Help understanding a paper on kernel density estimates

Question

I want to calculate a KDE (Kernel Density Estimate) with a confidence band for some data. I am using this paper to learn how to do this. However, I am having difficulty understanding some parts of it. Do you mind helping me make sense of some of it?

• On page 167 it explains the difference between confidence intervals and confidence bands (see screenshot below). Do you know what the highlighted $K$ symbol denotes here?

• Do you know what the $z_{1-\alpha/2}$ symbol denotes on the next page (see screenshot below)?

• Below that it talks about using empirical bootstrap to generate bootstrap sample $X_1^*$, ..., $X_n^*$. Can you explain in more detail how this is done? Or link to something that explains it?

Answer 1

1. $\mathbb{K}$ is their notation for the support. c.f. p.177-178.
2. $z_{1-\alpha/2}$ is standard notation for the $1-\alpha/2$ quantile of a standard normal, i.e. satisfying $\Phi(z_{1-\alpha/2})=1-\alpha/2$, where $\Phi$ is the standard normal CDF.
3. See wiki or these MIT notes