A direct calculation of moments of the sample variance (Q419424)

Let \(V\) be the sample variance of an i.i.d. sample \(X_1,\dots,X_N\). The authors propose a new method of representation of \(\mathbf{E} V^j\) as a polynomial from the moments \(\mathbf{E}(X_1)^i\). It is based on the representation of \(\mathbf{E} V^j\) as a weighted sum of \(\mathbf{E} \left(\sum_{j=1}^N X_j^2\right)^m\left(\sum_{j=1}^N X_j\right)^n \) and calculation of this expectation directly, using some new combinatorics results. This method can be considered as an alternative to the Polykays rules.NEWLINENEWLINEThe derivation of the Gauss formula for \(\text{Var} V\) is presented as an example.