find a kernel for the parameter $\theta = \mathbb{E}[(X_1 - \mathbb{E}X_1)^3]$

Question

let $(X_1,\dots,X_n)$ be an i.i.d sample,

find a kernel for the parameter $\theta = \mathbb{E}[(X_1 - \mathbb{E}X_1)^3]$, basically, find some numerical function $h : \mathbb{R^d} \to \mathbb{R} $ such that $d \leq n$ and $\theta = \mathbb{E} h(X_1,\dots,X_d)$

I tried $h(x_1, \dots ,x_n) = (x_1 - \frac{x_1+\dots+x_n}{n})^3 $

but it doesn't check out, $\mathbb{E}[(X_1 - \mathbb{E}X_1)^3]$ simplifies down to $EX_1^3-3EX_1EX_1^2+2E^3X_1$ while $\mathbb{E}[(X_1 - \overline{X}_n)^3]$ is a messy expression which can't be simplified.

any help will be greatly appreciated, thanks !

Answer 1

The kernel in this case is given by $$ h(x_1,x_2,x_3)=\frac{1}{3}\sum_{1\le i\le 3} x_i^3-\frac{1}{2}\sum_{\substack{1\le i,j\le 3, \\ i\ne j}}x_i^{2}x_j+2 x_1x_2x_3. $$

A more general result (for the $r$-th central moment) can be found in this paper (Theorem 1 on page 862).