Method of moments for symmetric mean zero distributions When using the method of moments to fit a symmetric mean zero distribution, does it make more sense to fit higher order moments or lower order absolute moments? I could not find any resources which discuss/compare the merits of these two approaches.
My intuition is that with fewer observations it would be preferable to fit the absolute lower order moments which will be a more stable estimation (just a guess). In practice (in industry not academia) I have seen both methods used (usually the method that is easier to solve analytically...).

I don't have much reputation for adding a comment. Hence I am adding an answer.
Some reasons:

*

*Analytical answer is much comprehensible and straightforward, as suggested by @whuber

*Higher-order moments are usually less robust and more prone to outliers, especially when the sample size is small. Think about the median absolute deviation and standard deviation of a sample from the normal zero-mean distribution.

*Higher-order moments might not exist.

*For some distribution, lower-order moments might give the corresponding MLE, for some, higher-order moments might give the corresponding MLE. (Laplace vs Normal distribution). Those would give asymptotically the best solution under the 'usual' conditions.

