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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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  • $\begingroup$ duplicate of math.stackexchange.com/questions/145466 $\endgroup$
    – Carlo Beenakker
    Commented Dec 29, 2013 at 13:06
  • $\begingroup$ Carlo, thank you for pointing to this previous question. It is indeed highly related. However, I don't believe it is the same question. $\endgroup$
    – Daniel Soudry
    Commented Dec 29, 2013 at 19:27

1 Answer 1

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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 orthant probability for which there is no known closed-form solution for $n-1>3$

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  • $\begingroup$ Arthur, thank you very much for your answer. Do you happen to have a reference for the last sentence? $\endgroup$
    – Daniel Soudry
    Commented Dec 31, 2013 at 7:29
  • $\begingroup$ Take your pick: google.com/… 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. $\endgroup$
    – Arthur B
    Commented Dec 31, 2013 at 16:46

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