Statistical estimation ($\hat{\theta }$)

Statistical estimation uses observed data to infer numerical properties of a population or model. A point estimator gives a value, while a confidence interval describes uncertainty with a specified coverage property. Maximum likelihood estimation is one approach; bias, variance, identifiability and the sampling or selection mechanism all affect an estimate's interpretation.

A statistical estimate is the realized numerical value of an estimator after observing a statistical sample. Before observation, the estimator is a random variable or random vector; its realized value is the estimate. Both are commonly written $\hat{\theta }$, so context distinguishes them.

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.