Fieller Ratio Confidence Sets

Worked Example

Let the estimate pair be (x̄, ȳ) = (2, 1) (the "Stable" case), with known variances v_XX = 0.04, v_YY = 0.09, zero covariance (v_XY = 0), and critical z = 2.

We test the null hypothesis H₀: θ = μ_Y / μ_X, which is equivalent to H₀: μ_Y - θμ_X = 0.

1. The test statistic numerator is w = ȳ - θx̄ = 1 - 2θ.
2. The variance of w is V(w) = v_YY - 2θv_XY + θ²v_XX = 0.09 + 0.04θ².
3. The set of accepted θ satisfies w² ≤ z²V(w):
   (1 - 2θ)² ≤ 4(0.09 + 0.04θ²).
4. Expanding this gives: 1 - 4θ + 4θ² ≤ 0.36 + 0.16θ².
5. Rearranging into the standard quadratic form Aθ² + Bθ + C ≤ 0, we get:
   (4 - 0.16)θ² - 4θ + (1 - 0.36) ≤ 0
   3.84θ² - 4.0θ + 0.64 ≤ 0.
6. Here, A = 3.84, B = -4.0, C = 0.64. Since A > 0, the parabola opens upwards. The roots are found using the quadratic formula, giving approximately: Lower boundary = 0.1974130271612495
   Upper boundary = 0.8442536395054172
7. The confidence set is the bounded interval between these roots.

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.

Limitations

This module assumes exact bivariate normality and known covariance structures to isolate the core geometric mechanism of Fieller's theorem. It does not cover cases where the covariance matrix must be estimated (e.g., using t-distributions), nor does it attempt to regularize or force the output into bounded intervals using delta-method approximations. "All real" means every finite θ is compatible with the test, not that the mathematical parameter 0/0 is well-defined.

References

For a comprehensive modern treatment and comparison of ratio estimation methods:

• Hirschberg, J., & Lye, J. (2007). A Geometric Comparison of the Delta and Fieller Confidence Intervals.