A quick question.
If I give you the mean vector and the covariance matrix of a multivariate gaussian distribution (call it G1),
is there a way to find a lognormal distribution which mimics the shape of G1 as much as possible?
I mean,
this lognormal distribution has an underlying multivariate gaussian distribution (call it G2).
is there an analytic solution, such that the mean and covariance of G1 can be converted to those of G2?
Is there any algorithm or Fortran code for that?
Thank you very much indeed in advance!
PS.
Of course, multivariate gaussian and lognormal are different, I just try to see if it is possible find a lognormal which mimic the shape of the multivariate gaussian as much as possible.
I asked a simialr question on math stackexchange just in case someone is interested.
A lognormal distribution can mimic a normal distribution well when the mean divided by the standard deviation is large, so that the data is almost always positive.
Nassim Taleb gives formulas for the mean and covariance of a bivariate lognormal distribution here. In general you would have a system of nonlinear equations to solve to match the 1st and 2nd order moments of a multivariate lognormal distribution to specified values.
where \sigma_X is the standard deviation of the lognormal, \mu_X is the mean of the lognormal;
where \sigma is the standard deviation of the underlying normal, \mu is the mean of the underlying normal.
I mean,
lognormal and its underlying normal will not really overlap even if \sigma_X \ll \mu_X , right?
Because when \sigma_X \ll \mu_X, we have \mu \approx \ln(\mu_X), not \mu \approx \mu_X.
I mean, say, if \sigma_X \approx 0, \mu_X=0.1, then \mu \approx \ln(0.1) \approx 1.1,
so the mean of the lognormal is 0.1, the mean of its underlying normal is 1.1. They will not overlap. But their shape can be similar.
