Delta Method for Ratio Metric Variance

Background

Ratio metrics (CTR, sessions-per-user, GMV-per-visit) don't have an obvious $t$-test: you can't just test the numerator's mean. The delta method linearises the ratio around the sample means via a first-order Taylor expansion, giving a usable variance estimate. The standard tool in production experimentation platforms for variance on ratio KPIs.

Problem statement

Implement delta_method_variance(numerator, denominator):

$$\mathrm{Var}\!\Bigl(\tfrac{\bar X}{\bar Y}\Bigr) \;\approx\; \tfrac{1}{n}\!\left(\tfrac{\sigma_X^2}{\bar Y^2} - 2\,\tfrac{\bar X\,\mathrm{Cov}(X,Y)}{\bar Y^3} + \tfrac{\bar X^2\,\sigma_Y^2}{\bar Y^4}\right)$$

Use sample variances and sample covariance (Bessel-corrected).

Input

• numerator — 1-D np.ndarray of floats, length $n$.
• denominator — 1-D np.ndarray of floats, length $n$, same length as numerator.

Output

Returns a Python float >= 0, the estimated variance of the ratio.

Examples

Example 1 — constant denominator collapses to Var(X) / n

Input:  numerator = random N(0, 1), denominator = ones
Output: ≈ Var(X) / n

Example 2 — both noisy → positive variance

Input:  both N(10, 2) and N(20, 3)
Output: > 0

Example 3 — length mismatch

Input:  shape (3,) and (4,)
Output: ValueError

Constraints

• numerator and denominator must have the same length, $\ge 2$.
• Use $n - 1$ Bessel correction for variance and covariance.
• Returns a Python float.

Notes

• Sensitivity to small $\bar Y$. If $\bar Y$ is near zero, the ratio's variance blows up. Production code adds a "clip $\bar Y$" or a guard before reporting.
• Pair with CUPED. Delta method on the ratio of CUPED-adjusted metrics is the standard ratio-A/B test at scale.