Statistica Sinica

Ulrike von Luxburg and Volker H. Franz

Abstract:We present a geometric method to determine confidence sets for the ratio of the means of random variables and . 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 exists and linear combinations of the form are well-behaved. Finally, our geometric method allows us to derive a very simple bootstrap approach for constructing conservative confidence sets for ratios that perform favorably in certain situations, in particular in the asymmetric heavy-tailed regime.

Key words and phrases:Bootstrap, confidence interval, Fieller's theorem, index variable, ratio.