# Generalize the usage of moments in method of moments?

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.

1. 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?
2. Also would it be better if we choose moments of other orders, instead of $i=1,...,k$?
• Related to your questions, but not exactly answering them, sometimes it is useful to consider the method of moments through functions of the observations. For example, in exponential families, the maximum likelihood estimators are method of moments estimators through the sufficient statistics. For example, if $T_1(X),\ldots,T_p(X)$ are the sufficient statistics, than the MLEs are found by solving the system of $p$ equations: $n^{-1} \sum_j T_i(X_j) = \mathbb E T_i(X)$. – cardinal Oct 16 '11 at 1:49