How to derive a nonparametric minimum volume tolerance region from a random sample? Given only a sample of IID variables with unknown density $f$, is there a way to derive a nonparametric tolerance region $T$ that contains at least a specified proportion $p$ of the population with at least a specified confidence $c$ and with minimum volume? Such a region would prioritize points of high density and not be one contiguous region in space:
$c\leq Pr(p\leq\int_Tf(x)dx:f(x:x\in T)\geq f(x:x\notin T))$
 A: I took a quick look at the papers of di Bucchianico et al., and of Li and Liu. (I couldn't get my hands on Chatterjee and Patra.) I see what you're saying about the data depth approach disallowing disjoint regions.
A quick idea that would build on the depth approach while allowing multiple, disconnected regions is the following:
(1) Use an unsupervised machine learning method like $K$-means clustering to partition the observed data into $K$ groups.
(2) For each group, use the depth approach to define a tolerance region.
(3) Take the union of the $K$ individual tolerance regions.
I'm finessing on details here: how to choose the number of groups $K$, and how to set $p$ and $c$ values for each group. These are definitely nontrivial. For choosing the number of groups, you could take a look at different ideas here. The dispersion within each group and the group size seem like important considerations for setting the individual $p$ values, so maybe you could use the covariance matrices of the groups to develop some criterion.
All that said, getting an asymptotically minimal volume would be a big question mark. I don't have enough expertise, nor is the method fleshed out enough.
