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Suppose you have a general random Gaussian vector $\mathbf{X}\sim\mathcal{N}\left(\boldsymbol{\mu},\boldsymbol{\Sigma}\right)$. I'm looking for the simple way to calculate the distribution of the maximal component: $P\left(\mathrm{argmax}_{i}X_{i}=k\right)$. Is there some closed-form expression, or a recursive formula for this?

Thanks in advance!

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duplicate of – Carlo Beenakker Dec 29 '13 at 13:06
Carlo, thank you for pointing to this previous question. It is indeed highly related. However, I don't believe it is the same question. – Daniel Soudry Dec 29 '13 at 19:27
up vote 2 down vote accepted

Without loss of generality, the problem is equivalent to computing the probability $P(X_1 \geq \max(X_2, \ldots, X_n))$. We can transform the coordinates as $X_2-X_1, X_3-X-1, \ldots, X_n-X_1$ which is a fully general multivariate normal distribution of degree $n-1$. The problem is thus equivalent to finding the probability of an orthant for which there is no known closed-form solution for $n-1>3$

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Arthur, thank you very much for your answer. Do you happen to have a reference for the last sentence? – Daniel Soudry Dec 31 '13 at 7:29
Take your pick:… I don't think there is a proof that no closed-form solution exists, it's just that none is known and it seems very unlikely that one would exist. – Arthur B Dec 31 '13 at 16:46

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