Exact confidence nets based on finite reflection groups

Confidence nets — that is, collections of confidence intervals that fill out parameter space and whose exact 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 interval probabilities. The proof makes elementary use of the theory of “buildings” and the Chevalley factorization theorem for the length distribution on Cayley graphs.