# zbMATH — the first resource for mathematics

Nonparametric empirical Bayes and compound decision approaches to estimation of a high-dimensional vector of normal means. (English) Zbl 1166.62005
Summary: We consider the classical problem of estimating a vector $$\mu=(\mu_1,\dots,\mu_n)$$ based on independent observations $$Y_i\sim N(\mu_i, 1)$$, $$i=1,\dots,n$$. Suppose $$\mu_i$$, $$i=1,\dots,n$$, are independent realizations from a completely unknown $$G$$. We suggest an easily computed estimator $$\widehat\mu$$, such that the ratio of its risk $$E(\widehat\mu-\mu)^2$$ with that of the Bayes procedure approaches 1. A related compound decision result is also obtained.
Our asymptotics is of a triangular array; that is, we allow the distribution $$G$$ to depend on $$n$$. Thus, our theoretical asymptotic results are also meaningful in situations where the vector $$\mu$$ is sparse and the proportion of zero coordinates approaches 1. We demonstrate the performance of our estimator in simulations, emphasizing sparse setups. In “moderately-sparse” situations, our procedure performs very well compared to known procedures tailored for sparse setups. It also adapts well to nonsparse situations.

##### MSC:
 62C12 Empirical decision procedures; empirical Bayes procedures 62C25 Compound decision problems in statistical decision theory 62H12 Estimation in multivariate analysis
##### Keywords:
empirical Bayes; compound decision; simulations
EBayesThresh
Full Text:
##### References:
 [1] Brown, L. D. (1971). Admissible estimators, recurrent diffusions, and insoluble boundary value problems. Ann. Math. Statist. 42 855-904. · Zbl 0246.62016 [2] Brown, L. D. (2008). In-season prediction of bating averages: A field test of simple empirical Bayes and Bayes methodologies. Ann. Appl. Statist. 2 113-152. · Zbl 1137.62419 [3] Copas, J. B. (1969). Compound decisions and empirical Bayes (with discussion). J. Roy. Statist. Soc. Ser. B 31 397-425. JSTOR: · Zbl 0186.51905 [4] Donoho, D. L. and Johnstone, I. M. (1994). Ideal spatial adaptation by wavelet shrinkage. Biometrika 81 425-455. JSTOR: · Zbl 0815.62019 [5] Efron, B. (2003). Robbins, empirical Bayes, and microarrays. Ann. Statist. 31 366-378. · Zbl 1038.62099 [6] Greenshtein, E. and Park, J. (2007). Application of nonparametric empirical Bayes to high-dimensional classification. · Zbl 1235.62010 [7] Greenshtein, E. and Ritov, Y. (2008). Asymptotic efficiency of simple decisions for the compound decision problem. In The 3rd Lehmann Symposium (J. Rojo, ed.). IMS Lecture Notes Monograph . · Zbl 1271.62022 [8] Johnstone, I. M. and Silverman, B. W. (2004). Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences. Ann. Statist. 32 1594-1649. · Zbl 1047.62008 [9] Robbins, H. (1951). Asymptotically subminimax solutions of compound decision problems. In Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950 131-148. Univ. California, Berkeley. · Zbl 0044.14803 [10] Robbins, H. (1956). An empirical Bayes approach to statistics. In Proc. Third Berkeley Symp. 157-164. Univ. California Press, Berkeley. · Zbl 0074.35302 [11] Robbins, H. (1964). The empirical Bayes approach to statistical decision problems. Ann. Math. Statist. 35 1-20. · Zbl 0138.12304 [12] Samuel, E. (1965). On simple rules for the compound decision problem. J. Roy. Statist. Soc. Ser. B 27 238-244. JSTOR: · Zbl 0295.62008 [13] van der Vaart, A. W. and Wellner, J. (1996). Weak Convergence and Empirical Processes . Springer, New York. · Zbl 0862.60002 [14] Wenhua, J. and Zhang, C.-H. (2007). General maximum likelihood empirical Bayes estimation of normal means. Manuscript. · Zbl 1168.62005 [15] Zhang, C.-H. (1997). Empirical Bayes and compound estimation of a normal mean. Statist. Sinica 7 181-193. · Zbl 0904.62008 [16] Zhang, C.-H. (2003). Compound decision theory and empirical Bayes methods (invited paper). Ann. Statist. 31 379-390. · Zbl 1039.62005 [17] Zhang C.-H. (2005). General empirical Bayes wavelet methods and exactly adaptive minimax estimation. Ann. Statist. 33 54-100. · Zbl 1064.62009
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.