What does the term "Uniformly" mean in UMP(Uniformly Most Powerful) test and in UMVUE(Uniformly Minimum Variance Unbiased Estimator)?

So far I have learnt, it means for all $\theta$(parameter). If it is, then for which $\theta$?

Please explain this.

  • 2
    $\begingroup$ Uniformly in these expressions means all over the parameter space, i.e., for all $\theta$'s. The domination of the UMP test or or the UMVU estimator over the competition is uniform in that there is no value of $\theta$ for which it does not occur. $\endgroup$ – Xi'an Oct 23 '20 at 5:35
  • $\begingroup$ @Xi'an Can you please tell me for the UMP test- is it "for all $\theta$'s" for the Alternative hypothesis? If it is, then why? $\endgroup$ – n.u2877 Oct 23 '20 at 6:06
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    $\begingroup$ I do not understand the question. The test $\varphi_0$ is UMP if its power is larger than the power of any other test, for all $\theta$'s. That is the definition of being UMP. So, why the "why"? $\endgroup$ – Xi'an Oct 23 '20 at 6:08
  • $\begingroup$ I meant for which $\theta$. That is, the parameter space of the Null hypothesis or Alternative hypothesis? $\endgroup$ – n.u2877 Oct 23 '20 at 6:15
  • 1
    $\begingroup$ Power is defined as the probability of a right decision under the alternative. $\endgroup$ – Xi'an Oct 23 '20 at 6:22

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