Maximal component of a multivariate Gaussian distribution

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!

• duplicate of math.stackexchange.com/questions/145466 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. Dec 29 '13 at 19:27

1 Answer

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 $$n-1>3$$

• Arthur, thank you very much for your answer. Do you happen to have a reference for the last sentence? Dec 31 '13 at 7:29
• 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. Dec 31 '13 at 16:46