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<p>The Hill estimator, based on order statistics, is one of the most widely used tools for estimating the tail index (thickness parameter) of the 2-parameter Pareto distribution with heavy tails. Distribution tails are of paramount importance for engineering design and reliability assessments. In practical applications, however, the Hill estimator must be modified to incorporate at least shift and scale parameters, as the original estimator is not shift- and scale-invariant. In practical engineering applications, however, distribution tails are not fat (heavy, thick) but instead thin, e.g., Weibull-type. Fortunately, the Hill-type estimator can be modified to serve Weibull-type distributions. This study addresses a novel 4-parameter Weibull distribution, which extends the classic 2-parameter Weibull distribution. The primary challenge is that computational effort, prediction inaccuracies, and numerical instabilities increase with the number of distribution parameters. Previously, the Levenberg-Marquardt Least-Squares (LMLS) optimization scheme was used to estimate four parameters of the 4-parameter Weibull distribution. However, LMLS optimization proved to be numerically unstable and lacked convergence criteria. This study presents a novel Maximum Likelihood Estimator (conditional MLE) as a step forward in both parameter convergence and, most importantly, in answering the main designer question: does the selected distribution model fit the underlying data? A renewable energy application (offshore wind speed dynamics) was chosen to benchmark and validate the proposed methodology. Novelty: presented the study for the first time and formulated a semi-analytical solution for the 4-parameter Weibull distribution.</p>

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Novel Estimator for 4-parameter Weibull Extrapolation Scheme: Computer Code Implementation and Wind Power Generation Application

  • 期刊论文

  • 发表日期：

    2026年5月11日

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  • Lixin Xu,
  • Zhiyang Zhang,
  • Oleg Gaidai,
  • Tao Zhang,
  • Shicheng He,
  • Qianlong Ma,
  • Ahmed Elkelity,
  • Salwa Noureldine

摘要

The Hill estimator, based on order statistics, is one of the most widely used tools for estimating the tail index (thickness parameter) of the 2-parameter Pareto distribution with heavy tails. Distribution tails are of paramount importance for engineering design and reliability assessments. In practical applications, however, the Hill estimator must be modified to incorporate at least shift and scale parameters, as the original estimator is not shift- and scale-invariant. In practical engineering applications, however, distribution tails are not fat (heavy, thick) but instead thin, e.g., Weibull-type. Fortunately, the Hill-type estimator can be modified to serve Weibull-type distributions. This study addresses a novel 4-parameter Weibull distribution, which extends the classic 2-parameter Weibull distribution. The primary challenge is that computational effort, prediction inaccuracies, and numerical instabilities increase with the number of distribution parameters. Previously, the Levenberg-Marquardt Least-Squares (LMLS) optimization scheme was used to estimate four parameters of the 4-parameter Weibull distribution. However, LMLS optimization proved to be numerically unstable and lacked convergence criteria. This study presents a novel Maximum Likelihood Estimator (conditional MLE) as a step forward in both parameter convergence and, most importantly, in answering the main designer question: does the selected distribution model fit the underlying data? A renewable energy application (offshore wind speed dynamics) was chosen to benchmark and validate the proposed methodology. Novelty: presented the study for the first time and formulated a semi-analytical solution for the 4-parameter Weibull distribution.