A New T-X Family of Distributions: Structural Properties and Applications

This study introduces a new class of probability distributions within the transformed-transformer (T - X) family framework, using the paralogistic distribution as the baseline generator. The proposed Paralogistic-X family provides a flexible model capable of capturing various distributional shapes such as skewness and heavy tails. General expressions for its structural properties including the density, distribution, quantile function, moments, and entropy measures are developed in terms of the baseline distribution. A special submodel, the Paralogistic--Burr III (PBIII) distribution, is examined in detail. Maximum likelihood estimation is used for parameter inference, and a simulation study is conducted to evaluate the estimators' performance under varying sample sizes. The results confirm the consistency and efficiency of the estimators across different settings. To assess its practical utility, the PBIII distribution is applied to real-world datasets and compared with four established competing distributions. The comparison, based on both graphical tools and model selection criteria such as AIC, BIC, and log-likelihood, shows that the proposed model offers superior fitting capabilities in the two cases. The findings highlight the versatility and robustness of the Paralogistic-X family for statistical modeling.