# Advantage of central moment over moment?

Is there any advantage of using "central moments" over "moments" when approximating a distribution to a known distribution using moment matching? I have noticed that in lot of papers.

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Since central moments and regular moments are related via linear transformations, there is no theoretical advantage of one form over the other. In some cases, there might be a small computational advantage in beginning with one form versus the other. – Dilip Sarwate Mar 17 '12 at 19:21
@DilipSarwate, that looks like it addresses the OP's question pretty directly. I wonder if you'd like to switch it to an answer from a comment. – gung Mar 18 '12 at 18:08

Following the suggestion by gung:

Central moments $\mu_n$ are related to moments $m_n$ through linear transformations:

$$\mu_n = E[(X-\mu)^n] = \sum_{i=0}^n \binom{n}{i}(-\mu)^{n-i}E[X^i] = \sum_{i=0}^n \binom{n}{i}(-\mu)^{n-i}m_i.$$

An adaptation, emendation, and incorporation of a proposal by @Flav for an improved statement:

One theoretical advantage to central moments is that they are invariant to translations along the $x$ axis: the central moments of $Y = X + a$ are the same as the central moments of $X$. From a computational perspective, in some cases, (non-central) moments are slightly easier to calculate than central moments, while in other cases, the central moments are slightly easier to calculate.

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 Thanks for the reply. I solved a moment matching problem and have seen that expression for central moment is much simpler than that of moment – prasenjit Mar 19 '12 at 16:35 @prasenjit: Don't forget to designate this as your answer if it turns out to be the best. – Wayne Jun 21 '12 at 20:54