The Beta Transmuted Fréchet Distribution: Statistical Inference and Applications

Abstract

This study introduces a novel probability distribution known as the Beta Transmuted Fréchet Distribution (BTFr), which integrates the Rank Transmutation Map (RTM) and Beta Generalized (B-G) techniques with the classical Fréchet (Fr) distribution to enhance the capability and flexibility of statistical modeling for complex data characteristics (e.g., heavy tails and asymmetric skewness). The proposed distribution encompasses several well-known distributions as special cases and is characterized by five parameters, making it more flexible in representing asymmetric skewness and heavy tails compared to classical distributions like the original Fréchet (Fr) and its common generalizations (BFr, TFr, EFr). The study derives the mathematical properties of the new distribution, including survival and hazard functions, moments, quantiles, moment-generating functions, random number generation, and order statistics. Parameter estimation is performed using the maximum likelihood method (MLE), with the observed information matrix employed to estimate approximate variances of the estimators and construct confidence intervals. A simulation study evaluates the performance of the estimators in finite samples and analyzes their asymptotic efficiency, demonstrating their consistency and efficiency. The study also includes a practical application to two real-world datasets to illustrate the significance, efficiency, and utility of the proposed distribution in modeling complex data. The performance of BTFr is compared to its sub-distributions using well-known goodness-of-fit tests and likelihood ratio (LR) tests. The results confirm the superior fit and statistical significance of BTFr in representing and modeling extreme-value data, with the additional parameters providing a meaningful improvement in flexibility compared to its sub-distributions. The study recommends using this distribution in fields such as financial risk analysis and engineering reliability.