ABSTRACT
The Weibull distribution function has been historically used in probabilistic modeling of wind, wave, fatigue load and many other types of data. In some cases the Weibull model has limited accuracy in the tail region of distribution, which can influence design (extreme value) estimates obtained from the tail extrapolation. In order to improve the modeling of distribution tail, the use of quadratic/cubic polynomials of Weibull distributed random variable has been proposed in the literature. The idea is to improve quality of distribution fitting by preserving the first three or four moments of sample data through the polynomial transformation. In particular, the quadratic Weibull polynomial has been shown to be effective in modeling the distributions of actual rainflow-counted range data.
A practical difficulty associated with application of moments-based models is large sampling uncertainty associated with skewness and kurtosis estimated from limited data. Obviously, any model relying on poor moment estimates would lead to erroneous predictions of extreme design values. To overcome this deficiency of moment-based modeling, the paper present the Weibull polynomial models that is derived using the probability-weighted moments (PWMs) of data. PWMs are essentially expectations of order statistics and their estimation is much more robust than that of traditional moments. A PWM-based Weibull polynomial would also preserve the linear analogues of skewness and kurtosis of data, referred to as L-moments in the literature. The implementation of the proposed approach is extremely simple, which is a significant advantage over the moment-based approach. Examples presented in the paper show that PWM-based approach is superior to moment-based model in terms of reducing the sampling error from estimation.
