- The use of multi-parameter distribution functions that incorporate empirically derived parameters to more accurately capture the nature of data being studied is investigated. Improving the accuracy of these models is especially important for predicting the extreme values of the non-linear random variables. This study was motivated by problems commonly encountered in the design of offshore systems where the accurate modeling of the distribution tail is of significant importance. A four-parameter Weibull probability distribution model whose structural form is developed using a quadratic transformation of linear random variables is presented. The parameters of the distribution model are derived using the method of linear moments. For comparison, the model parameters are also derived using the more conventional method of moments. To illustrate the behavior of these models, laboratory data measuring the time series of wave run-up on a vertical column of a TLP structure and wave crests interacting in close proximity with an offshore platform are utilized. Comparisons of the extremal predictions using the four-parameter Weibull model and the three-parameter Rayleigh model verify the ability of the new formulation to better capture the tail of the sample distributions. 2013 Elsevier Ltd.