A Geometric Approach to Confidence Sets for Ratios: Fieller's Theorem, Generalizations, and Bootstrap

TL;DR

This paper introduces a geometric method for constructing confidence sets for the ratio of means of two variables, generalizing Fieller's theorem and enabling bootstrap approaches for diverse distributions.

Contribution

It presents a novel geometric framework that simplifies ratio confidence set construction, extending Fieller's theorem and supporting bootstrap methods for broad distribution classes.

Findings

Confidence sets coincide with Fieller's in normal cases

Generalizations allow for exact and conservative sets in broad distributions

Bootstrap approach performs well with heavy-tailed data

Abstract

We present a geometric method to determine confidence sets for the ratio E(Y)/E(X) of the means of random variables X and Y. This method reduces the problem of constructing confidence sets for the ratio of two random variables to the problem of constructing confidence sets for the means of one-dimensional random variables. It is valid in a large variety of circumstances. In the case of normally distributed random variables, the so constructed confidence sets coincide with the standard Fieller confidence sets. Generalizations of our construction lead to definitions of exact and conservative confidence sets for very general classes of distributions, provided the joint expectation of (X,Y) exists and the linear combinations of the form aX + bY are well-behaved. Finally, our geometric method allows to derive a very simple bootstrap approach for constructing conservative confidence sets for ratios which perform favorably in certain situations, in particular in the asymmetric heavy-tailed regime.