I was reading Hogg/Craig and my course notes and encountered a subsection on simulation to find out Confidence Intervals. The method expressed is to generate a random sample, calculate the MLE, repeat a billion times, find upper and lower $\alpha/2$ quantiles for the empirical distribution of the MLE. (Call them $l$ and $u$.)
Then, it said that a crude CI for the MLE would be $\hat{\theta}\pm z_{\alpha/2} SE(\hat{\theta})$. I understood everything till her. Now, they said, that a better estimate would be :
$$\hat{\theta} \in (2\hat{\theta}-u,2\hat{\theta}-l)$$ and added that this is called "reflecting quantiles". I don't appreciate
- Why this is a CI in the first place?
- How is this better?
- What is the rationale behind this?
