An Interpretable Approximation for The Threshold Parameter of Weibull Probability Distributions

Music City Center 

The location or threshold parameter of the three-parameter Weibull distribution is often the most important parameter in many applications that require an estimate of a minimum value. However, traditional methods for estimating this parameter rely on complex numerical procedures that hinder interpretability. In this work, we propose a novel, closed-form approximation that expresses the Weibull threshold as a function of the first three statistical moments: mean, standard deviation, and skewness. This approach enhances understanding of how these common statistical measures influence Weibull threshold behavior and simplifies computation. By prioritizing interpretability, a framework is provided that reveals fundamental relationships between statistical moments and Weibull threshold values, offering insights into the behavior of Weibull random variables across a wide range of skew. This proposed approximation is compared to classical estimation methods, demonstrating its effectiveness in capturing the threshold behavior with minimal mathematical complexity and high interpretability. This work serves as a step toward developing practical methods for minimum value estimation.

mean, μ

standard deviation, σ

skew coefficient

threshold parameter, γ

random variable

Weibull distribution