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I am trying to find the CDF $Z = \max(X_1,X_2,\dots,X_N)$ and in my case $X_i$ are correlated. Is there any transfer domain or one to one function where I can derive the CDF and invert back to current domain ? Or how to handle this type of problem ?

Edit: Since the correlated joint distribution is not known ( but marginal distribution is known), I was wandering if we can transfer the variables in other domain e.g. multiplying with another matrix (whitening) so that the joint distribution would be simple multiplication of marginal distribution. But I am not sure about the equivalent one to one transforming function for maximum.

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  • $\begingroup$ $$F_Z(z) = P\{Z \leq z\} = P\{X_1\leq z, X_2\leq z, \cdots, X_N\leq z\} = F_{X_1,X_2,\cdots, X_N}(z,z,\cdots, z)$$ where just knowing the correlations between the $X_i$ is not enough to determine the right hand side. $\endgroup$
    – Dilip Sarwate
    Commented Oct 23, 2014 at 16:37
  • $\begingroup$ I changed \text{max} to \max. This is standard usage and not only prevents italicization, but also does some things that \text{max} does not do: (1) It results in proper spacing in expressions like $a\max b$; and (2) In a "displayed" rather than "inline" context, it affect the positions of subscripts in things like $\displaystyle\max_{a\in A} f(a)$. ${}\qquad{}$ $\endgroup$
    – Michael Hardy
    Commented Oct 23, 2014 at 17:11
  • $\begingroup$ You need more information about the joint distribution than what is given here. $\endgroup$
    – Michael Hardy
    Commented Oct 23, 2014 at 17:12

1 Answer 1

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If you know the joint distribution $F(\{X_i\leq a_i\})$, then in general:

If $Z:=\max\{X_i\}$ then $F(z)=F(\{x_i\leq z\})$

Since the variables are correlated, that's about as far as you can get without a specific distribution.

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  • $\begingroup$ Since the correlated joint distribution is not known ( but marginal distribution is known), I was wandering if we can transfer the variables in other domain e.g. multiplying with another matrix (whitening) so that the joint distribution would be simple multiplication of marginal distribution. But so far I have no luck. $\endgroup$
    – upol94
    Commented Oct 23, 2014 at 21:04
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    $\begingroup$ @upol94 the closest thing I can think of to what you want are copulas. $\endgroup$
    – user76844
    Commented Oct 24, 2014 at 5:48
  • $\begingroup$ How would you proceed if you knew the joint distribution between any two r.v ,i.e the distribution of every random variable? (for example you know that they all follow a normal distribution and you know the mean and variance of each) $\endgroup$
    – DariusTheGreat
    Commented Jul 10, 2019 at 8:38

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