# order statistics

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 order statistics  of $X_{1},\ldots,X_{n}$. If all the orderings are strict, then $X_{(1)},\ldots,X_{(n)}$ are the order statistics of $X_{1},\ldots,X_{n}$. Furthermore, each $X_{(i)}$ is called the $i$th order statistic of $X_{1},\ldots,X_{n}$.

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 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.

Title order statistics OrderStatistics 2013-03-22 14:33:30 2013-03-22 14:33:30 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 62G30