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In any hypothesis test, we have a false positive rate (a.k.a. $\alpha$; a.k.a. type 1 error rate) and false negative rate (a.k.a. $\beta$; a.k.a. type 2 error rate). It's clear that we can set the $\alpha$ to whatever we like. But the smaller the $\alpha$, the larger the $\beta$ so there is a tradeoff. If we consider some effect size (difference between modes of yellow and purple curve in the figure below), the distribution of the statistic under the null hypothesis shown in the yellow curve and the distribution of the alternate hypothesis shown in the purple curve, this tradeoff looks something like this:

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Now my question: All such $\alpha$-$\beta$ curve's I've seen are decreasing (of course) but also convex as long as the distributions of the null and alternate hypothesis are the same but simply shifted. This implies that we can never have one test that has a lower $\beta$ for some values of $\alpha$, but a higher one for other values. I'm looking for a proof of this claim or the outline of one.

Rohit Pandey
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    It depends on how you set up your rejection region, but for UMP tests this is true by definition. By the way, the curve looks convex to me (it lies above support lines). – user10354138 Jun 03 '19 at 04:03
  • Yes, I should have said convex. Just fixed the question. Does it say anywhere in the link that the $\alpha$-$\beta$ curve for UMP tests must be convex? Also, can you think of any counter-example. A rejection region set up in such a way that the $\alpha$-$\beta$ curve is not convex? – Rohit Pandey Jun 03 '19 at 04:10
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    $\alpha$-$\beta$ curve must be convex, because between $\alpha_1$ and $\alpha_2$ you can linearly interpolate as a rejection strategy (use a Bernoulli distributed random variable to decide on rejection), so the curve cannot go above its chords. Of course, if you set up your rejection region the other way (a silly case: purple on the right, yellow on the left) you get a concave curve instead. – user10354138 Jun 03 '19 at 04:39
  • That's a nice argument. Not rigorous proof, but still very nice. Could be an answer. – Rohit Pandey Jun 03 '19 at 05:01

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