The Cornish-Fisher Expansion provides a way to estimate the quantiles of a distribution based on moments. (In this sense, I see it as a complement to the Edgeworth Expansion, which gives an estimate of the cumulative distribution based on moments.) I would like to know in which situations would one prefer the Cornish-Fisher expansion for empirical work over the sample quantile, or vice-versa. A few guesses:

- Computationally, sample moments can be computed online, whereas online estimation of sample quantiles is difficult. In this case, the C-F 'wins'.
- If one had the ability to forecast moments, the C-F would allow one to leverage these forecasts for quantile estimation.
- The C-F Expansion can possibly give estimates of quantiles outside the range of observed values, whereas the sample quantile probably should not.
- I am not aware of how to compute a confidence interval around the quantile estimates given by C-F. In this case, sample quantile 'wins'.
- It seems like the C-F Expansion requires one to estimate multiple higher moments of a distribution. The errors in these estimates probably compound in such a way that the C-F Expansion has a higher standard error than the sample quantile.

Any others? Does anybody have experience using both of these methods?

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#### Best Answer

I have never seen C-F used for empirical estimates. Why bother? You have outlined a good set of reasons why not. (I don't think C-F "wins" even in case 1 due to the instability of estimates of higher-order cumulants and their lack of resistance.) It is intended for theoretical approximations. Johnson & Kotz, in their encyclopedic work on distributions, routinely use C-F expansions to develop approximations to distribution functions. Such approximations were useful to supplement tables (or even create them) before powerful statistical software was widespread. They can still be useful on platforms where appropriate code is not available such as quick-and-dirty spreadsheet calculations.

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