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: $c\leq Pr(p\leq\int_Tf(x)dx:f(x:x\in T)\geq f(x:x\notin T))$

• Two questions: (1) How strict is the requirement that $T$ have minimal volume? That is, could you live with a region $T$ that satisfied the proportion and confidence requirements but didn't necessarily have the smallest volume possible? (2) Have you looked into kernel density estimation so far as a general strategy for getting at $f$? Feb 17, 2018 at 21:23
• (1) Ideally asymptotically minimal volume with increasing sample size. (2) I have come across Chatterjee and Patra's approach (1980), but as di Bucchianico et al. (2001) noted, the approach is overly conservative (ironically, Li and Liu (2008) would find that di Bucchianico's approach OVERestimates the tolerance coverage denoted by 'p' in my post). Li and Liu's data depth approach is close to what I want, but it inherently disallows disjoint regions. Feb 18, 2018 at 22:25
• Right now I'm down this rabbit hole of univariate sample spacings (the differences between adjacent order statistics). I figure that at least for the univariate case, the sample spacings be ordered, and a tolerance region could be constructed from a union of the smaller sample spacings. No idea what the distribution of the ordered sample spacings would be, though, nor even if it could be independent of the sample variables' distribution. Feb 18, 2018 at 22:38

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.

• I finally tracked down a paper (Frey, 2009, Data-driven nonparametric tolerance sets) whose abstract claims to have developed a method to do what I want, if only for univariate data. Problem is the full text isn't available online, but I've put a request for it through ResearchGate. Hopefully they respond to my normie ass. Feb 20, 2018 at 1:50
• So I got the paper, and it claims in part to have derived a way to compute a tolerance region that is a union of a specified number of shortest inter-order statistic intervals. When I got to the specifics though, what they have done is derived the mean and variance of the tolerance. In my own simulation from a univariate Gaussian mixture, it turns out their formulas are correct only if you sort by interval coverage, not length; in their proofs they noted that the two sorts were identical in the uniform case and claimed it extended to any continuous distribution. Mar 1, 2018 at 2:16