Estimation precision of lower- vs. higher-order moments 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.
 A: Here is what I believe might be a counterexample if the intuition were a general claim, or at least a result that seems to indicate that the answer to 2. might be "not really". The measure of the precision of an estimator of a certain moment that I use here is the variance.
It is well known that the variance of the sample variance, when sampling from a normal population, is $\frac{2\sigma^4}{n-1}$, and that that of the mean is $\sigma^2/n$.
So, the former is larger if
$$\frac{2\sigma^4}{n-1}>\frac{\sigma^2}{n}$$ or $$\sigma^2>\frac{n-1}{2n},$$
which evidently need not be the case.
n <- 10
sigma.sq <- 4/10 # 9/20 or 4.5/10 would be cutoff here

sim.mean.s2 <- function(n){
  x <- rnorm(n, sd=sqrt(sigma.sq))
  xbar <- mean(x)
  s2 <- var(x)
  return(list(xbar, s2))
}

sims <- matrix(unlist(replicate(1e6, sim.mean.s2(n))), nrow=2)

var(sims[1,]) # may also try moments::moment(sims[1,],2, central=T) to simulate population variance, but does not matter at many replications
sigma.sq/n

var(sims[2,])
2*sigma.sq^2/(n-1)

