PlanetMath: estimator

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (210)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (38)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

estimator

(Definition)

Let $X_1,X_2,\ldots,X_n$ be samples (with observations ) from a distribution with probability density function $f(X\mid\theta)$ , where $\theta$ is a real-valued unknown parameter in . 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 is an estimator of $\sigma^2$ . In addition, sample median, sample mode, sample trimmed mean are all estimators of $\mu$ . The statistic defined by