# Properties of Nonparametric Test Statistics $B$

## 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.

• 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.
– whuber
Feb 20, 2019 at 19:06
• @whuber Thank you for your feedback; I have edited my question to narrow the scope of what's confusing me. Feb 20, 2019 at 19:35
• "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. Feb 21, 2019 at 0:52
• @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. Feb 21, 2019 at 1:32
• $\mathbb{I}$ is an indicator function. What's the distribution of $\psi_i$ in general? What is it under the null? Feb 21, 2019 at 7:17

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:

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