# Distribution of the maximum of a multivariate normal random variable

Suppose there is a vector of collectively generally dispersed arbitrary variables $X \sim \mathcal{N}(\mu_X, \Sigma_X)$. What is the circulation of the maximum amongst them? To put it simply, I want this chance $P(max(X_i) < x), \forall i$.

Thanks.

Regards,. Ivan

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2022-07-25 20:42:35
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For general $(\mu_X,\Sigma_X)$ The problem is quite difficult, even in 2D. Clearly $P(\max(X_i)<x) = P(X_1<x \wedge X_2<x \cdots \wedge X_d <x)$, 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:

• Ker, Alan P., , Extremes 4, No. 2, 185–190 (2001).

• Aksomaitis, A.; Burauskaitė-Harju, A., , Information Technology and Control 38, No. 4, 301–302 (2009)

• Ross, Andrew M., , ISE Working Paper 03W-004 (2003)

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2022-07-25 22:19:34
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