Hello,(adsbygoogle = window.adsbygoogle || []).push({});

This time my question is not about Catalan numbers but something much more interesting (to me at least;))

I was wondering how the maximum of a multinormal random vector is distributed, for example let

[tex]X \approx N(\mu_1,\sigma_1^2)[/tex]

[tex]Y \approx N(\mu_2,\sigma_2^2)[/tex]

be normally distributed random variables.

Let further

[tex]Z=max(X,Y)[/tex]

What is the distribution of Z. Maybe it doesn't even have a distribution which can be put in closed form..so what can be said about the expectation value...?

What about the more general case, if you have a random vector

[tex](X_1,X_2,\dots,X_n) \approx N(\vec{\mu},\vec{\vec{\Sigma}})[/tex]

with mean [tex]\vec{\mu}[/tex] and covariance matrix [tex]\vec{\vec{\Sigma}}[/tex].

Can anything be sais about the distribution of [tex]Z=max(X_1,X_2,\dots,X_n)[/tex]?

And talking about multivariate normaldistributions there comes yet another question to my mind. If the covariance matrix [tex]\vec{\vec{\Sigma}}[/tex] is not positive definite but only positive semi-definite, then [tex]det(\vec{\vec{\Sigma}})=0[/tex] so the there is no probability density function.

Is there nonetheless a good way to draw samples from such a multivariate normal distribution? (The most obvious way via the Cholesky decomposition of [tex]\vec{\vec{\Sigma}}[/tex] and a suitable linear transformation of a standard normally disitributed random vector doesn't work if the determinant is zero)

So a bunch of questions, thanks for any answers.

Cheers,

Pere

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# Expectation of maximum of a multinormal random vector

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