order statistics

Let $X_1,\mathrm{\dots },X_n$ be random variables with realizations in $ℝ$. Given an outcome $\omega$, order $x_i=X_i(\omega )$ in non-decreasing order so that

$$x_{(1)}\le x_{(2)}\le \mathrm{\cdots }\le x_{(n)}.$$

Note that $x_{(1)}=\min (x_1,\mathrm{\dots },x_n)$ and $x_{(n)}=\max (x_1,\mathrm{\dots },x_n)$. Then each $X_{(i)}$, such that $X_{(i)}(\omega )=x_{(i)}$, is a random variable. Statistics defined by $X_{(1)},\mathrm{\dots },X_{(n)}$ are called order statistics of $X_1,\mathrm{\dots },X_n$. If all the orderings are strict, then $X_{(1)},\mathrm{\dots },X_{(n)}$ are the order statistics of $X_1,\mathrm{\dots },X_n$. Furthermore, each $X_{(i)}$ is called the $i$th order statistic of $X_1,\mathrm{\dots },X_n$.

Remark.If $X_1,\mathrm{\dots },X_n$ 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 $(X_{(1)},\mathrm{\dots },X_{(n)})$ (with strict orderings), then one can show that the joint probability density function $f_{\text{𝐙}}$ of the order statistics is:

$$f_{\text{𝐙}}(𝒛)=n!\prod _{i=1}^n{f}_X(z_i),$$

where $𝒛=(z_1,\mathrm{\dots },z_n)$ and .