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I want to calculate higher-order joint cumulants for 2 variables. I calculated the higher order single-variable and bivariate moments numerically. Now I need to combine them into cumulants (upto the 6th order cumulant, eg k_{3,3})

However, surprisingly, I was unable to find the equations for this anywhere online. Wikipedia (see link above) gives the general formula and provides a few examples for joint cumulants. Also, the summary wolfram site gives a few more examples for bivariate cumulants. However, I was not able to find any comprehensive table that gives all the equations. I feel I understand the formula and the partition concept, but it is tedious to do all the calculations by hand and I dont trust myself not to make an error.

My questions are:

1) Do these equations exist anywhere online?

2) Is there an easy way to get these equations using computing software like Python or Matlab?

3) If not, what else can I do to get them?

Thanks!

DankMasterDan
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    Stuart & Ord (*Kendall's Advanced Theory of Statistics,* Fourth Edition) give these relations in formula (3.81). It expresses the bivariate cumulants in terms of the bivariate central moments. For instance, it includes $$\kappa_{33} = \mu_{33} - 3\mu_{31}\mu_{02} - \cdots + 12 \mu_{11}^3.$$ *Mathematica* will [compute them](http://mathworld.wolfram.com/Cumulant.html). – whuber Jan 31 '20 at 17:53
  • @whuber, thanks for comment! Unfortunately I don't have access to the Stuart & Ord book, nor do I have a Mathematica/MathStatica license. I was hoping there would be something available online or in open-source software... – DankMasterDan Feb 03 '20 at 16:48

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so I found what I was looking for in some academic papers. For example, see:

Abdelmutalab, Ameen, Khaled Assaleh, and Mohamed El-Tarhuni. "Automatic modulation classification based on high order cumulants and hierarchical polynomial classifiers." Physical Communication 21 (2016): 10-18.

They provide equations for cumulants of complex variables upto the 6th order:

enter image description here

DankMasterDan
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  • What does M stand for in the above relations? M is not the bivariate central moments - and I am at a loss as to what the table above is showing at all!? – wolfies Aug 29 '21 at 14:27