# Kurtosis expressed in raw moments

It is possible to express central moments in terms of raw moments.

E.g. for skewness we get: {\begin{aligned}\gamma _{1}&=\operatorname {E} \left[\left({\frac {X-\mu }{\sigma }}\right)^{3}\right]\\&={\frac {\operatorname {E} [X^{3}]-3\mu \operatorname {E} [X^{2}]+3\mu ^{2}\operatorname {E} [X]-\mu ^{3}}{\sigma ^{3}}}\\&={\frac {\operatorname {E} [X^{3}]-3\mu (\operatorname {E} [X^{2}]-\mu \operatorname {E} [X])-\mu ^{3}}{\sigma ^{3}}}\\&={\frac {\operatorname {E} [X^{3}]-3\mu \sigma ^{2}-\mu ^{3}}{\sigma ^{3}}}.\end{aligned}}

I was wondering how to express kurtosis in terms of raw moments? I came this far:

$$\beta_4= {E} \left[\left({\frac {X-\mu }{\sigma }}\right)^{4}\right]= \frac{E(X^4)-4E(X^3)\mu+6E(X^2)\mu^2-3\mu^4}{(\sigma^2)^2}$$

Hereby I used Binomial expansion to write out the brackets. If this is correct, can the nominator perhaps be more simplified (like in the skewness example where the variance was taken out in step 3)?

• Instead of saying you seek expression in terms of 'non-central' moments, you appear to be after the conversion from central moments to raw moments. Note further that you have left your denominator expressed as a function of $\sigma^2$, which is still the 2nd central moment ... so it appears that you are not really seeking expression in terms of raw moments, but something else. Sep 29, 2016 at 13:26
• @wolfies The expression for variance in raw moments is known (E(X^2)-E(X)^2). Sep 29, 2016 at 13:43
• Yes - but my point is that the OP has chosen to keep expressing his solution in terms of $\sigma^2$ ... i.e. a central measure. The conversion of all central moments to raw is 'known'. Sep 29, 2016 at 13:45
• I would think that the widely known numerical cancellation issues with the comparable variance formula would only be worse here. Sep 29, 2016 at 13:57
• @GeoMatt22 These are population moments (not sample moments), so the issue you raise would not be applicable to the theoretical distribution moments. It is true that the same relations do also hold for sample moments ... but even then, I wonder whether the cancellation issues you refer to have been rendered obsolete by arbitrary precision numerical math packages. I would be interested to see if the same old examples still pose problems with current arbitrary precision numerical software. Sep 29, 2016 at 19:05

There are many alternative methods to convert from central to raw moments ... the text by Stuart and Ord, Kendall's Advanced Theory of Statistics, volume 1, provides a good treatment. There are also automated functions to do this. Here is an automated conversion of the numerator: the $$4^{\text{th}}$$ central moment $$\mu_4$$ in terms of raw moments:
using the CentralToRaw function from the mathStatica package.