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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.

Questions

  1. Show that $B$ has binomial distribution and specify its parameters.
    • The equation given for $B$ makes me think it's the test statistic for the SIGN.test in ${\tt R}$, but I don't understand the notation: $\psi_i = \mathbb{I}(Z_i > \theta)$, where $\mathbb{P}(\psi_i=1)=\mathbb{P}(Z_i > \theta)=p$
    • My guess is that $\mathbb{P}(\psi_i=1)=\mathbb{P}(Z_i > \theta)=p$ is saying that the probability of $\psi_i$ being $1$ is $p$, and then I need to relate $\mathbb{P}(Z_i > \theta)$ to $\mathbb{I}(Z_i > \theta)$ in order to show that $B$ has a binomial distribution?
  2. Write down the null distribution of $B$.
    • I'm not familiar with the term null distribution, but I suspect null indicates something similar to the term null model, which is the model based solely on $\beta_0$ --- Perhaps the null distribution is the distribution based only on one of the parameters of $B$?
    • Upon further reading, it looks like the null distribution is the distribution under the assumption that $H_0$ is true. Since the test statistic $B$ is (supposed to be) binomial, I would guess that I just need to write $B$ as $B\overset{iid}{\sim} \text{Bernoulli}(n, p)$, where $n$ is the sample size and $p$ is as given?

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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  • $\begingroup$ That there are five questions here makes it too broad for our format. Especially with self-study questions we recommend you focus your post by explaining what steps you have taken towards solving a single clearly articulated question and where you are stuck. $\endgroup$ – whuber Feb 20 '19 at 19:06
  • $\begingroup$ @whuber Thank you for your feedback; I have edited my question to narrow the scope of what's confusing me. $\endgroup$ – inkalchemist1994 Feb 20 '19 at 19:35
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    $\begingroup$ "we recommend you focus your post by explaining what steps you have taken towards solving a single clearly articulated question". While several parts look like a better fit for self-study type questions, this still appears to be a laundry-list of questions rather than focusing on one issue. $\endgroup$ – Glen_b -Reinstate Monica Feb 21 '19 at 0:52
  • $\begingroup$ @Glen_b I've narrowed down my question and provided some more of my thoughts. My key question is how to show that $B$ has a binomial distribution, which requires making sense of the formulas that are given. $\endgroup$ – inkalchemist1994 Feb 21 '19 at 1:32
  • $\begingroup$ $\mathbb{I}$ is an indicator function. What's the distribution of $\psi_i$ in general? What is it under the null? $\endgroup$ – Glen_b -Reinstate Monica Feb 21 '19 at 7:17
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  1. Show that $B$ has binomial distribution and specify its parameters.

$\mathbb{I}$ is an indicator function (sometimes referred to as an indicator variable). The notation $\psi_i = \mathbb{I}(Z_i > \theta)$ is saying that $\psi_i$ will be $1$ when $Z_i > 0$, and $0$ in all other cases. Thus, we can represent $B$ as

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

Clearly, $B$ has a binomial distribution. Binomial distributions have two parameters: the number of trials/observations, $n$, and the probability of success for each trial, $p$. Both $n$ and $p$ are given in the problem description.

  1. Write down the null distribution of $B$.

The null distribution is just the distribution of the test statistic under the assumption that $H_0$ is true:

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

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