次序统计量

$X_1,\cdots,X_n$ 独立同分布，则次序统计量 $X_ {(1)},\cdots,X_ {(n)}$ 为

$$\begin{aligned} \left\{ \begin{array}{l} X_{(1)}=\min X_i\\ X_{(2)}=\min_{i_1<i_2} \max(X_{i_1},X_{i_2})\\ \vdots\\ X_{(k)}=\min_{i_1<\cdots<i_k} \max(X_{i_1},\cdots,X_{i_k})\\ \vdots\\ X_{(n)}=\max X_i \end{array} \right. \end{aligned}$$

其联合概率密度表达式

$$\begin{aligned} f_{X_{(1)},\cdots,X_{(n)}}(\mu_1,\cdots,\mu_n)=n!f(\mu_1)\cdots f(\mu_n) \end{aligned}$$

边缘概率密度表达式

$$\begin{aligned} f_{X_{(k)}}(t)=\frac{n!}{(k-1)!(n-k)!}f(t)(F(t))^{k-1}(1-F(t))^{n-k} \end{aligned}$$