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Suppose there is a vector of jointly normally distributed random variables $X \sim \mathcal{N}(\mu_X, \Sigma_X)$. What is the distribution of the maximum among them? In other words, I am interested in this probability $P(max(X_i) < x), \forall i$.

Thank you.

Regards, Ivan

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    Have you tried to solve the 2-dime$n$sio$n$ case?2012-05-15

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For general $(\mu_X,\Sigma_X)$ The problem is quite difficult, even in 2D. Clearly $P(max(X_i), so to get the distribution function of the maximum one must integrate the joint density over that region... but that's not easy for a general gaussian, even in two dimensions. I'd bet there is no simple expression for the general case.

Some references:

http://www.springerlink.com/content/ca94xg2tdy7evdpb/

http://itc.ktu.lt/itc384/Aksom384.pdf

http://people.emich.edu/aross15/q/papers/bounds_Emax.pdf

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    @Ivan: well, the estimation seems quite straightforward to me: produce some sample values and use the common sample estimators of mean and variance. are you thinking of something else?2012-05-28
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Multivariate Skew-Normal Distributions and their Extremal Properties

Rolf Waeber February 8, 2008 Abstract In this thesis it is established that the distribution is a skew normal dist.

A paper by Nadarajah and Samuel Kotz gives the expression for the max of any bivariate normal F(x,y). IEEE TRANSACTIONS ON VERY LARGE SCALE INTEGRATION (VLSI) SYSTEMS, VOL. 16, NO. 2, FEBRUARY 2008 Exact Distribution of the Max/Min of Two Gaussian Random Variables Saralees Nadarajah and Samuel Kotz If F(x,y) is a standard normal (means=0 and variances=1, r>0) the dist of the maximum is a skew normal.

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    @AsafKaragila I will explore the options you recommended, thanks. $S$omehow, the $S$E software can keep things hidden from me for months before I find out some sort of feature exists.2012-10-26
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Please see paper by Reinaldo B. Arellano-Vallea and Marc G. Genton:

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

Link: https://stsda.kaust.edu.sa/Documents/2008.AG.SPL.pdf

Corollary 4 (page 31) gives the general form for the distribution of the maximal of a multivariate Gaussian. Discussion following the corollary says that the distribution of maximal is skew-normal when $X$ is bivariate normal.