Distribution of order statistics for multivariate distribution

Question

Suppose random vector $(X_1, X_2, \ldots, X_n)$ has a continuous multivariate distribution as F. I want to obtain the distribution of order statistics vector $(X_{(1)}, X_{(2)}, \ldots, X_{(n)})$. Can you help me?

Answer 1

This is not a proof, merely a plausibility argument.

Let $f(x_1,\dots,x_n)$ denote the density of $(X_1,X_2,\dots,X_n)$ at $x_1,\dots,x_n$, and let $X_{(1)},\dots,X_{(n)}$ denote the $X_1,\dots,X_n$ in increasing order.

Choose $ -\infty < x_{(1)} < x_{(2)} < \dots < x_{(n)} < \infty$, let $dx_{(1)}, \dots, dx_{(n)}$ be very tiny quantities so that the following holds $ -\infty < x_{(1)} < x_{(1)} + dx_{(1)} < x_{(2)} + dx_{(2)} < \dots < x_{(n)} + dx_{(n)} < \infty.$

We have, $P\left(X_{(1)} \in (x_{(1)},x_{(1)} + dx_{(1)}),X_{(2)}\in(x_{(2)},x_{(2)}+dx_{(2)}),\dots,X_{(n)}\in(x_{(n)},x_{(n)}+dx_{(n)})\right) = \sum_{\pi \in S_n}P\left(X_1 \in (x_{(\pi(1))},x_{(\pi(1))}+dx_{(\pi(1))}),\dots,X_n \in (x_{\pi(n)},x_{\pi(n)}+dx_{\pi(n)})\right)$

where the sum on the RHS is over all permulations of $1,2,\dots,n$. This holds because the intervals $(x_{(i)},x_{(i)}+dx_{(i)})$ are disjoint and each $X_{(i)}$ must correspond to some $X_j$.

The RHS is upto a close approximation $\sum_{\pi \in S_n} f(x_{\pi(1)},\dots,x_{\pi(n)})dx_{(1)}dx_{(2)} \dots dx_{(n)}$. This "implies" the required density is $\sum_{\pi \in S_n} f(x_{\pi(1)},\dots,x_{\pi(n)})$ for any $x_1 < x_2 \dots < x_n$.