Statistical Applications of the Ujlayan–Dixit Fractional Lomax Probability Distribution

The Ujlayan–Dixit (UD) fractional calculus provides a powerful fractional extension of the Lomax distribution, offering a suitable framework for representing complex behaviors beyond classical approaches. In this paper, we adopt the UD fractional Lomax distribution and establish its statistical theory. Based on the adopted density, we derive closed-form expressions for the cumulative distribution, survival, and hazard functions, as well as the mode. Several UD fractional statistical measures of the Lomax random variable are derived, including the fractional moments, fractional information theoretic measures, including UD fractional Shannon and Tsallis entropy measures, and the probability density function of the kth order statistic under the UD fractional framework. Finally, a real data application concerning the time to break down an insulating fluid is used to illustrate the usefulness of the proposed distribution in modeling real data applications. The fitting performance of the suggested model is compared with several extensions of the Lomax distribution. The comparative results show that the UD fractional Lomax distribution outperforms several well-known extensions of Lomax distribution. This framework provides researchers with many robust tools for advanced reliability assessment, uncertainty quantification, and risk modeling, providing insights into phenomena not captured by the classical Lomax distribution. Moreover, when the fractional parameter q→1−, the proposed approach converges to the classical Lomax results, bridging fractional and classical perspectives.