estimator

Let $X_1,X_2,\mathrm{\dots },X_n$ be samples (with observations $X_i=x_i$) from a distribution with probability density function $f(X\mid \theta )$, where $\theta$ is a real-valued unknown parameter (http://planetmath.org/StatisticalModel) in $f$. Consider $\theta$ as a random variable and let $\tau (\theta )$ be its realization.

An estimator for $\theta$ is a statistic ${\eta }_{\theta }={\eta }_{\theta }(X_1,X_2,\mathrm{\dots },X_n)$ that is used to, loosely speaking, estimate $\tau (\theta )$. Any value ${\eta }_{\theta }(X_1=x_1,X_2=x_2,\mathrm{\dots },X_n=x_n)$ of ${\eta }_{\theta }$ is called an estimate of $\tau (\theta )$.

Example.Let $X_1,X_2,\mathrm{\dots },X_n$ be iid from a normal distribution $N(\mu ,{\sigma }^2)$. Here the two parameters are the mean $\mu$ and the variance ${\sigma }^2$. The sample mean $\overline{X}$ is an estimator of $\mu$, while the sample variance $s^2$ is an estimator of ${\sigma }^2$. In addition, sample median, sample mode, sample trimmed mean are all estimators of $\mu$. The statistic defined by

$$\frac{1}{n-1}\sum _{i=1}^n{(X_i-m)}^2,$$

where $m$ is a sample median, is another estimator of ${\sigma }^2$.