In the literature on distribution theory, numerous probability distributions are applied to predict and model real-world phenomena across diverse applied domains, including the medical and healthcare sectors. In the present paper, we develop a new distributional method, referred to as the new exponential power function distribution. The introduced model is incorporated using the transformed-transformer (T-X) approach. The density function of the newly presented distribution is investigated graphically, revealing three distinct patterns, namely symmetric, asymmetric, complex, and skewed shapes. Similarly, the hazard function patterns of the proposed model are also illustrated, which capture the increasing, unimodal, decreasing, and increasing shapes. Further, we developed several key properties such as the moments, quantile function, moment-generating function, and order statistics. Several classical and Bayesian estimation parameters of the new distribution are provided. A Monte Carlo simulation study is conducted to assess the efficiency of these estimators. Lastly, the practicality and efficiency of the novel distribution were validated using four datasets. It is found that the proposed distribution efficiently analyzed these datasets compared with competitive distributions.

Uncertainty measures are widely used in various statistical applications, including hypothesis testing and characterizations. Numerous generalizations of information measures with different extensions have been developed. Inspired by this, our study introduced the principle of the fractional generalized entropy measure and investigated its properties through stochastic comparisons and characterizations using order statistics and upper random variables. We explored the monotonicity and symmetry properties of the fractional generalized entropy, emphasizing conditions under which it uniquely identified the parent distribution. In the case of distributions that were completely continuous, The symmetrical nature of order statistics suggested that symmetry of the underpinning distribution. Based on the fractional generalized entropy measure in non-parametric estimate of order statistics, a new test for the symmetry hypothesis was put forward. This test offered the supremacy of not requiring the symmetry center to be specified. Additionally, an example of real-world data was shown to illustrate how the suggested technique might be applied.