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Suppose we have a random variable $X$ and a smooth function $g$. We want to calculate the expectation value $\mathbb{E}(g(X))$. To be able to write down at least an approximate solution, we perform a Taylor expansion $g_T$ of $g$ up to second order around the mean of $X$ and use $\mathbb{E}(g_T(X))$ as the approximation.

Now, what would be a reasonable estimate of how good this approximation is? Or an estimate that tells me for which $X$ the truncation after second order is justified?

Whereas for a real valued function this is clear, in this case I am looking for an adequate statistical dispersion measure. I have simulation data, so can I could actually calculate the measure and then want to make an appropriate approximation to get an analytical insight into my system.

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    $\begingroup$ en.wikipedia.org/wiki/Cumulant#Some_properties_of_cumulants $\endgroup$
    – Steve Huntsman
    Commented Aug 16, 2011 at 22:35
  • $\begingroup$ "up to second order" including or excluding the quadratic term? $\endgroup$
    – user5810
    Commented Aug 17, 2011 at 2:51
  • $\begingroup$ including the quadratic term $\endgroup$
    – madison54
    Commented Aug 17, 2011 at 12:47
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    $\begingroup$ The usual error estimates for Taylor expansions let you estimate the error in terms of the sup of $g'''$ and the third absolute moment of $X$. Is this adequate for your purpose? $\endgroup$
    – Mark Meckes
    Commented Aug 17, 2011 at 13:49

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