PlanetMath: order statistics

(more info)

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order statistics

(Definition)

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

$\displaystyle x_{(1)}\leq x_{(2)}\leq\cdots\leq x_{(n)}.$

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

th order statistic of $X_1,\ldots,X_n$ .

Remark. If $X_1,\ldots,X_n$ are iid as with probability density function (assuming 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: