method of moments for estimation

When trying to estimate an unknown $k$-dimensional parameter $\theta =({\theta }_1,\dots ,{\theta }_k)$ (statistical inference) the method of moments is an old school method (though it’s recently seen a revival in interest), which equates the first $k$ theoretical moments with the first $k$ empirical moments.

That is, we solve $\frac{1}{n}{\sum }_{i\le n}{X}_i^j={\mathbb{E}}_{\hat{\theta }}[X^j]$ for $j=1,\dots ,k$. Note this is a purely frequentist approach .

For exponential families, the MoM estimator coincides with the MLE.