Celebratio Mathematica — Blackwell

Statistics

Works connected to Jerzy Neyman

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D. Blackwell: “Comparison of experiments,” pp. 93–102 in Proceedings of the second Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 31 July–12 August 1950). Edited by J. Neyman. University of California Press (Berkeley and Los Angeles), 1951. MR 0046002 Zbl 0044.14203 inproceedings

Bohnenblust, Shapley, and Sherman [unpublished] have introduced a method of comparing two sampling procedures or experiments; essentially their concept is that one experiment \( \alpha \) is more informative than a second experiment \( \beta \), \( \alpha\supset\beta \), if, for every possible risk function, any risk attainable with \( \beta \) is also attainable with \( \alpha \). If \( \alpha \) is a sufficient statistic for a procedure equivalent to \( \beta \), \( \alpha\succ\beta \), it is shown that \( \alpha\supset\beta \). In the case of dichotomies, the converse is proved. Whether \( \succ \) and \( \supset \) are equivalent in general is not known. Various properties of \( \succ \) and \( \supset \) are obtained, such as the following: if \( \alpha\succ\beta \) and \( \gamma \) is independent of both, then the combination \( (\alpha,\gamma)\succ(\beta,\gamma) \). An application to a problem in \( 2{\times}2 \) tables is discussed.