Estimator 2004-12-09 20:25:04 2005-08-03 18:35:09 Definition estimate % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \PMlinkescapeword{strictly} 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 \PMlinkname{parameter}{StatisticalModel} in $f$. Consider $\theta$ as a random variable and let $\tau(\theta)$ be its realization. An \emph{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 \emph{estimate} of $\tau(\theta)$. \textbf{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 $\sigma^2$.