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Given a data set $x = \{ x(n) \mid n=1,\ldots,m\}$ how do I calculate the $p^{th}$-order cumulant? In particular I need to calculate the 4th-order cumulant. I found that I can calculate the cumulant from moments, but I also don't know how to calculate moments. I found how to calculate the 4th central moment (kurtosis) but I am not sure if/how they are related.

Since I have to implement the calculation of the cumulant in C++ I would be interested in a general explanation (to understand what I am doing) but also hints on how to implement the calculation efficiently would be appreciated. How can one interpret the 4th-order cumulant?

I am sorry if this is a trivial question but I don't have a strong mathematical background and was not able to find the answer online.

Michael Hardy
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Techy
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1 Answers1

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You can calculate the cumulants using the following recursion formula : http://upload.wikimedia.org/math/9/b/2/9b2b9e42c18cd141a6236fa04b6715a5.png where the $\kappa$ are the cumulants and the $m$ are the moments.

The $p$-th order moment is given by $$m_p=\frac{1}{n}\sum_{k=1}^nx_k^p$$

Augustin
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  • Thanks for your answer, I will try it and accept your answer. Since the image you linked to is from wikimedia, is there a corresponding article where the image is used? Could you provide the link to it? – Techy May 20 '15 at 09:48
  • Sure, it's from the Cumulants page : http://en.wikipedia.org/wiki/Cumulant – Augustin May 20 '15 at 09:50