# 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. – wolfies Sep 29 '16 at 13:26
• @wolfies The expression for variance in raw moments is known (E(X^2)-E(X)^2). – emcor Sep 29 '16 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'. – wolfies Sep 29 '16 at 13:45
• I would think that the widely known numerical cancellation issues with the comparable variance formula would only be worse here. – GeoMatt22 Sep 29 '16 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. – wolfies Sep 29 '16 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: