Distribution of Order Statistics

The following theorem is proved in HC for $k=3$.

Theorem 5..1

If $X_1, X_2, \dots , X_n$ is a random sample of size $n$ from a continuous population with pdf $f(x), a<x<b$, then the order statistic ${\bf Y} = (Y_1, Y_2, \dots , Y_n)$ has joint pdf given by

$$f_{\bf Y}(y_1,y_2,\ldots,y_n) = n! \prod^n_{i=1} f(y_i), \ a<y_1<y_2< \ldots<y_n<b.$$ (5.1)

Comment: The proof essentially uses the change of variable technique for the case when the transformation is not one to one. That is, we have a transformation of the form

$Y_1 =$ smallest observation in $(X_1,X_2, \dots , X_n)$
$Y_2 =$ second smallest observation in $(X_1,X_2, \dots , X_n)$
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$Y_n =$ largest observation in $(X_1,X_2, \dots , X_n)$. This has $n!$ possible inverse transformations. You should read and understand the proof given in HC.

The Jacobian for the transformation is:-

$$\left\vert \begin{array}{rrrrr} n & 0 & 0 & \ldots & 0 \\ ... ...ts & \vdots \\ 0 & & & & 1 \\ \end{array} \right\vert = n!$$