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# 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 $z_{1}<\cdots<z_{n}$.

## Mathematics Subject Classification

62G30*no label found*

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