Residual and Past Discrete Tsallis and Renyi Extropy with an Application to Softmax Function

Taghreed M Jawa; Nahid Fatima; Neveen Sayed-Ahmed; Ramy Aldallal; Mohamed Said Mohamed

doi:10.3390/e24121732

Residual and Past Discrete Tsallis and Renyi Extropy with an Application to Softmax Function

Entropy (Basel). 2022 Nov 27;24(12):1732. doi: 10.3390/e24121732.

Authors

Taghreed M Jawa¹, Nahid Fatima², Neveen Sayed-Ahmed¹, Ramy Aldallal³, Mohamed Said Mohamed⁴

Affiliations

¹ Department of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia.
² Department of Mathematics & Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia.
³ Department of Accounting, College of Business Administration in Hawtat Bani Tamim, Prince Sattam bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia.
⁴ Department of Mathematics, Faculty of Education, Ain Shams University, Cairo 11341, Egypt.

Abstract

In this paper, based on the discrete lifetime distribution, the residual and past of the Tsallis and Renyi extropy are introduced as new measures of information. Moreover, some of their properties and their relation to other measures are discussed. Furthermore, an example of a uniform distribution of the obtained models is given. Moreover, the softmax function can be used as a discrete probability distribution function with a unity sum. Thus, applying those measures to the softmax function for simulated and real data is demonstrated. Besides, for real data, the softmax data are fit to a convenient ARIMA model.

Keywords: ARIMA model; Renyi extropy; Tsallis entropy; extropy; past Tsallis entropy; past extropy; residual Tsallis entropy; residual extropy; softmax function.

Grants and funding

This study was supported by the Taif University Researchers Supporting Project number (TURSP-2020/318), Taif University, Taif, Saudi Arabia. The author Nahid Fatima would like to acknowledge the support of Prince Sultan University for paying the Article Processing Charges (APC) of this publication.