Abstract
In this article, we consider confidence set estimation of the mean vector of a p-dimensional normal distribution, based on n iid observations from it, with the covariance matrix \(\Sigma \) assumed positive definite but completely unknown. We consider two regimes, p fixed and p growing with n. The coverage properties as well as the expected volume and the girth of the classic Hotelling confidence set and Bayesian HPD sets under normal priors are worked out in a series of theorems. Of specific interest are theorems on the existence of an explicit phase transition in the coverage probability of the Hotelling confidence set and subexponential bounds on the noncoverage probability of HPD sets when the mean vector is sparse in \(\mathcal {L}_0\) and \(\mathcal {L}_1\) norms. The results are illustrated with discussions and two sets of computations.
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References
Abramowitz, M., & Stegun, I. A. (1964). Handbook of mathematical functions. National Bureau of Standards.
Anderson, T. W. (2003). An introduction to multivariate statistical analysis. Wiley-Interscience.
Baranchik, A. J. (1970). A family of minimax estimators of the mean of a multivariate normal distribution. Annals of Mathematical Statistics, 41, 641–645.
Brown, L. D. (1978). A contribution to Kiefer’s theory of conditional confidence procedures. Annals of Statistics, 6, 59–71.
Bühlmann, P., & van de Geer, S. A. (2011). Statistics for high dimensional data. Springer.
Cai, T. (2010). Lectures on high-dimensional statistical inference. ENSAE-CREST.
Casella, G., & Hwang, J. T. (1982). Minimax confidence sets for the mean of a multivariate normal distribution. Annals of Statistics, 10, 868–881.
DasGupta, A., & Johnstone, I. M. (2014). Asymptotic risk and Bayes risk of thresholding and superefficient estimates. Contemporary developments in statistical theorySpringer.
Dembo, A., & Zeitouni, O. (2009). Large deviation inequalities and applications. Springer.
Joshi, V. M. (1967). Inadmissibility of the usual confidence sets for the mean of a multivariate normal distribution. Annals of Mathematical Statistics, 38, 1868–1875.
Maruyama, Y., & Strawderman, W. E. (2005). A new class of generalized Bayes minimax ridge regression estimators. Annals of Statistics, 33, 1753–1770.
Patel, J., & Read, C. (1996). Handbook of the normal distribution. CRC Press.
Rao, C. R. (1973). Linear statistical inference and its applications. Wiley.
Slud, E. (1977). Distribution inequalities for the binomial law. Annals of Probability, 5, 404–412.
Varadhan, S. R. S. (1984). Large deviations and applications. SIAM.
Vershynin, R. (2018). High dimensional probability. Cambridge University Press.
Wasserman, L. (2006). All of nonparametric statistics. Springer.
Wellner, J. A. (1992). Empirical processes in action: a review. IS Review, 60, 247–269.
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DasGupta, A. Asymptotics of coverages of HD confidence sets and recentering at shrinkage estimates: phase transitions, large deviations. Jpn J Stat Data Sci (2024). https://doi.org/10.1007/s42081-023-00236-9
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DOI: https://doi.org/10.1007/s42081-023-00236-9