Advancing Statistical Model Flexibility through the Development of the Exponentiated Generalized Topp Leone-Fréchet Distribution

Abstract

We propose a new flexible statistical model called the Exponentiated Generalized Topp Leone-Fréchet (EGTL-F) distribution, obtained by compounding the Exponentiated Generalized Topp Leone-G (EGTL-G) family with the heavy-tailed Fréchet distribution. The EGTL-F distribution combines the skewness-control properties of the Topp Leone transformation with the extreme-value tail behaviour of the Fréchet distribution, making it suitable for modelling positively skewed data with heavy tails. We derive its cumulative distribution function, probability density function, survival function, hazard function, quantile function, ordinary moments, order statistics, and Rényi entropy. Parameters are estimated using the method of Maximum Likelihood Estimation (MLE). A Monte Carlo simulation study shows that the bias and mean squared error (MSE) of the MLEs decrease as the sample size increases from n =10 to n = 100, confirming the consistency of the estimators. Two real datasets, the strength of 1.5 cm glass fibres and the relief times following an analgesic treatment, are used to compare the EGTL-F distribution against three existing competitor distributions: the Exponentiated Fréchet distribution (EFD), the Fréchet distribution (FD), and the Exponential distribution (ED). In both applications, the EGTL-F distribution achieves the highest log-likelihood and the lowest Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) values, indicating a superior fit. These results show that the EGTL-F distribution is a competitive and flexible model for right-skewed, heavy-tailed lifetime data.