LH-Moments Parameter Estimation of Weibull Distribution

Natural disasters such as sudden floods, storms, severe snowfall, and droughts are major problems in the world. Generally the distributions of extreme values are heavy-tailed distributions, and an important heavy-tailed distribution is the Weibull distribution, especially for non-linear behaviors. Therefore, accurately estimation of the occurrence of disasters is required to deal with such situations in a timely and efficient manner. Several methods can be used to estimate the parameters, for example, moments estimate, maximum likelihood estimate, linear of moment, and high-order L-moments. The objectives of this article are to estimate the parameters of the four-parameter Weibull distribution with weak non-linear effects (W4DN) based on the LH-moments method, and to propose a new parameter estimation formula. The proposed formula is classified into two cases based on the coefficient of the second-order term (δ): Case 1, where the coefficient is positive (δ > 0) and Case 2, where the coefficient is negative (δ < 0). In both cases, the corresponding estimation formulas are derived β_r and λ_r^p for p=1, 2, ... and r=1, 2, ..., respectively. The parameter estimations (γ ̂,α ̂,δ ̂,ϕ ̂ and κ ̂) are then optimized using the augmented Lagrangian adaptive barrier minimization algorithm. These formulas provide a practical approach for parameter estimation that is essential for forecasting extreme events in various disciplines, including hydrology, meteorology, insurance, finance, and engineering.