# Bootstrap hypothesis testing p-value confusion

I missed one of the lectures in my stats class and there was one slide that I could not understand.

In this example where $$p^* = \sum_{j=1}^n l{\{...}\}/B$$, I'm more familiar with the l indicating the function as a likelihood function for MLE. However, what does the l mean in this case for bootstrapping?

• If you look carefully, you'll see that's an "I", not a lower-case "L". – jbowman Aug 17 '19 at 14:03

I don't know why it was notated like this but what is meant is that you count the elements in the set that fulfills the condition in {}. That is, for

$$p* = \frac{\Sigma_{j=1}^n I\{\bar{X}^*_j ≥ \bar{X}\}}{B}$$

You count, how many $$\bar{X}^*_j$$ are greater or equal to $$\bar{X}$$. The typical notation in the statistical literature on bootstraps is:

$$p* = \frac{\#\{\bar{X}^*_j ≥ \bar{X}\}}{B}$$ where $$\#$$ denotes the counting operation.

From this knowledge, I infer that what is meant on the slide is that by doing $$I\{\bar{X}^*_j ≥ \bar{X}\}$$ you create a binary variable, like this:

$$I\{\bar{X}^*_j ≥ \bar{X}\} = \begin{cases} 1, & \text{if } \bar{X}^*_j ≥ \bar{X}\\\ 0, & \text{otherwise} \end{cases}$$

Summing over this variable is a technical definition of counting.

Does this make sense for you?