Ensemble minimaxity of James‐Stein estimators

This article considers the estimation of a multivariate normal mean based on heteroscedastic observations. Under heteroscedasticity, estimators that shrink more on coordinates with larger variances seem desirable. Although they are not necessarily minimax in the ordinary sense, we show that certain James‐Stein type estimators can be ensemble minimax, that is, minimax with respect to the ensemble risk considered in the empirical Bayes perspective of Efron and Morris.

[1]  Yuzo Maruyama,et al.  A new class of generalized Bayes minimax ridge regression estimators , 2004, math/0508282.

[2]  J. Berger,et al.  Choice of hierarchical priors: admissibility in estimation of normal means , 1996 .

[3]  George Casella,et al.  Condition Numbers and Minimax Ridge Regression Estimators , 1985 .

[4]  C. Stein.  Estimation of the Mean of a Multivariate Normal Distribution , 1981 .

[5]  George Casella,et al.  Minimax Ridge Regression Estimation , 1980 .

[6]  J. Berger.  A Robust Generalized Bayes Estimator and Confidence Region for a Multivariate Normal Mean , 1980 .

[7]  J. Berger,et al.  Generalized Bayes Estimators in Multivariate Problems , 1978 .

[8]  Lawrence D. Brown,et al.  Estimation with Incompletely Specified Loss Functions (the Case of Several Location Parameters) , 1975 .

[9]  B. Efron,et al.  Stein's Estimation Rule and Its Competitors- An Empirical Bayes Approach , 1973 .

[10]  B. Efron,et al.  Empirical Bayes on vector observations: An extension of Stein's method , 1972 .

[11]  B. Efron,et al.  Limiting the Risk of Bayes and Empirical Bayes Estimators—Part II: The Empirical Bayes Case , 1972 .

[12]  B. Efron,et al.  Limiting the Risk of Bayes and Empirical Bayes Estimators—Part I: The Bayes Case , 1971 .

[13]  L. Brown.  Admissible Estimators, Recurrent Diffusions, and Insoluble Boundary Value Problems , 1971 .

[14]  W. Strawderman.  Proper Bayes Minimax Estimators of the Multivariate Normal Mean , 1971 .

[15]  L. Brown,et al.  Ensemble Minimax Estimation for Multivariate Normal Means , 2011 .

[16]  J. Neyman,et al.  INADMISSIBILITY OF THE USUAL ESTIMATOR FOR THE MEAN OF A MULTIVARIATE NORMAL DISTRIBUTION , 2005 .

[17]  C. Stein,et al.  Estimation with Quadratic Loss , 1992 .

[18]  R. Berger.  Gamma minimax robustness of bayes rules , 1979 .

[19]  J. Berger.  Admissible Minimax Estimation of a Multivariate Normal Mean with Arbitrary Quadratic Loss , 1976 .