OrderStatistics 2004-08-24 20:26:28 2006-09-15 15:44:07 Definition % 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 Let $X_1,\ldots,X_n$ be random variables with realizations in $\mathbb{R}$. Given an outcome $\omega$, order $x_i=X_i(\omega)$ in non-decreasing order so that $$x_{(1)}\leq x_{(2)}\leq\cdots\leq x_{(n)}.$$ Note that $x_{(1)}=\operatorname{min}(x_1,\ldots,x_n)$ and $x_{(n)}=\operatorname{max}(x_1,\ldots,x_n)$. Then each $X_{(i)}$, such that $X_{(i)}(\omega)=x_{(i)}$, is a random variable. Statistics defined by $X_{(1)},\ldots,X_{(n)}$ are called \emph{order statistics} of $X_1,\ldots,X_n$. If all the orderings are strict, then $X_{(1)},\ldots,X_{(n)}$ are \emph{the} order statistics of $X_1,\ldots,X_n$. Furthermore, each $X_{(i)}$ is called the $i$th order statistic of $X_1,\ldots,X_n$. \par \textbf{Remark.} If $X_1,\ldots,X_n$ are iid as $X$ with probability density function $f_X$ (assuming $X$ is a continuous random variable), Let $\textbf{Z}$ be the vector of the order statistics $(X_{(1)},\ldots,X_{(n)})$ (with strict orderings), then one can show that the joint probability density function $f_{\textbf{Z}}$ of the order statistics is: $$f_{\textbf{Z}}(\boldsymbol{z})=n!\prod_{i=1}^{n}f_X(z_i),$$ where $\boldsymbol{z}=(z_1,\ldots,z_n)$ and $z_1<\cdots<z_n$.