Hello everyone,
I need to calculate high order cross cumulant for my research. The attached pictures shows how I calculated the second order cross cumulant. However, when I tried to solve for third order, I am stuck with the Expectation of three variables.
So, is their any simple way to solve it or can you please suggest me other technique to obtain the value?
Thank you in advance,
Samundra
It is unclear what you are asking. If you are seeking formulae relating cumulants to ordinary moments, then you could start with a basic source such as;
Kendall, M.G. & Stuart A.(1969) "The Advanced Theory of Statistics, Vol 1, Distribution Theory" (3rd Edition) Griffin, London (or later editions possibly). see Chapter 3 "Moments and Cumulants"
If you are looking for more general approaches then you could look at the (multivariate) cumulant generating function, which is the logarithm of the (multivariate) characteristic function. This might let you avoid starting from the moments.
It rather depends on what your starting point is and where you are trying to get to.
You are correct. It is more difficult to deal with the third-order cross cumulant when you consider the correlations between different variables. The numerical way should be easier than the analytical way. You could simply generate the Pi1, Pi2 and Pi3 samples with consideration of their own distribution and mutual correlations. And then by using the formula you presented in the picture, it is quite straight away to calculate. I believe this is the simple way and could be applied to even higher-order cumulants as well.
Please refer to my previous paper for further reference:
Article Probabilistic Analysis of Small-Signal Stability of Large-Sc...