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Does there exist any agreement on what must rejection regions look like topologically? If we identify the region of "acceptance" with the corresponding confidence interval (or confidence region in dimensions greater than 1) then, according to this question, it seems that rejection regions must be open. On the other hand, the rejection region given by the Neyman-Pearson lemma is usually defined in terms of a non-strict inequality, so at least in the continuous case it will be closed. (I know that this is a somewhat bizantine question, at least from a practical point of view.)

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  • $\begingroup$ The question you reference does not appear to assert that rejection regions must be open. In fact, it claims that opennness vs. closedness does not matter (with continuous distributions). The underlying problem with this question is that topology is not really relevant when discussing probability measures: one is not the same as the other. $\endgroup$
    – whuber
    Commented Mar 26, 2012 at 18:30
  • $\begingroup$ Maybe "topology" is too strong a word here. I was simply wondering whether it is more reasonable or not (for technical reasons or merely because of tradition) to include the boundary in the rejection region. There seems to be such a convention when dealing with confidence intervals. Of course in the continuous case all boundaries with practical interest have zero probability. Many thanks for your comment @whuber $\endgroup$ Commented Mar 27, 2012 at 10:33
  • $\begingroup$ Fair enough, Xavier: now I understand the thrust of your question better. Thank you for the clarification. And this issue actually has more than academic (or "byzantine") applications. $\endgroup$
    – whuber
    Commented Mar 27, 2012 at 13:50

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