Properties of fractional generalized entropy in ordered variables and symmetry testing

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.