Fieller Ratio Confidence Sets

The Fieller method construct confidence sets for the ratio of two normal means, inverting a bivariate normal mean test. Unlike the delta method, which always produces bounded intervals based on a linear approximation, Fieller's theorem solves a quadratic inequality derived directly from the exact joint distribution.

Experiment

Explore the exact geometry of Fieller's confidence sets using the Interactive module:
https://riano.app/module/fieller-ratio-confidence-sets

1. Select the "Uncertain denominator" Estimate pair with Covariance 0. Notice that the resulting quadratic coefficient A becomes negative (-0.15). The parabola opens downwards, leading to a confidence set formed by two exterior rays instead of a bounded interval.
2. Change the Estimate pair to "Both near zero". Now A = -0.16 and the parabola lies entirely below the axis, meaning all real numbers are accepted. The confidence set is "All real".
3. Select "Linear boundary" and observe how A = 0 exactly. The quadratic reduces to a linear inequality, resulting in a single bounded half-line.
4. Return to "Stable" and adjust the Covariance to 0.03. Observe how a positive covariance between the numerator and denominator estimates shifts the accepted interval.