Good exact confidence sets for a multivariate normal mean

Tseng, Yu-Ling; Brown, Lawrence D.

doi:10.1214/aos/1069362396

Cited by 16 publications

(14 citation statements)

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“…For dimensions p >2, Tseng and Brown proposed the pseudo‐empirical Bayes 100(1− α ) % confidence region

C^{T B} (X) = \{θ : | | X - θ (1 + γ (| | θ | {false|}^{2})) | {false|}^{2} ⩽ χ_{1 - α, p}^{2} (| | θ | {false|}^{2} γ^{2} (| | θ | {false|}^{2}))\},

where

γ false(false| false| bold-italicθ false| |^{2} false) = \frac{1}{A + B false| false| bold-italicθ false| |^{2}}

.…”

Section: Methodsunclassified

“…The TOST procedure itself is easy to extend by applying the intersection-union principle, 12 and other multivariate equivalence tests are available as well, 13,14 but there have been controversies about how to construct a simultaneous confidence region around the vector of estimated PK measures. A number of ideas have been put forward in the past two decades: Various authors suggested methods tailored to (bio-)equivalence problems, [15][16][17][18][19] but also approaches that were not specifically designed for application in pharmaceutical statistics 20,21 may prove useful. For 3 and more PK parameters, methods that involve shrinkage (see Casella and Hwang 22 for an overview) seem appealing due to Stein's paradox.…”

Section: Introductionmentioning

confidence: 99%

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Simultaneous confidence regions for multivariate bioequivalence

Pallmann

Jaki

2017

Statistics in Medicine

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Demonstrating bioequivalence of several pharmacokinetic (PK) parameters, such as AUC and C , that are calculated from the same biological sample measurements is in fact a multivariate problem, even though this is neglected by most practitioners and regulatory bodies, who typically settle for separate univariate analyses. We believe, however, that a truly multivariate evaluation of all PK measures simultaneously is clearly more adequate. In this paper, we review methods to construct joint confidence regions around multivariate normal means and investigate their usefulness in simultaneous bioequivalence problems via simulation. Some of them work well for idealised scenarios but break down when faced with real-data challenges such as unknown variance and correlation among the PK parameters. We study the shapes of the confidence regions resulting from different methods, discuss how marginal simultaneous confidence intervals for the individual PK measures can be derived, and illustrate the application to data from a trial on ticlopidine hydrochloride. An R package is available.

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“…For dimensions p >2, Tseng and Brown proposed the pseudo‐empirical Bayes 100(1− α ) % confidence region

C^{T B} (X) = \{θ : | | X - θ (1 + γ (| | θ | {false|}^{2})) | {false|}^{2} ⩽ χ_{1 - α, p}^{2} (| | θ | {false|}^{2} γ^{2} (| | θ | {false|}^{2}))\},

where

γ false(false| false| bold-italicθ false| |^{2} false) = \frac{1}{A + B false| false| bold-italicθ false| |^{2}}

.…”

Section: Methodsunclassified

Section: Introductionmentioning

confidence: 99%

Simultaneous confidence regions for multivariate bioequivalence

Pallmann

Jaki

2017

Statistics in Medicine

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“…Tseng and Brown (1997) have given an excellent review of the earlier literature on the multivariate normal confidence set problem, which, in addition to those references already given, includes Stein (1962, 1981) and Hwang and Casella (1982, 1984). Tseng and Brown themselves proposed somewhat egg‐shaped sets which have exact coverage probability and they also found sufficient conditions under which their sets uniformly dominate C 0 ( X ) in terms of volume.…”

Section: Introductionmentioning

confidence: 99%

“…However, Berger (1980) has given a heuristic argument suggesting that this choice may not be so clear cut. In fact, Faith (1976), Shinozaki (1989) and Tseng and Brown (1997) have all also proposed non-spherical confidence regions. There is a consensus that an acceptable confidence set should be at least connected, though this still seems to be quite a weak requirement.…”

