I have a vague feeling that for a fixed sample size, lower-order moments of a distribution would typically be estimated more precisely than higher-order moments. E.g. mean would be estimated more precisely than the second moment.
- How do I phrase this formally?*
- Is this correct (perhaps under certain conditions)? What are the conditions for this to hold?
The question is motivated by the following thread at Quantitative Finance Stack Exchange: "Why is asset volatility easier to estimate than the asset mean if it contains the mean?"
*There are different measures of precision and I wonder which one(s) may make most sense; perhaps there is a standard way of thinking about this problem that I am not aware of.