I have been trying to understand and classify how much
information is lost when we model signals using Gaussian models
and throw away higher order statistical information content.
Clearly this is fine when signals are Gaussian, but if
when signals are not Gaussian.
Example : if I want to estimate a variable x (Multi-dimensional)
from observations of a related signal y (Multi-dimensional),
assuming the signals are Gaussian best linear estimation of x
is x_estimate=Rxy * Inverse(Ry) * y where Ry=E(xx*) is Cov Matrix of
y and similarly Rxy=E(xy*). The estimate of x is optimal in the
least square sense; but if we estimate x using higher order
statistical information content (cumulants beyond 2nd order) then
estimate is not optimal in the least square sense, yet it is
more consistent with the statistical properties of the signals.
When should be use higher order statistics and when does it actually
make a difference and how much beyond the usual 2n'd order estimation
Any references would be greatly appreciated.