zbMATH — the first resource for mathematics

Admissible predictive density estimation. (English) Zbl 1216.62012
Summary: Let \(X|\mu \sim N_p(\mu , v_xI)\) and \(Y|\mu \sim N_p(\mu , v_yI)\) be independent \(p\)-dimensional multivariate normal vectors with common unknown mean \(\mu \). Based on observing \(X=x\), we consider the problem of estimating the true predictive density \(p(y|\mu )\) of \(Y\) under expected Kullback-Leibler loss. Our focus here is the characterization of admissible procedures for this problem. We show that the class of all generalized Bayes rules is a complete class, and that the easily interpretable conditions of L.D. Brown and J.T. Hwang [Statistical decision theory and related topics III, 205–230 (1982; Zbl 0585.62016)] are sufficient for a formal Bayes rule to be admissible.

62C15 Admissibility in statistical decision theory
62C07 Complete class results in statistical decision theory
62C10 Bayesian problems; characterization of Bayes procedures
Full Text: DOI arXiv
[1] Aitchison, J. (1975). Goodness of prediction fit. Biometrika 62 547-554. JSTOR: · Zbl 0339.62018
[2] Berger, J. O. (1985). Statistical Decision Theory and Bayesian Analysis , 2nd ed. Springer, New York. · Zbl 0572.62008
[3] Brown, L. D. (1971). Admissible estimators, recurrent diffusions, and insoluble boundary value problems. Ann. Math. Statist. 42 855-903. · Zbl 0246.62016
[4] Brown, L. D. (1986). Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory . IMS, Hayward, CA. · Zbl 0685.62002
[5] Brown, L. D. and Hwang, J. (1982). A unified admissibility proof. In Statistical Decision Theory and Related Topics III (S. S. Gupta and J. O. Berger, eds.) 1 205-230. Academic Press, New York. · Zbl 0585.62016
[6] Eaton, M. L. (1982). A method for evaluating improper prior distributions. In Statistical Decision Theory and Related Topics III (S. S. Gupta and J. O. Berger, eds.) 1 329-352. Academic Press, New York. · Zbl 0581.62005
[7] Eaton, M. L. (1992). A statistical diptych: Admissible inferences-recurrence of symmetric Markov chains. Ann. Statist. 20 1147-1179. · Zbl 0767.62002
[8] Eaton, M. L., Hobert, J. P., Jones, G. L. and Lai, W.-L. (2007). Evaluation of formal posterior distributions via Markov chain arguments. Preprint. Available at http://www.stat.ufl.edu/ jhobert/. · Zbl 1274.62078
[9] Gatsonis, C. A. (1984). Deriving posterior distributions for a location parameter: A decision theoretic approach. Ann. Statist. 12 958-970. · Zbl 0544.62008
[10] George, E. I., Liang, F. and Xu, X. (2006). Improved minimax prediction under Kullback-Leibler loss. Ann. Statist. 34 78-91. · Zbl 1091.62003
[11] Komaki, F. (2001). A shrinkage predictive distribution for multivariate normal observations. Biometrika 88 859-864. JSTOR: · Zbl 0985.62024
[12] Liang, F. (2002). Exact minimax procedures for predictive density estimation and data compression. Ph.D. dissertation, Dept. Statistics, Yale Univ.
[13] Liang, F. and Barron, A. (2004). Exact minimax strategies for predictive density estimation, data compression and model selection. IEEE Trans. Inform. Theory 50 2708-2726. · Zbl 1315.94022
[14] Murray, G. D. (1977). A note on the estimation of probability density functions. Biometrika 64 150-152. JSTOR: · Zbl 0347.62035
[15] Ng, V. M. (1980). On the estimation of parametric density functions. Biometrika 67 505-506. JSTOR: · Zbl 0451.62006
[16] Stein, C. (1974). Estimation of the mean of a multivariate normal distribution. In Proceedings of the Prague Symposium on Asymptotic Statistics (J. Hajek, ed.) 345-381. Univ. Karlova, Prague. · Zbl 0357.62020
[17] Stein, C. (1981). Estimation of a multivariate normal mean. Ann. Statist. 9 1135-1151. · Zbl 0476.62035
[18] Strawderman, W. E. (1971). Proper Bayes minimax estimators of the multivariate normal mean. Ann. Math. Statist. 42 385-388. · Zbl 0222.62006
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.