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I am currently trying to work out how to get from the Edgeworth expansion to the Cornish-Fisher expansion. I use van-der-Vaarts "Asymptotics Statistics" and Hall's book on Edgeworth expansions and the bootstrap. Unfortunately neither help much to get the details right (vdVaart p 338).

tldr: Edgeworth expansions can be inverted to get Cornish Fisher expansions - but how?

The paragraph im having trouble with is the following

Let $\Phi, \phi$ be the normal cdf, pdf, $p_1$ a polynomial, $z_\alpha, \hat{\xi}_{n,\alpha}$ the normal and bootstrap upper $\alpha$ Quantiles.

The book states that

$$ 1-\alpha = \Phi(\hat{\xi}_{n,\alpha})+\frac{p_1(\hat{\xi}_{n,\alpha}\mid\hat{P}_n)\phi(\hat{\xi}_{n,\alpha})}{\sqrt n} + O_P(n^{-1}) $$

can be somehow inverted to obtain

$$ \hat{\xi}_{n,\alpha}=z_\alpha-\frac{p_1(z\alpha \mid P)}{\sqrt n}+ O_P(n^{-1}) $$

apparently by Taylor expanding $\Phi, \phi$ and $p_1$ around $z_\alpha$

I guess it all makes sense and I can accept that it is possible, but I'd like to see formal justification.

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  • $\begingroup$ Getting to Cornish Fisher expansions relies on Laplace Inversion. See Jaschke for the derivation: jaschke-net.de/papers/CoFi.pdf or Lee and Lin. $\endgroup$
    – shabbychef
    Commented Feb 4, 2015 at 17:38

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Lengthy formal justification is found in Cornish and Fisher, 1937 and to some degree in Halls "Bootstrap and Edgeworth expansions", where it is explained why the first polynomials are the same.

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