Is it possible to calculate the 5th, 6th, 7th, 8th and higher-order central robust moments? Is there any other methodology and implementation?
How are these comparable to regular sample moments?
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Note that higher order moments are inherently unstable (with maybe some exceptions, as noted in comments). That means that higher order moments depend on properties of distributions (like extreme tails) that only extremely large samples will contain much information about. I have discussed that at What is the difference between finite and infinite variance.
So, one question is: Does it even make much sense to ask for "robust estimation" of quantities which, by its very definition, are inherently unstable, in that very small perturbations of the model can change this quantities appreciably? Such ideas might be a reason there is little information about robust estimation of higher order moments: the robust way is to avoid their use!
But you could look into L-moments, see https://stats.stackexchange.com/search?q=L-moments+