[Math] Power Function for the uniform distribution

hypothesis testingstatisticsuniform distribution

Completely stuck on this homework question, I think my knowledge of the power function is nowhere near good enough coming up to finals!

Consider the following alternative testing problem: the
two hypothesis are $H_0 : θ = θ_0$ versus
$H_1 : θ > θ_0$ (note that the alternative hypothesis is composite). Since we know already from the notes that:
$m_n =$max${X_1, . . . , X_n}$ is the MLE, we use it as our test statistic. We reject $H_0$
in favour of $H_1$ if $m_n > t$ for some threshold t.

Compute the power function of the test for arbitrary threshold t.

Can anyone help?

Best Answer

  1. Derive the likelihood to see why $m_n$ is the MLE.
  2. Derive the distribution of $m_n$ (hint: what is the probability that the maximum of $X_i$s is less than $a$? How does it relate to the probability that each individual $X_i<a$?)
  3. Apply the definition of the power of the test: for a given alternative paramter $\theta_a>\theta_0$, determine ${\rm Prob}[m_n>t;\theta=\theta_a]$. Viewed as a function of $\theta_a$, this is your power function.

Oh, and step 0: Don't rely on the notes.

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