Comparison of Experiments

Abstract

Bohnenblust, Shapey, and Sherman [2] 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.