Wind and wave loads on offshore structures show nonlinear effects, which require non-Gaussian statistical models. Here we critically review the behavior of various non-Gaussian models. We first survey moment-based models; in particular, the four-moment “Hermite” model, a cubic transformation often used in wind and wave applications. We then derive an “L-Hermite” model, an alternative cubic transformation calibrated by the response “L-moments” rather than its ordinary statistical moments. These L-moments have recently found increasing use, in part because they show less sensitivity to distribution tails than ordinary moments. We find here, however, that these L-moments may not convey sufficient information to accurately estimate extreme response statistics. Finally, we show that four-moment maximum entropy models, also applied in the literature, may be inappropriate to model broader-than-Gaussian cases (e.g., responses to wind and wave loads).