"Building" exact confidence nets

TL;DR

This paper introduces a method for constructing exact confidence nets in nonparametric statistics using invariance under finite reflection groups, providing a formula for coverage probabilities based on root systems.

Contribution

It develops a novel approach to exact confidence intervals leveraging group invariance and the theory of buildings, with explicit formulas for coverage probabilities.

Findings

Derived a formula for coverage interval probabilities

Connected confidence nets to root systems of reflection groups

Utilized Chevalley factorization and building theory in proofs

Abstract

Confidence nets, that is, collections of confidence intervals that fill out the parameter space and whose exact parameter coverage can be computed, are familiar in nonparametric statistics. Here, the distributional assumptions are based on invariance under the action of a finite reflection group. Exact confidence nets are exhibited for a single parameter, based on the root system of the group. The main result is a formula for the generating function of the coverage interval probabilities. The proof makes use of the theory of "buildings" and the Chevalley factorization theorem for the length distribution on Cayley graphs of finite reflection groups.