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




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.

Author(s): von Luxburg, U. and Franz, VH.
Journal: Statistica Sinica
Volume: 19
Number (issue): 3
Pages: 1095-1117
Year: 2009
Month: July
Day: 0

Department(s): Empirical Inference
Bibtex Type: Article (article)

Language: en
Organization: Max-Planck-Gesellschaft
School: Biologische Kybernetik

Links: PDF


  title = {A Geometric Approach to Confidence Sets for Ratios: Fieller's Theorem, Generalizations, and Bootstrap},
  author = {von Luxburg, U. and Franz, VH.},
  journal = {Statistica Sinica},
  volume = {19},
  number = {3},
  pages = {1095-1117},
  organization = {Max-Planck-Gesellschaft},
  school = {Biologische Kybernetik},
  month = jul,
  year = {2009},
  month_numeric = {7}