The \(k\)th order statistic is the \(k\)th smallest value in a sample.
\(x_{(1)}\) is the smallest value in a sample, the minimum.
\(x_{(n)}\) is the largest value in a sample, the maximum.
The probability distribution of order statistics depends on the underlying probability distribution.
If we have:
\(Y=\max \mathbf X\)
The probability distribution is:
\(P(Y\le y)=P(X_1\le y, X_2\le y,...,X_n\le y)\)
If these are iid we have:
\(P(Y\le y)=\prod_i P(X_i\le y)\)
\(F_y(y)=F_X(y)^n\)
The density function is:
\(f_y(y)=nF_X(y)^{n-1}f_x(y)\)
If we have:
\(Y=\min \mathbf X\)
The probability distribution is:
\(P(Y\le y)=P(X_1\ge y, X_2\ge y,...,X_n\ge y)\)
If these are iid we have:
\(P(Y\le y)=\prod_i P(X_i\ge y)\)
\(F_y(y)=[1-F_X(y)]^n\)
The density function is:
\(f_y(y)=-n[1-F_X(y)]^{n-1}f_x(y)\)