DEVELOPMENT OF THE WEIBULL EXPONENTIATED FRÉCHET DISTRIBUTION (WEFD): STATISTIC AND APPLICATIONS

Abstract

This research introduced and studied the Weibull Exponentiated Fréchet Distribution (WEFD), a new five-parameter continuous probability distribution constructed by applying the Weibull exponentiated generator to the classical Fréchet baseline. The proposed model is designed to accommodate a broad range of distributional shapes and hazard rate behaviours, including those characterised by negative skewness and heavy tails, which existing Fréchet-type models cannot adequately represent. A comprehensive treatment of the mathematical and statistical properties of the WEFD is provided, including the probability density function, cumulative distribution function, quantile function, moments, and related descriptive measures. The parameters of the distribution were estimated by maximum likelihood, and the finite-sample performance of the estimators was assessed through an extensive Monte Carlo simulation study comprising 1,000 replications across multiple sample sizes. The simulation results confirmed that the estimators are consistent, with bias and mean squared error both decreasing toward zero as the sample size increases. The practical relevance and superiority of the WEFD are demonstrated through application to two real datasets: glass fiber strength measurements and a Civil Engineering hailing dataset. In both cases, the WEFD is compared against four competing distributions, namely the Fréchet, Exponentiated Fréchet, Kumaraswamy Fréchet, and Marshall–Olkin Fréchet distributions, using the negative log-likelihood, Akaike Information Criterion, Bayesian Information Criterion, Kolmogorov–Smirnov statistic, Cramér–von Mises statistic, and Anderson–Darling statistic as model selection criteria, and the results show that it consistently gives the best fit. The WEFD outperformed all competing models on the combined evidence from these criteria across both datasets, demonstrating its flexibility, goodness of fit, and suitability for complex lifetime and failure-time datasets.