Section: Introductionmentioning

confidence: 99%

“…Robinson (1979b), Lu and Berger (1989), Robert and Casella (1994) and Wang (2000) all took a complementary approach and discussed how to improve the reported confidence statements for the usual confidence set C 0 .X/. Tseng and Brown (1997) have given an excellent review of the earlier literature on the multivariate normal confidence set problem, which, in addition to those references already given, includes Stein (1962Stein ( , 1981 and Casella (1982, 1984). Tseng and Brown themselves proposed somewhat egg-shaped sets which have exact coverage probability and they also found sufficient conditions under which their sets uniformly dominate C 0 .X/ in terms of volume.…”

Section: Introductionmentioning

confidence: 99%

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Small Confidence Sets for the Mean of a Spherically Symmetric Distribution

Samworth

2005

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Suppose that "X" has a "k"-variate spherically symmetric distribution with mean vector "&thgr;" and identity covariance matrix. We present two spherical confidence sets for "&thgr;", both centred at a positive part Stein estimator . In the first, we obtain the radius by approximating the upper "&agr;"-point of the sampling distribution of by the first two non-zero terms of its Taylor series about the origin. We can analyse some of the properties of this confidence set and see that it performs well in terms of coverage probability, volume and conditional behaviour. In the second method, we find the radius by using a parametric bootstrap procedure. Here, even greater improvement in terms of volume over the usual confidence set is possible, at the expense of having a less explicit radius function. A real data example is provided, and extensions to the unknown covariance matrix and elliptically symmetric cases are discussed. Copyright 2005 Royal Statistical Society.

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Shrinkage confidence intervals for the normal mean: Using a guess for greater efficiency

Willink¹

2008

Can J Statistics

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If the unknown mean of a univariate population is sufficiently close to the value of an initial guess then an appropriate shrinkage estimator has smaller average squared error than the sample mean. This principle has been known for some time, but it does not appear to have found extension to problems of interval estimation. The author presents valid two-sided 95% and 99% "shrinkage" confidence intervals for the mean of a normal distribution. These intervals are narrower than the usual interval based on the Student distribution when the population mean lies in such an "effective interval." A reduction of 20% in the mean width of the interval is possible when the population mean is sufficiently close to the value of the guess. The author also describes a modification to existing shrinkage point estimators of the general univariate mean that enables the effective interval to be enlarged.Intervalles de confianceà rétrécisseur pour l'espérance d'une loi normale : utilisation d'une valeur initiale au profit d'une meilleure efficacité Résumé : Si l'espérance inconnue d'une population univariée est assez proche d'une valeur initiale, l'erreur quadratique moyenne d'un bon estimateurà rétrécisseur sera plus faible que celle de la moyenneéchantillonnale. Ce principe est connu depuis longtemps, mais il ne semble pas avoirété appliquéà l'estimation par intervalle. L'auteur présente des intervalles de confiance "à rétrécisseur" bilatérauxà 95% et 99% pour l'espérance d'une loi normale. Ces intervalles sont plus courts que l'intervalle habituel fondé sur la loi de Student dès que l'espérance de la population se situe dans un "intervalle effectif." On peut observer une réduction de 20% de la longueur moyenne de l'intervalle quand l'espérance de la population est assez proche de la valeur initiale. L'auteur montre aussi comment certains estimateurs ponctuelsà rétrécisseur d'une espérance univariée quelconque peuventêtre modifiés de façonàélargir l'intervalle effectif.

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Good exact confidence sets for a multivariate normal mean

Cited by 16 publications

References 16 publications

Simultaneous confidence regions for multivariate bioequivalence

Simultaneous confidence regions for multivariate bioequivalence

Small Confidence Sets for the Mean of a Spherically Symmetric Distribution

Shrinkage confidence intervals for the normal mean: Using a guess for greater efficiency

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