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Given

Consider the upper tail $H_0: \theta \leq 0 \,\,\, \text{vs} \,\,\, H_A: \theta > 0$ sign test with the test statistic $B = \sum_{i=1}^n{\psi_i}$ where $\psi_i = \mathbb{I}(Z_i > \theta)$, where $\mathbb{P}(\psi_i=1)=\mathbb{P}(Z_i > \theta)=p$. With the assumption that the $n$ observations are iid, answer the following questions.

$$B = \sum_{i=1}^{n} \psi_i \hspace{1em} \text{where} \hspace{1em} \psi_i = \left\{\begin{array}{ll} 1 & \texttt{if} \,\, Z_i > \theta \\ 0 & \texttt{otherwise}\end{array}\right.$$

$$B \overset{iid}{\sim} {\tt Binom}(n, p)$$

Question

Write down the expression of the power of the upper tail sign test when the alternative has $p_0$.

  • So, I know that the power of a hypothesis test is the probability of correctly rejecting $H_0$ when $H_A$ is true. i.e.

$$ {\tt power} = \mathbb{P}(\text{reject } H_0 | H_A \text{ is true})$$

  • However, since the question specifically mentions $p_0$, I think I need to express power in terms of $p$, $p_0 = 1 - p$, $n$, $\psi_i$, $\theta$, and/or $Z_i$ -- I'm not sure how to begin doing that, as I can't find a concrete formula for the power of a binomial.

Note

  • The $\mathbb{I}$ should be a $1$ with two lines, but SO wouldn't let me use \mathbbm{1}.

Thanks in advance for any help you can provide! I don't just want answers, I want to understand how to find the answers.

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