Problem of computing standard error of higher moments

Question

currently, I am trying to understand how one calculates the standard errors on higher moments using Rao's book [1].

On page 437, he defines

$$ O_r = \frac{1}{n} \sum x_i^r, \, v_r = E[x^r], \, \mathrm{and} \; \mu_r = E[(x-v_1)^r]. $$

I am stuck at computing $E[O_2 O_1^2]$. In the book it is claimed on p. 438 that

$$ E[O_2 O_1^2] = \frac{\mu_4 + (n - 1) \mu_2^2}{n^2} $$

assuming that the origin is where the population mean is; claiming that this is without loss of generality. However, when I try to compute this

\begin{eqnarray} E[O_2^{} O_1^2] &=& \frac{1}{n^3} E\left[\sum_i x_i^2 \sum_j x_j \sum_k x_k\right] \\ &=& \frac{1}{n^3} E\left[ \sum_i x_i^2 \left( \sum_l x_l^2 + \sum_{k \neq j} x_k x_j \right) \right] \\ &=& \frac{1}{n^3} E\left[\sum_m x_m^4 + \sum_{i \neq l} x_i^2 x_l^2 + \sum_{i} x_i^2 \sum_{k \neq j} x_k x_j \right] \\ &=& \frac{v_4 + (n - 1) v_2^2}{n^2} + \frac{1}{n^3} E\left[ \sum_{i} x_i^2 \sum_{k \neq j} x_k x_j \right] \end{eqnarray}

I cannot see how this would go over to the form Rao uses. If my calculations are correct the missing term is

\begin{eqnarray} E\left[ \sum_{i} x_i^2 \sum_{k \neq j} x_k x_j \right] &=& E \left[ 2\sum_{l \neq m} x_l^3 x_m + \sum_{i \neq j \neq k \neq i} x_i^2 x_j x_k\right] \\ &=& 2 n (n - 1) v_3 v_1 + (n^2 (n - 1) - 2 n (n - 1)) v_2 v_1^2 \end{eqnarray}

which is quite complicated. Are my calculations wrong here or do I miss something? Does this also mean the formula for the variance of the variance only holds for the assumption about the mean??

Thanks in advance!

[1] Linear statistical interference and its application

Answer 1

Here is a quick check using the moments of moments functions from the mathStatica package for Mathematica. That uses power sum notation where $s_r = \sum_{i=1}^n X_i^r$. Your problem is to find $E[O_2 O_1^2]$, which denotes the 1st RawMoment of $\frac{s_2}{n} \frac{s_1^2} {n^2}$:

where the solution is expressed in terms of Central Moments $\mu_r$ of the population. In the special case where the mean of the population is 0, this simplifies to:

... as stated in Rao.