A Convex Combination of Fréchet and Weibull Distributions: Model, Estimation, and Applications

Technical Summary: A Convex Combination of Fréchet and Weibull Distributions

Problem Statement
Real-world data in fields such as hydrology, reliability engineering, and extreme value analysis often exhibit heterogeneous tail behaviors, where extreme events coexist with regular occurrences. Capturing this variability using a single parametric distribution frequently results in poor model fit and misleading inferences. While the Fréchet distribution effectively models heavy-tailed phenomena (e.g., rare, high-impact events), and the Weibull distribution is versatile for lighter-tailed data and varying hazard rates, neither alone can adequately represent the full spectrum of tail behaviors found in complex, heterogeneous populations.

Methodology
The paper proposes a two-component finite mixture model formed by the convex combination of Fréchet and Weibull probability density functions. The model is defined as:
$$f(x) = w f_{Fr}(x; \alpha, s, m) + (1-w) f_{Weib}(x; \lambda, k)$$
where $w \in [0, 1]$ is the mixing proportion, and the parameters represent shape ($\alpha, k$), scale ($s, \lambda$), and location ($m$) for the respective components.

To estimate the parameter vector $\theta = (w, \alpha, s, m, \lambda, k)$, the authors develop a Maximum Likelihood Estimation (MLE) procedure utilizing the Expectation–Maximization (EM) algorithm. The methodology involves:

1. E-step: Computing posterior probabilities (fractional weights) that each observation belongs to the Fréchet component.
2. M-step: Updating parameters iteratively. The mixture weight $w$ is updated via a closed-form average of posterior probabilities. The Weibull scale $\lambda$ and Fréchet scale $s$ are updated via closed-form weighted MLEs given fixed shape parameters. The shape parameters ($k, \alpha$) and the Fréchet location parameter $m$ are updated via one-dimensional numerical optimization (e.g., Newton–Raphson). To ensure numerical stability for the location parameter $m$, a reparameterization ($m = \min(x_i) - e^t$) is employed.

While the paper outlines Method of Moments (MME) and Least Squares Estimation (LSE) as alternative approaches, the primary focus and implementation details are centered on the EM algorithm for MLE.

Key Contributions

• Model Formulation: Introduction of a flexible parametric framework that interpolates between heavy-tailed (Fréchet) and light-to-moderately heavy-tailed (Weibull) regimes, capable of modeling multimodality and skewness in heterogeneous data.
• Estimation Procedure: Development of a robust EM algorithm that combines closed-form updates with low-dimensional numerical optimization, addressing specific challenges such as the unconstrained optimization of the Fréchet location parameter.
• Numerical Stability: Implementation of strategies to ensure convergence and stability, including specific reparameterizations and convergence criteria based on log-likelihood changes.

Results

• Simulation Study: A comprehensive simulation study was conducted with 1,000 replications and a sample size of $n=200$. The results demonstrated that the proposed estimator yields small biases and low Root Mean Squared Errors (RMSE) for all parameters ($w, \alpha, s, m, \lambda, k$), indicating satisfactory finite-sample performance.
• Real Data Application: The model was applied to a dataset of 365 daily maximum wind speeds. The fitted model assigned approximately 36% weight to the Fréchet component and 64% to the Weibull component. The estimated parameters ($\alpha \approx 2.15$, $k \approx 1.47$) successfully captured both the heavy-tailed extremes and the moderate body of the distribution. The model achieved an Akaike Information Criterion (AIC) of 512.4, suggesting a favorable trade-off between goodness-of-fit and model complexity compared to single-component alternatives.

Significance and Claims
The paper claims that the proposed mixture model offers a tractable yet flexible approach for analyzing complex datasets with heterogeneous tail structures. It bridges the gap between extreme value modeling and light-tailed distribution analysis. The authors assert that the model is particularly valuable in applications where catastrophic events (modeled by Fréchet) coexist with routine variations (modeled by Weibull), such as in hydrology, finance, and environmental science. The study concludes that the EM-based estimation procedure is effective for practical implementation, providing robust fits for both simulated and real-world observations.

Note: The paper explicitly states that no competing interests exist and that no external funding was received for this research.