I am trying to approximate a set of quantiles from the estimated mean, variance, skewness and kurtosis of a random variable with unknown distribution. I tried to apply the Cornish-Fisher expansion of level 3, however, I run over the problem of non-monotonic patterns of estimated quantiles. According to Chernozhukov, I. Fernandez-Val and A. Galichon (2010) (https://arxiv.org/abs/0708.1627), I should rearrange the expansion to correct for the approximation error and impose monotonic pattern in estimated quantiles. However, I failed to conduct the code to correct the issue. I am just wondering is there any code (MatLab or R) that could help me ease this problem?

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    $\begingroup$ Could you provide the four moment estimates? That would help add some concreteness to the problem. ... also, kurtosis has two definitions, one having "3" subtracted from the other (which sets it equal to 0 for the Normal distribution), so telling us which one you use will help. $\endgroup$ – jbowman Sep 7 '17 at 18:11
  • $\begingroup$ Possibly of interest here: karlin.mff.cuni.cz/~rokyta/vyuka/Mgr-statnice/diplomky/… "[I]t is virtually impossible to calculate the non-increasing rearrangement of a concrete given function precisely. For this reason, it is quite desirable to have instead at least some numerical algorithms ... in order to remedy the lack of the precise formula. Our main objective in this thesis is to develop such algorithms." $\endgroup$ – eric_kernfeld Sep 11 '17 at 18:02

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