shifting the center slightly [48,49]. Furthermore, other constructions similar to an egg [50] or the non-convex Pascal limaçon [51] are known to outperform the standard ellipsoids. Nevertheless, none of these constructions is known to be optimal and, to the best of the authors' knowledge, no optimal confidence region for multivariate Gaussians in higher dimensions is known.…”

Section: Confidence Regions From Linear Inversionmentioning

confidence: 99%

Error regions in quantum state tomography: computational complexity caused by geometry of quantum states

Suess¹,

Rudnicki²,

Maciel³

et al. 2017

New J. Phys.

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The outcomes of quantum mechanical measurements are inherently random. It is therefore necessary to develop stringent methods for quantifying the degree of statistical uncertainty about the results of quantum experiments. For the particularly relevant task of quantum state tomography, it has been shown that a significant reduction in uncertainty can be achieved by taking the positivity of quantum states into account. However-the large number of partial results and heuristics notwithstandingno efficient general algorithm is known that produces an optimal uncertainty region from experimental data, while making use of the prior constraint of positivity. Here, we provide a precise formulation of this problem and show that the general case is NP-hard. Our result leaves room for the existence of efficient approximate solutions, and therefore does not in itself imply that the practical task of quantum uncertainty quantification is intractable. However, it does show that there exists a non-trivial trade-off between optimality and computational efficiency for error regions. We prove two versions of the result: one for frequentist and one for Bayesian statistics.

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“…x is that of Shinozaki (1989). Shinozaki worked with the xsection of the confidence set, starting with the set C 0 x .…”

Section: Reducing Volume and Increasing Coveragementioning

confidence: 99%

Shrinkage Confidence Procedures

Casella¹,

Hwang²

2012

Statist. Sci.

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The possibility of improving on the usual multivariate normal confidence was first discussed in Stein (1962). Using the ideas of shrinkage, through Bayesian and empirical Bayesian arguments, domination results, both analytic and numerical, have been obtained. Here we trace some of the developments in confidence set estimation.Comment: Published in at http://dx.doi.org/10.1214/10-STS319 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Improved confidence sets for the mean of a multivariate normal distribution

Cited by 9 publications

References 11 publications

Optimized recentered confidence spheres for the multivariate normal mean

Optimized recentered confidence spheres for the multivariate normal mean

Error regions in quantum state tomography: computational complexity caused by geometry of quantum states

Shrinkage Confidence Procedures

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