Ratio estimator ($\hat{\theta }=U_n/V_n$)

A ratio estimator divides one statistic by another, for example an estimated joint moment by an estimated marginal probability. Its definition must specify what happens when the denominator is zero. A denominator bounded away from zero permits direct error bounds through $U_n-\theta V_n$.

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.