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

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##### Citation

von Luxburg, U., & Franz, V. (2009). A Geometric Approach to Confidence Sets for
Ratios: Fieller‘s Theorem, Generalizations, and Bootstrap.* Statistica Sinica,* *19*(3),
1095-1117. Retrieved from http://www3.stat.sinica.edu.tw/statistica/J19N3/j19n312/j19n312.html.

Cite as: http://hdl.handle.net/11858/00-001M-0000-0013-C3D9-7

##### 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.