Order Statistic

Definition

Let $X_1,X_2,\dots ,X_n$ be Random Sample from a population with a PDF $f(x)$ and CDF $F(x)$. And let $Y_1$ be the smallest of $X_i$‘s, $Y_2$ be the 2nd smallest of $X_i$‘s , $\dots$, and $Y_n$ be the largest of $X_i$‘s, then $Y_1<Y_2<\cdots <Y_n$ is called the order statistic of $X_1,X_2,\dots ,X_n$

Properties

Joint PDF

$g(y_1,y_2,\dots ,y_n)=n!{\prod }_{i=1}^{n}f(y_i)$ where $y_1<y_2<\cdots y_n$

Marginal PDF

$g_k(y_k)=\frac{n!}{(k-1)!(n-k)!}(F(y_k){)}^{k-1}(1-F(y_k){)}^{n-k}f(y_k)$

Joint PDF of $Y_i$ and $Y_j$

Suppose $i<j$ $g_{ij}(y_i,y_j)=\frac{n!}{(i-1)!(j-i-1)!(n-j)!}(F(y_i){)}^{i-1}(F(y_j)-F(y_i){)}^{j-i-1}(1-F(y_j){)}^{n-j}f(y_i)f(y_j)$

Joint PDF of $Y_{l_1},Y_{l_2},\dots ,Y_{l_k}$

Suppose $l_1<l_2<\cdots <l_k$ $g_{l_1,l_2,\dots ,l_k}(y_{l_1},y_{l_2},\dots ,y_{l_k})=\frac{n!}{{\prod }_{i=1}^{k+1}(l_i-l_{i-1}-1)!}\prod _{i=1}^{k}f(y_{l_i})\prod _{i=1}^{k+1}[F(y_{l_i})-F(y_{l_{i-1}}){]}^{l_i-l_{i-1}-1}$ where $l_0=0,l_{k+1}=n+1,x_{(l_0)}=-\infty ,x_{(l_{k+1})}=\infty$