On the exact distribution of the maximum of absolutely continuous dependent random variables

We derive the exact probability density function of the maximum of arbitrary absolutely continuous dependent random variables and of absolutely continuous exchangeable random variables. We show this density is related to the family of fundamental skew distributions. In particular, we examine the case where the random variables have an elliptically contoured distribution. We study some particular examples based on the multivariate normal and multivariate Student t distributions, and discuss numerical computation issues. We illustrate our results on a genetic selection problem and on an autoregressive time series model of order one.