Moment

mnp:=1n∑i=1nxipm_n^p:=\frac{1}{n} \sum_{i=1}^n x_i^p

The Expectation EF(Xp)E_F(X^p) is then called the pthp^{th} moment of FF.

We have

1n∑i=1nxir⟶EF(Xr)\frac{1}{n} \sum_{i=1}^n x_i^r \longrightarrow \mathbb{E}_F\left(X^r\right)

hence the moment can be used to estimate the Expectation of an ECDF.

We also get a 95% Confidence Interval

∣mnp−EF(Xp)∣≤2VF(Xp)n\left|m_n^p-\mathbb{E}_F\left(X^p\right)\right| \leq \frac{2 \sqrt{\mathbb{V}_F\left(X^p\right)}}{\sqrt{n}}