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?