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Minimum volume confidence regions for a multivariate normal mean vector

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  • Bradley Efron

Abstract

Summary. Since Stein's original proposal in 1962, a series of papers have constructed confidence regions of smaller volume than the standard spheres for the mean vector of a multivariate normal distribution. A general approach to this problem is developed here and used to calculate a lower bound on the attainable volume. Bayes and fiducial methods are involved in the calculation. Scheffé‐type problems are used to show that low volume by itself does not guarantee favourable inferential properties.

Suggested Citation

  • Bradley Efron, 2006. "Minimum volume confidence regions for a multivariate normal mean vector," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(4), pages 655-670, September.
    • Handle: RePEc:bla:jorssb:v:68:y:2006:i:4:p:655-670
    DOI: 10.1111/j.1467-9868.2006.00560.x
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    Cited by:

    1. J. T. Gene Hwang & Jing Qiu & Zhigen Zhao, 2009. "Empirical Bayes confidence intervals shrinking both means and variances," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(1), pages 265-285, January.
    2. Taras Bodnar & Stepan Mazur & Nestor Parolya, 2019. "Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix‐variate location mixture of normal distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 46(2), pages 636-660, June.
    3. Bedbur, S. & Lennartz, J.M. & Kamps, U., 2020. "On minimum volume properties of some confidence regions for multiple multivariate normal means," Statistics & Probability Letters, Elsevier, vol. 158(C).
    4. Jin Zhang, 2017. "Minimum volume confidence sets for parameters of normal distributions," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 101(3), pages 309-320, July.
    5. Ahmed, S. Ejaz & Volodin, Andrei I. & Volodin, Igor N., 2009. "High order approximation for the coverage probability by a confident set centered at the positive-part James-Stein estimator," Statistics & Probability Letters, Elsevier, vol. 79(17), pages 1823-1828, September.
    6. Bar, Haim & Wells, Martin T., 2023. "On graphical models and convex geometry," Computational Statistics & Data Analysis, Elsevier, vol. 187(C).

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