Estimator

An estimator is a rule that tells how to calculate an Estimate based on the measurements contained in a sample. For example, the ``sample Mean'' Average $\bar x$ is an estimator for the population Mean $\mu$.

The mean square error of an estimator $\tilde\theta$ is defined by

$${\rm MSE} \equiv \left\langle{(\tilde\theta-\theta)^2}\right\rangle{}.$$

Let $B$ be the Bias, then

$${\rm MSE} = \left\langle{[(\tilde\theta-\left\langle{\tilde\theta}\right\rangle{})+B(\tilde\theta)]^2}\right\rangle{}$$

$$= \left\langle{(\tilde\theta-\left\langle{\tilde\theta}\right\rangle{})^2}\right\rangle{}+B^2(\tilde\theta) \equiv V(\tilde\theta)+B^2(\tilde\theta),$$

where $V$ is the estimator Variance.

See also Bias (Estimator), Error, Estimate, k-Statistic