8
$\begingroup$

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.

$\endgroup$
4
  • 1
    $\begingroup$ en.wikipedia.org/wiki/Cumulant#Some_properties_of_cumulants $\endgroup$ Aug 16, 2011 at 22:35
  • $\begingroup$ "up to second order" including or excluding the quadratic term? $\endgroup$
    – user5810
    Aug 17, 2011 at 2:51
  • $\begingroup$ including the quadratic term $\endgroup$
    – madison54
    Aug 17, 2011 at 12:47
  • 3
    $\begingroup$ 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? $\endgroup$ Aug 17, 2011 at 13:49

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.