Variance of the estimators of the first central moments

Question

I want to know what is the variance of the unbiased estimators of the first 8 central moments. The variables are i.i.d. and the distribution is unknown.

Although answering this question seems to require a lot of work, I understand that it is possible to solve it using software, as was done in this question where the second moment case was specifically asked.

Answer 1

By definition, the h-statistic $h_r$ is an unbiased estimator of the $r^\text{th}$ central moment of the population $\mu_r$. That is:

$$E\left[h_r\right]=\mu _r$$

I am using here the mathStatica package for Mathematica of which I am one of the authors which has:

• an HStatistic function that generates the desired unbiased estimator, and
• functions for taking symbolic moments of moments. For example, CentralMomentToRaw[3, blah] finds the $3^\text{rd}$ central moment of blah, expressing the solution in terms of Raw moments of the population. The variance operator is just a special case of this function: $\text{Var}[blah]$ is the $2^\text{nd}$ central moment of $blah$.

Combining the above, $\text{Var}[h_2]$ can be found with:

where the solutions is expressed in terms of central moments of the population $\mu_i$, assuming they exist of course. This result is quite well known. The other cases you have requested are little known, and it would probably be difficult to find them all published.

Here is $\text{Var}[h_3]$:

And here are the remaining variances you request. As some of the output is quite long, each is given separately:

Here is $\text{Var}[h_4]$:

Here is $\text{Var}[h_5]$:

Here is $\text{Var}[h_6]$:

Here is $\text{Var}[h_7]$:

And finally here is $\text{Var}[h_8]$: