## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Optimality
- 2009, 266-275

### Asymptotic Efficiency of Simple Decisions for the Compound Decision Problem

Eitan Greenshtein and Ya’acov Ritov

#### Abstract

We consider the compound decision problem of estimating a vector of *n* parameters, known up to a permutation, corresponding to *n* independent observations, and discuss the difference between two symmetric classes of estimators. The first and larger class is restricted to the set of all permutation invariant estimators. The second class is restricted further to simple symmetric procedures. That is, estimators such that each parameter is estimated by a function of the corresponding observation alone. We show that under mild conditions, the minimal total squared error risks over these two classes are asymptotically equivalent up to essentially *O*(1) difference.

#### Chapter information

**Source***Optimality: The Third Erich L. Lehmann Symposium* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2009)

**Dates**

First available in Project Euclid: 3 August 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1249305334

**Digital Object Identifier**

doi:10.1214/09-LNMS5716

**Mathematical Reviews number (MathSciNet)**

MR2681676

**Zentralblatt MATH identifier**

1271.62022

**Subjects**

Primary: 62C25: Compound decision problems

Secondary: 62C12: Empirical decision procedures; empirical Bayes procedures 62C07: Complete class results

**Keywords**

compound decision simple decision rules permutation invariant rules

**Rights**

Copyright © 2009, Institute of Mathematical Statistics

#### Citation

Rojo, Javier. Asymptotic Efficiency of Simple Decisions for the Compound Decision Problem. Optimality, 266--275, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2009. doi:10.1214/09-LNMS5716. https://projecteuclid.org/euclid.lnms/1249305334

#### References

- [1] Berger, J. O. (1985).
*Statistical Decision Theory and Bayesian Analysis*, 2nd ed. Springer, New York. - [2] Brown, L. D. and Greenshtein, E. (2009). Non parametric empirical Bayes and compound decision approaches to estimation of a high dimensional vector of normal means.
*Ann. Statist.***37**1685–1704. - [3] Copas, J. B. (1969). Compound decisions and empirical Bayes (with discussion).
*J. Roy. Statist. Soc. Ser. B***31**397–425. - [4] Diaconis, P. and Freedman, D. (1980). Finite exchangeable sequences.
*Ann. Probab.***8**745–764. - [5] Hannan, J. F. and Robbins, H. (1955). Asymptotic solutions of the compound decision problem for two completely specified distributions.
*Ann. Math. Statist.***26**37–51. - [6] Lehmann, E. L. (1986).
*Testing Statistical Hypothesis*, 2nd ed. Wiley, New York. - [7] Robbins, H. (1951). Asymptotically subminimax solutions of compound decision problems. In
*Proc. Third Berkeley Symp.*157–164. - [8] Robbins, H. (1983). Some thoughts on empirical Bayes estimation.
*Ann. Statist.***11**713–723. - [9] Samuel, E. (1965). On simple rules for the compound decision problem.
*J. Roy. Statist. Soc. Ser. B***27**238–244. - [10] Zhang, C. H. (2003). Compound decision theory and empirical Bayes methods. (Invited paper.)
*Ann. Statist.***31**379–390. - [11] Wenuha, J. and Zhang, C. H. (2009). General maximum likelihood empirical Bayes estimation of normal means.
*Ann. Statist.*To appear.