# estimator

Let $X_{1},X_{2},\ldots,X_{n}$ 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   $\eta_{\theta}=\eta_{\theta}(X_{1},X_{2},\ldots,X_{n})$ that is used to, loosely speaking, estimate $\tau(\theta)$. Any value $\eta_{\theta}(X_{1}=x_{1},X_{2}=x_{2},\ldots,X_{n}=x_{n})$ of $\eta_{\theta}$ is called an estimate of $\tau(\theta)$.

Example. Let $X_{1},X_{2},\ldots,X_{n}$ 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 $\sigma^{2}$.

Title estimator Estimator 2013-03-22 14:52:22 2013-03-22 14:52:22 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 62A01 estimate