Algorithms for Approximating Conditionally Optimal Bounds

This work develops algorithms for non-parametric confidence regions for samples from a univariate distribution whose support is a discrete mesh bounded on the left. We generalize the theory of Learned-Miller to preorders over the sample space. In this context, we show that the lexicographic low and lexicographic high orders are in some way extremal in the class of monotone preorders. From this theory we derive several approximation algorithms: 1) Closed form approximations for the lexicographic low and high orders with error tending to zero in the mesh size; 2) A polynomial-time approximation scheme for quantile orders with error tending to zero in the mesh size; 3) Monte Carlo methods for calculating quantile and lexicographic low orders applicable to any mesh size.

💡 Research Summary

The paper tackles the problem of constructing non‑parametric confidence regions for samples drawn from a univariate distribution whose support lies on a left‑bounded discrete mesh. Building on the Learned‑Miller framework, which originally handled only total orders, the authors extend the theory to arbitrary preorders (reflexive, transitive relations that may contain equivalence classes). They formalize the notation x ≲₍R₎ y, x <₍R₎ y and x ∼₍R₎ y, and distinguish between partial orders, total preorders, and total orders. By invoking the Szpilrajn extension theorem they prove that every preorder R admits at least one total order T that “agrees” with R (i.e., x <₍R₎ y ⇒ x <₍T₎ y). This bridge allows them to transfer results known for total orders to the more general preorder setting.

Two canonical preorders are introduced: the low lexicographic order Tℓ (standard lexicographic) and the high lexicographic order Th (reverse lexicographic). Both are shown to be strongly monotone, meaning that component‑wise ordering of samples implies ordering under the preorder. The authors prove that these two orders are extremal: for any monotone preorder R, the conditionally optimal bound evaluated at any sample lies between the bounds obtained from Tℓ and Th. This extremality is crucial because it reduces the analysis of an entire class of preorders to just two representative cases.

The central statistical object is the conditionally optimal bound B*₍R₎(x) = min{E