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Letting $\Phi$ be the CDF of the standard normal, and $\Phi^{-1}$ be the quantile function of the normal, I am looking for the Taylor series expansion of $$ \Phi^{-1}\left(\Phi\left(x\right) + \epsilon\right). $$ Using the Taylor series expansion of the inverse error function, I believe I get something like $$ \Phi^{-1}\left(\Phi\left(x\right) + \epsilon\right) = x + \sqrt{\frac{\pi}{2}}\left[2\epsilon + \frac{\pi}{12}\left(3\left(2\Phi\left(x\right)-1\right)^2 2\epsilon + 3\left(2\Phi\left(x\right)-1\right) 4\epsilon^2 + 8\epsilon^3\right) + \ldots \right]. $$ This is a bit ugly for my tastes, and would prefer to collect the terms in powers of $\epsilon$. Is this a known expansion?

(FWIW, I am getting the $\epsilon$ from the Edgeworth expansion.)

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  • $\begingroup$ I think maybe I should just use a Cornish-Fisher expansion instead... $\endgroup$ – shabbychef May 29 '14 at 17:12

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