An estimator is a function that uses sample data to infer the value of an unknown population parameter. Estimators are commonly evaluated by their bias, variance, standard error, consistency, and mean squared error.

If the observed data are $X_1,\dots,X_n$, an estimator has the form:

$$ \hat\theta=g(X_1,\dots,X_n) $$

where:

• $\theta$ is the unknown population parameter being estimated
• $\hat\theta$ is the estimator, considered a random variable before the data are observed
• The numerical value obtained after observing the data is called an estimate

Example

The population mean $\mu$ is commonly estimated by the sample mean:

$$ \hat\mu=\bar X=\frac12\sum^n_{i=1} X_i $$