In Method of Moments for estimation, if there are $k$ parameters to estimate, we usually consider $i$-th moments, $i=1,...,k$, so that we have k equations for k unknowns.
- I wonder if it is wise to consider more moments of different orders, i.e. $i$-th moments, $i=1,...,n>k$, so that we are to solve a over-determined linear system? Why?
- Also would it be better if we choose moments of other orders, instead of $i=1,...,k$?