Order statistics of equal correlated continuous random variables Suppose that $X_1, \ldots, X_n$ are mutlivariate normal with equal correlation
$\rho$ and each of them are marginally
distributed as $N(0,1)$. Let $X_{(1)}, \ldots, X_{(n)}$
be the corresponding order statistics. The distribution of 
$X_{(1)}$ and $X_{(n)}$ are easily found. What about the 
distribution of the other order statistics? Can anyone 
give some reference on this?
Thank you. Hanna
 A: As shown in Tong, Y. L. (1990). Multivariate normal distribution. Springer-Verlag., ch. 6, for the setup described in the question and for non-negative correlation coefficient $\rho\in [0,\;1)$, the distribution function (cdf) and the density of an order statistic $X_{(i)}$ are (where $\phi()$ and $\Phi()$ are the standard normal pdf and cdf)
$$G_{(i)}(x) = \int_{-\infty}^{\infty}F_{(i)}\left(\frac{x+\sqrt{\rho}z}{\sqrt{1-\rho}}\right)\phi(z)dz$$
and differentiating,
$$g_{(i)}(x) = \int_{-\infty}^{\infty}\frac 1{\sqrt{1-\rho}}f_{(i)}\left(\frac{x+\sqrt{\rho}z}{\sqrt{1-\rho}}\right)\phi(z)dz$$
where
$$f_{(i)}(y) = \frac{n!}{(i-1)!(n-i)!}[\Phi(y)]^{i-1}[\Phi(-y)]^{n-i}\phi(y)$$
and
$$F_{(i)}(y) = \sum_{j=i}^n {n \choose j}[\Phi(y)]^{j}[\Phi(-y)]^{n-j}$$
i.e $f_{(i)}(y)$ and $F_{(i)}(y)$ are the pdf and cdf of the order statistic $(i)$ from an i.i.d. standard normal random sample.
For the corresponding results when the correlation coefficient is negative, the author refers to the book "Order Statistics", by H.A. David & H.N. Nagaraja ch. 5 (now in its 3d edition, 2003).
