Absolute error bound for a ratio estimator with a controlled denominator ($\mathbb{E}|U_n/V_n-\theta |\le {\delta }^{-1}(|\mathbb{E}{G}_n|+\sqrt{\mathrm{Var}{G}_n})+r_n$)

Set $G_n=U_n-\theta V_n$, and define the estimator on zero-denominator events as part of the model. On $V_n>\delta >0$, its error is at most $|G_n|/\delta$. Centering $G_n$ and applying the Cauchy-Schwarz inequality gives the displayed bound, where $r_n$ is the expected error on $V_n\le \delta$. This controls a ratio estimator without falsely assuming that its numerator and denominator are independent.