Celebratio Mathematica — Blackwell

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Works connected to Peter J. Bickel

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P. J. Bickel and D. Blackwell: “A note on Bayes estimates,” Ann. Math. Stat. 38 : 6 (1967), pp. 1907–1911. MR 0219175 Zbl 0155.26103 article

Throughout this paper we are concerned with the problem of estimating a real parameter when the loss function is such that the Bayes estimate exists, is unique, and satisfies a simple Equation, (1.5). If the estimate is unbiased (in the general sense of [Lehmann 1951]) we show under weak conditions that it must satisfy another Equation, (1.14). The main result of Section 1, Theorem 1.3, shows that, in general, these two equations are incompatible unless the Bayes risk is 0. This extends Theorem 11.24 of [Blackwell and Girshick 1954] which states that in estimation with quadratic loss, unbiased Bayes estimates have Bayes risk 0. Some counter-examples at the end of the section indicate the limits of this incompatibility result.

D. Blackwell: “A hypothesis-testing game without a value,” pp. 79–82 in A Festschrift for Erich L. Lehmann. Edited by P. J. Bickel, K. A. Doksum, and J. L. Hodges. Wadsworth Statistics/Probability Series. Wadsworth (Belmont, CA), 1983. MR 689739 Zbl 0525.62006 incollection

Kjell Andreas Doksum	Related
Joseph Lawson Hodges, Jr.	Related
Erich Leo Lehmann	Related
Peter J. Bickel	Related