Multivariate normal distribution
I have some data for a multivariate normal distribution. It is:
The mean of each of the marginal distributions. mu1, mu2, ... , mun: in vector mu
The 99th percentile for each of the marginal distributions. VaR1, VaR2, ..., VaRn: in vector VaR
The correlation matrix between each of the marginal distributions. sigma
From this information I am able to calculate the standard deviation for each of the marginal distributions. sd1, sd2, ..., sdn: in vector sd
Then calculate the mean and standard deviation for the multivariate normal distribution MU and SD respectively.
MU = sum (mu1,...,mun)
SD = sqrt(transpose(sd)*sigma*sd)
Then calculate the 99th percentile for the multivariate (which is what I really want) using this MU and SD and the normal distribution. Call this 99VaR.
I have seen that one can get the same answer as 99VaR by using:
I am just not sure of the maths as to why?
If this works for the multivariate normal - does it work for other multivariate distributions?
Any comments are most welcome