# Types of moments used in the method of moments?

In Wikipedia, the method of moments uses only ordinary moments:

One starts with deriving equations that related the population moments (i.e., the expected values of powers of the random variable under consideration) to the parameters of interest

What if the method of moments uses other types of moments, such as central moments or standardized moments? If they are not used or preferred, why?

Thanks!

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$$\mu_n = \int (x-\mu)^nf(x)dx = \int \sum_{k=0}^n \binom{n}{k}x^n(-1)^{n-k}\mu^{n-k} f(x)dx.$$
So, the $n$th central moment is a linear combination of the first $n$ moments, where $\mu$ appears in the linear factors.
An intuitive argument for not using them is that the error in the asymptotic convergence is "cumulative". This is, there is a double approximation, the first one for approximating $\mu$, with $\bar{x}$, and the second one for approximating the central moment itself $\sum (x_i-\bar{x})^n/n$.