## Statistical Science

### Comment: Empirical Bayes, Compound Decisions and Exchangeability

#### Abstract

We present some personal reflections on empirical Bayes/ compound decision (EB/CD) theory following Efron (2019). In particular, we consider the role of exchangeability in the EB/CD theory and how it can be achieved when there are covariates. We also discuss the interpretation of EB/CD confidence interval, the theoretical efficiency of the CD procedure, and the impact of sparsity assumptions.

#### Article information

Source
Statist. Sci., Volume 34, Number 2 (2019), 224-228.

Dates
First available in Project Euclid: 19 July 2019

https://projecteuclid.org/euclid.ss/1563501637

Digital Object Identifier
doi:10.1214/19-STS709

Mathematical Reviews number (MathSciNet)
MR3983324

Keywords
$f$-modeling $g$-modeling sparsity

#### Citation

Greenshtein, Eitan; Ritov, Ya’acov. Comment: Empirical Bayes, Compound Decisions and Exchangeability. Statist. Sci. 34 (2019), no. 2, 224--228. doi:10.1214/19-STS709. https://projecteuclid.org/euclid.ss/1563501637

#### References

• Brown, L. D. and Greenshtein, E. (2009). Nonparametric empirical Bayes and compound decision approaches to estimation of a high-dimensional vector of normal means. Ann. Statist. 37 1684–1704.
• Brown, L. D., Greenshtein, E. and Ritov, Y. (2013). The Poisson compound decision problem revisited. J. Amer. Statist. Assoc. 108 741–749.
• Cohen, N., Greenshtein, E. and Ritov, Y. (2013). Empirical Bayes in the presence of explanatory variables. Statist. Sinica 23 333–357.
• Efron, B. and Morris, C. (1973). Stein’s estimation rule and its competitors—An empirical Bayes approach. J. Amer. Statist. Assoc. 68 117–130.
• Fay, R. E. III and Herriot, R. A. (1979). Estimates of income for small places: An application of James–Stein procedures to census data. J. Amer. Statist. Assoc. 74 269–277.
• Greenshtein, E., Mansura, A. and Ritov, Y. (2018). Nonparametric empirical Bayes improvement of common shrikage estimators. Submitted.
• Greenshtein, E., Park, J. and Ritov, Y. (2008). Estimating the mean of high valued observations in high dimensions. J. Stat. Theory Pract. 2 407–418.
• Greenshtein, E. and Ritov, Y. (2009). Asymptotic efficiency of simple decisions for the compound decision problem. In Optimality. Institute of Mathematical Statistics Lecture Notes—Monograph Series 57 266–275. IMS, Beachwood, OH.
• Hannan, J. F. and Robbins, H. (1955). Asymptotic solutions of the compound decision problem for two completely specified distributions. Ann. Math. Stat. 26 37–51.
• Jiang, W. and Zhang, C.-H. (2009). General maximum likelihood empirical Bayes estimation of normal means. Ann. Statist. 37 1647–1684.
• Koenker, R. and Mizera, I. (2014). Convex optimization, shape constraints, compound decisions, and empirical Bayes rules. J. Amer. Statist. Assoc. 109 674–685.
• Spearman, C. (1904). The proof and measurement of association between two things. The American Journal of Psychology 15 72–101.
• Weinstein, A., Ma, Z., Brown, L. D. and Zhang, C.-H. (2018). Group-linear empirical Bayes estimates for a heteroscedastic normal mean. J. Amer. Statist. Assoc. 113 698–710.