I am required for a homework question to find moment and maximum likelihood estimators, and in class we have only looked at finding MLEs and using already known moment estimators, hence I have no idea how to find the moment estimator myself. I have looked it up and done a little research, but I simply have no idea how go start. Is there some formula?
Sorry I don't want to post the question on fear of plagiarism concerns.
If there are parameters you are trying to estimate, the presumably each tuple of values of those parameters corresponds to some known probability distribution. For example, suppose you have a Gamma distribution
$$\frac{1}{\Gamma(\alpha)} \left(\frac x \beta\right)^{\alpha−1} e^{−x/\beta} \left(\frac{dx} \beta \right).$$
You want to estimate $\alpha$ and $\beta$. Given the density, you can find the population moments $\mathbb E(X)$ and $\mathbb E(X^2)$ as functions of $\alpha$ and $\beta$. Set those equal to the sample moments $(X_1+\cdots+X_n)/n$ and $(X_1^2+\cdots+X_n^2)/n$, then solve the resulting equations for $\alpha$ and $\beta$.
For the given Gamma distribution, the population moments are $\mathbb E(X)=αβ$ and $\mathbb E(X^2)=α(α+1)\beta^2$. So you have $αβ=(X_1+\cdots+X_n)/n$ and $\alpha(\alpha+1)\beta^2=(X_1^2+\cdots+X_n^2)/n$. Solve those for $\alpha$ and $\beta$ and that gives you the estimates of $\alpha$ and $\beta$.