A new addition to the statistical Hermite moment model of extremes is introduced for use on processes with high skewness and near-Gaussian kurtosis. The monotone limits of the existing model are expressed as ellipses in response moment space and a new methodology is introduced that combines hardening and softening models to overcome these limits. The result is that any fractile of a distribution described by its first four statistical moments can be transformed to or from the Gaussian. An example application to a tension leg platform is presented.

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