## Asymptotic efficiency of simple decisions for the compound decision problem

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Citations: | 4 - 3 self |

### BibTeX

@MISC{Greenshtein_asymptoticefficiency,

author = {Eitan Greenshtein},

title = {Asymptotic efficiency of simple decisions for the compound decision problem},

year = {}

}

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### Abstract

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.

### Citations

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Statistical Decision Theory and Bayesian Analysis
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Citation Context ...) + (n − K)f0(Y1) (Y2:n) − 1 )) −1 , ( fK fK−1 (Y2:n) − 1 )) −1 where, with some abuse of notation fk(Y2:k) is the joint density of Y2:n conditional on ∑ n j=2 µj = k, and the random variable γ is in =-=[0, 1]-=-. We prove now that fK/fK−1(Y2:n) converges to 1 in the mean square. We use Lemma 2.1 (with ψ the square) to compare the testing of fK(Y2:k) vs. fK−1(Y2:k) to an easier problem, from which the origina... |

53 |
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Citation Context ...asymptotically similar to E(M1|Y1,...,Yn). Quantifying this similarity as n grows, has to do with the rate of convergence in De Finetti’s theorem. Such rates were established by Diaconis and Freedman =-=[4]-=-, but are not directly applicable to obtain our results.Simple Estimators for Compound Decisions 269 After quoting a simple result in the following section, we consider in Section 3 the special impor... |

41 |
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Citation Context ...rther restricted to use procedures from DS . Obviously Oracle I has an advantage, our results quantify this advantage and show that it is asymptotically negligible. Furthermore, starting with Robbins =-=[7]-=- various oracle-inequalities were obtained showing that one can achieve nearly the risk of Oracle II, by a ‘legitimate’ statistical procedure. See, e.g., the survey Zhang [10], for oracle-inequalities... |

12 | Nonparametric empirical Bayes and compound decision approaches to estimation of a high-dimensional vector of normal means
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(Show Context)
Citation Context ...ieve nearly the risk of Oracle II, by a ‘legitimate’ statistical procedure. See, e.g., the survey Zhang [10], for oracle-inequalities regarding the difference in risks. See also Brown and Greenshtein =-=[2]-=-, and Wenuha and Zhang [11] for oracle inequalities regarding the ratio of the risks. However, Oracle II is limited, and hence, these claims may seem to be too weak. Our equivalence results, extend ma... |

12 |
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Citation Context ...denote its risk by r PI n Obviously D S ⊂D PI , and whence r S n = E||Δ PI μ (Y1:n) − μ1:n|| 2 . ≥ rPI n . Still, ‘folklore’, theorems in the spirit of De Finetti, and results like Hannan and Robbins =-=[5]-=-, imply that asymptotically (as n →∞)ΔPI μn and ΔSμ n will have ‘similar’ mean risks: rS n −rPI n = o(n). Our main result establishes conditions that imply the stronger claim, rS n − rPI n = O(1). To ... |

10 |
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Citation Context ...S μ − Δ PI μ ‖ 2 . We now briefly review some related literature and problems. On simple symmetric functions, compound decision and its relation to empirical Bayes, see Samuel [9], Copas [3], Robbins =-=[8]-=-, Zhang [10], among many other papers. Hannan and Robbins [5] formulated essentially the same equivalence problem in testing problems, see their Section 6. They show for a special case an equivalence ... |

10 | Compound decision theory and empirical Bayes methods. Ann. Statist. 31 379–390. Dedicated to the memory of Herbert
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(Show Context)
Citation Context ...re, starting with Robbins [7] various oracle-inequalities were obtained showing that one can achieve nearly the risk of Oracle II, by a ‘legitimate’ statistical procedure. See, e.g., the survey Zhang =-=[10]-=-, for oracle-inequalities regarding the difference in risks. See also Brown and Greenshtein [2], and Wenuha and Zhang [11] for oracle inequalities regarding the ratio of the risks. However, Oracle II ... |

7 | Testing Statistical Hypothesis. 2nd edition - Lehmann - 1986 |

5 |
Compound decisions and empirical Bayes (with discussion
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(Show Context)
Citation Context ... PI n +Eμ ‖Δ S μ − Δ PI μ ‖ 2 . We now briefly review some related literature and problems. On simple symmetric functions, compound decision and its relation to empirical Bayes, see Samuel [9], Copas =-=[3]-=-, Robbins [8], Zhang [10], among many other papers. Hannan and Robbins [5] formulated essentially the same equivalence problem in testing problems, see their Section 6. They show for a special case an... |

4 |
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(Show Context)
Citation Context ... |Y1:n} = r PI n +Eμ ‖Δ S μ − Δ PI μ ‖ 2 . We now briefly review some related literature and problems. On simple symmetric functions, compound decision and its relation to empirical Bayes, see Samuel =-=[9]-=-, Copas [3], Robbins [8], Zhang [10], among many other papers. Hannan and Robbins [5] formulated essentially the same equivalence problem in testing problems, see their Section 6. They show for a spec... |

3 |
General maximum likelihood empirical Bayes estimation of normal means
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(Show Context)
Citation Context ...acle II, by a ‘legitimate’ statistical procedure. See, e.g., the survey Zhang [10], for oracle-inequalities regarding the difference in risks. See also Brown and Greenshtein [2], and Wenuha and Zhang =-=[11]-=- for oracle inequalities regarding the ratio of the risks. However, Oracle II is limited, and hence, these claims may seem to be too weak. Our equivalence results, extend many of those oracle inequali... |