A Property of some Type a Regions

Abstract

In a test of an hypothesis one may regard a sample in the critical region as evidence that the hypothesis is false. Let us assume that for some reason it is desired to increase the critical size of the test, i.e., to make rejection of the hypothesis more probable. Then one may expect that an observation which led to rejection in the first test should still lead to rejection in the new test. In other words, one should expect $W_\alpha \supset W_{\alpha'}$ if $\alpha > \alpha'$, where $W_\alpha$ is the critical region for the test of size $\alpha$. An example is given where regions of type $A$ are uniquely specified except for sets of measure zero, but fail to have this property.