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Consider the following Hypothesis Testing problem:

Hypothesis $H_0$ : $X \sim N(\mu_0, \sigma_0)$. Mean $\mu_0$ is known but only upper and lower bounds on $\sigma_0$ are known.

Hypothesis $H_1$ : $X \sim N(\mu_1, \sigma_1)$. Mean $\mu_1$ is known but only upper and lower bounds on $\sigma_1$ are known.

How could one solve this problem? Do uniformly most powerful tests exist for this problem?

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  • $\begingroup$ Consider the wiki page on Hypothesis Testing. "Specifically, the null hypothesis allows to attach an attribute: it should be chosen in such a way that it allows us to conclude whether the alternative hypothesis can either be accepted or stays undecided as it was before the test." Are your null and alternative hypotheses chosen in this way? If we reject $H_{0}$, must we necessarily accept $H_{1}$? $\endgroup$ – preferred_anon Apr 13 '13 at 12:18
  • $\begingroup$ Thank you Daniel for your quick response. But available material does not explain any hypothesis testing method fo a given set up of problem. I know only upper and lower bound on variance. Unimornly most powerful (UMP) tests are available when range of mean is known. But I am not able to find material for finding UMP tests when range of variance is known. $\endgroup$ – Arti Apr 14 '13 at 8:42

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