The G-Rank-Mapped Transmuted Fréchet Weibull Distribution under Cubic Model: Mathematical Theory, Computational Statistics, and Sustainability Data Analysis

Abstract

This work introduces and examines a new probability distribution from the G-rank-mapped transmuted Fréchet-Weibull (G-RTFW) family, using both quadratic and cubic models.  The Quadratic Rank Transmuted Fréchet-Weibull (QRTFW) and Cubic Rank Transmuted Fréchet-Weibull (CRTFW) distributions are specifically investigated, with the current study focused solely on the CRTFW model. We calculated and investigated a wide range of statistical and mathematical aspects of the CRTFW distribution, including its moments, moment generating function, characteristic function, quantile function, mode, random variate generation, hazard rate function, entropy, and order statistics.  The suggested model is highly adaptable, capable of simulating both unimodal and bimodal behaviors, as well as tolerating symmetric and asymmetric data structures with variable degrees of kurtosis. The maximum likelihood method was used for parameter estimation, and a full Monte Carlo simulation study was carried out to assess the estimators' performance and efficiency under various circumstances.  The simulation results showed that the estimators performed effectively, with bias and mean square error reducing as the sample size increased. To demonstrate the CRTFW distribution's practical application, two real-world sustainability-related datasets were examined.  The CRTFW model outperformed numerous well-known lifetime and reliability distributions in terms of flexibility and goodness-of-fit, emphasizing its use for both theoretical and applied data modeling.