Estimate variance for ratio metrics

Suppose your KPI is ARPU, defined as $\text{ARPU} = \text{Revenue}/\text{ActiveUsers}$ and computed within each bucket, with $\text{corr}(\text{Revenue}, \text{ActiveUsers}) = 0.6$. Use the delta method to obtain an approximate expression for $\text{Var}(\text{ARPU})$, keeping the covariance term, and explain how the required inputs would be estimated from sample moments. Contrast directly testing ARPU against modeling log-Revenue with log-ActiveUsers included as a covariate (or applying ANCOVA). Under what conditions is a t-test appropriate, when would a bootstrap be preferable, and how would you deal with heavy-tailed revenue — for instance through winsorization or robust M-estimators?

Overview: This question assesses statistical inference and experimental-analysis ability in data scientists: variance estimation for ratio metrics through the delta method, covariance estimation from sample moments, model-based comparisons (ANCOVA or log models with offsets), considerations around bootstrap and t-test inference, and techniques for making heavy-tailed revenue more robust, such as winsorization and robust M-estimators. It is frequently posed to gauge both theoretical derivation and the practical trade-offs of estimation in A/B or experimental contexts; the category is Statistics & Math, and the abstraction level ranges from conceptual understanding to practical application.