Kato, Kengo Improved prediction for a multivariate normal distribution with unknown mean and variance. (English) Zbl 1332.62030 Ann. Inst. Stat. Math. 61, No. 3, 531-542 (2009). Summary: The prediction problem for a multivariate normal distribution is considered where both mean and variance are unknown. When the Kullback-Leibler loss is used, the Bayesian predictive density based on the right invariant prior, which turns out to be a density of a multivariate \(t\)-distribution, is the best invariant and minimax predictive density. In this paper, we introduce an improper shrinkage prior and show that the Bayesian predictive density against the shrinkage prior improves upon the best invariant predictive density when the dimension is greater than or equal to three. Cited in 13 Documents MSC: 62C10 Bayesian problems; characterization of Bayes procedures 62C20 Minimax procedures in statistical decision theory 62J07 Ridge regression; shrinkage estimators (Lasso) Keywords:Bayesian prediction; Kullback-Leibler divergence; multivariate normal distribution; multivariate \(t\)-distribution; right invariant prior; shrinkage prior; star ordering PDFBibTeX XMLCite \textit{K. Kato}, Ann. Inst. Stat. Math. 61, No. 3, 531--542 (2009; Zbl 1332.62030) Full Text: DOI References: [1] Aitchison J. (1975). Goodness of prediction fit. Biometrika 62:545–554 · Zbl 0339.62018 [2] Brown, L.D., George, E. I., Xu, X. (2007). Admissible predictive density estimation. Annals of Statistics, to appear. · Zbl 1216.62012 [3] Geisser S. (1993). Predictive inference: an introduction. New York, Chapman and Hall. · Zbl 0824.62001 [4] George E.I., Liang F., Xu X. (2006). Improved minimax predictive densities under Kullback–Leibler loss. Annals of Statistics 34:78–91 · Zbl 1091.62003 [5] Jeon J., Kochar S., Park C.G. (2006). Dispersive ordering-some applications and examples. Statistical Papers 47:227–247 · Zbl 1105.62014 [6] Komaki F. (1996). On asymptotic properties of predictive distributions. Biometrika 83:299–313 · Zbl 0864.62007 [7] Komaki F. (2001). A shrinkage predictive distribution for multivariate normal observables. Biometrika 88:859–864 · Zbl 0985.62024 [8] Liang F., Barron A. (2004). Exact minimax strategies for predictive density estimation, data compression, and model selection. IEEE Transactions on Information Theory 50:2708–2726 · Zbl 1315.94022 [9] Lin P.E., Tsai H.L. (1973). Generalized Bayes minimax estimations of the multivariate normal mean with unknown covariance matrix. Annals of Statistics 1:142–145 · Zbl 0254.62006 [10] Robert C.P. (2001). The Bayesian choice (2nd. ed). New York, Springer [11] Shaked M. (1982). Dispersive ordering of distribution. Journal of Applied Probability 19:310–320 · Zbl 0481.60022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.