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**On equivariance and the compound decision problem.**
*(English)*
Zbl 0797.62006

Summary: This paper obtains some extensions of D. C. Gilliland and J. Hannan’s [IMS Lect. Notes Monogr. Ser. 8, 129-145 (1986; Zbl 0689.62011)] results on equivariance and the compound decision problem. Consider a compound decision problem with restricted component risk and component distributions in a norm compact set of mutually absolutely continuous probability measures. Then the method of proof of a theorem of Gilliland and Hannan translates the results of the author [Stability of symmetrized probabilities and compact equivariant compound decisions. Ph. D. diss., Dpt. Stat. Probab., Michigan State Univ. (1990)] on symmetrization of product measures into uniform convergence to zero of the excess of the simple envelope over the equivariant envelope.

Our envelope results strengthen, among other things, the results of S. Datta [see ibid. 19, No. 1, 338-365 (1991; Zbl 0741.62004 and Zbl 0741.62005)] who obtained admissible asymptotically optimal solutions to the compound estimation problem for a large subclass of the real one parameter exponential family under squared error loss.

Sufficient conditions for asymptotic optimality of “delete bootstrap” rules are given and, for squared error loss estimation of continuous functions and for finite action space problems with continuous loss functions, the problem of treating the asymptotic excess compound risk of Bayes compound rules is reduced to the question of \(L_ 1\)-consistency of certain mixtures. Examples of estimates satisfying the above consistency condition are provided.

Our envelope results strengthen, among other things, the results of S. Datta [see ibid. 19, No. 1, 338-365 (1991; Zbl 0741.62004 and Zbl 0741.62005)] who obtained admissible asymptotically optimal solutions to the compound estimation problem for a large subclass of the real one parameter exponential family under squared error loss.

Sufficient conditions for asymptotic optimality of “delete bootstrap” rules are given and, for squared error loss estimation of continuous functions and for finite action space problems with continuous loss functions, the problem of treating the asymptotic excess compound risk of Bayes compound rules is reduced to the question of \(L_ 1\)-consistency of certain mixtures. Examples of estimates satisfying the above consistency condition are provided.

### MSC:

62C25 | Compound decision problems in statistical decision theory |

62A01 | Foundations and philosophical topics in statistics |