1Department of Statistics, University of Peshawar, Peshawar, Pakistan
2Institute of Numerical Sciences, Kohat University of Science and Technology, Kohat, Pakistan
3Department of Mathematics, College of Science, Princess Nourah Bint Abdulrahman University, P.O. Box84428, 11671 Riyadh, Saudi Arabia
4Department of Mathematics, College of Science, Qassim University, 51452 Buraydah, Saudi Arabia
5Department of Operations and Management Research, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, 12613 Egypt
© The Author(s) 2025
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.