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Testing the null hypothesis $H_0: \theta = 0$ against the alternative $H_1: \theta \neq 0$ is equivalent to constructing a confidence interval and checking whether or not zero belongs to it.

But what about the alternative $H_1: \theta > 0$. What is the "equivalent confidence interval?".

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The corresponding confidence interval would be a one sided interval, which can be constructed by inverting the test. I.e., it is the set of all parameter values that would not have led to the rejection of the null. While the agreement of a test and a confidence interval is of importance when testing hypotheses, it is unnatural for the purpose of estimation.

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