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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 $i=1,...,k$?
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  • $\begingroup$ 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: $n^{-1} \sum_j T_i(X_j) = \mathbb E T_i(X)$. $\endgroup$ – cardinal Oct 16 '11 at 1:49
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You can do either of these and create an estimator, but I would expect it to be of inferior quality, as higher moments are generally less well estimated. Also, you could imagine a case where the first and third moments are 0, and so with two parameters you'd need to look at the 2nd and 4th moments.

If you have a particular case in mind, try out your ideas and check their quality by simulation.

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  • $\begingroup$ Thanks, Karl! I wonder what do you mean by "you could imagine a case where the first and third moments are 0, and so with two parameters you'd need to look at the 2nd and 4th moments"? $\endgroup$ – Tim Oct 16 '11 at 0:04
  • $\begingroup$ @Tim - suppose you have some family of distributions symmetric around 0 but with two free parameters you wish to estimate. The symmetry around 0 would make all odd moments strictly 0, and so to use MM you'd need to look at even moments. $\endgroup$ – Karl Oct 16 '11 at 0:27

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