Inverse hill statistic for the novel four-parameter extrapolation Weibull scheme, an algorithmic design approach

One of the most popular methods for assessing the distribution tail thickness parameter (tail index) of a two-parameter PD is the Hill estimator, derived from order statistics. For reliability evaluations and engineering design, an accurate assessment of the distribution’s tails is crucial. Since the original Hill estimator is not shift- and scale-invariant, it must be modified in practice to include at least shift and scale parameters. Note that in a real-world engineering application, distribution tails—such as the Weibull type—are thin rather than thick (heavy). Fortunately, Weibull-type distributions can be estimated using a modified Hill-type estimator. The original two-parameter Weibull-type distribution suffers from the drawback that it resembles the two-parameter PD; without incorporating a shift model parameter, it lacks a shift parameter and is not shift-invariant. By adding shift parameters, the basic two-parameter Weibull distribution is transformed into a three-parameter form. Even so, fitting distributions with only three-parameters does not provide sufficient flexibility for most engineering and design applications; in other words, they often do not fit the underlying raw data. In many facets of statistical inference for rare events, selecting the threshold for the tail distribution is crucial. A novel method for selecting tail distribution thresholds is presented in this study. This study introduces a tailor-made four-parameter Weibull distribution with added $$\:q$$ -parameter that bridges it with PD and Extreme Value Theory (EVT) power parameters and thus enables Inverse Hill Statistic (IHS) extension from thick to thin tail distributions. We present a semi-analytical solution that improves accuracy of the design values and failure probability $$\:{P}_{F}$$ estimates, when the underlying data sample is of limited size. Offshore wind speed dynamics from industrial applications in renewable energy were selected for comparison and validation of the proposed model parameter estimators. Accurate modeling of the ambient wind speed distribution is critical for estimating in situ wind energy potential.