Properties of Nonparametric Test Statistics $B$

Question

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.

Answer 1

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.

2. 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:

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