Assume you have a sample drawn i.i.d. from some underlying distribution. Let and . Our goal is to estimate the ratio of means and provide a level- confidence set for it, i.e. a set such that . Fieller’s confidence sets are one way to do this. (The earliest reference to Fieller’s work is listed as Reference 1. The exposition here follows that in von Luxburg & Franz 2009 (Reference 2).)
Define the standard estimators for the means and covariances:
Let be the -quantile of the distribution with degrees of freedom. Compute the quantities
The confidence set is defined as follows:
The following result is often known as Fieller’s theorem:
Theorem (Fieller). If is jointly normal, then is an exact confidence region of level for , i.e. .
This is Theorem 3 of Reference 2, and there is a short proof of the result there. Of course, if is not jointly normal (which is almost always the case), then Fieller’s confidence sets are no longer exact but approximate.
Fieller’s theorem is also valid in the more general setting where we are given two independent samples and (rather than paired samples), and use unbiased estimators for the means and independent unbiased estimators for the covariances. Reference 2 notes that the degrees of freedom for the distribution needs to be chosen appropriately in this case.
I like the way Reference 2 explicitly lays out the 3 possibilities for the Fieller confidence set:
The first case corresponds to the setting where both and are close to 0: here, we can’t really conclude anything about the ratio. The second case corresponds to the setting where is close to zero while is not: here the ratio is something like or where is a big constant while is small. The final case corresponds to the setting where both quantities are not close to zero. The Wikipedia article for Fieller’s theorem describes the confidence set using a single formula. Even though all 3 cases above are contained within this formula, I found the formula a little misleading because on the surface it looks like the confidence set is always of the form for some real numbers and .
- Fieller, E. C. (1932). The distribution of the index in a normal bivariate population.
- Von Luxburg, U., and Franz, V. H. (2009). A geometric approach to confidence sets for ratios: Fieller’s theorem, generalizations and bootstrap.