Properties of fractional generalized entropy in ordered variables and symmetry testing

Mohamed Said Mohamed; Muqrin A. Almuqrin; Mohamed Said Mohamed; Muqrin A. Almuqrin

doi:10.3934/math.2025053

AIMS Mathematics

2025, Volume 10, Issue 1: 1116-1141. doi: 10.3934/math.2025053

Previous Article Next Article

Research article Special Issues

Properties of fractional generalized entropy in ordered variables and symmetry testing

Mohamed Said Mohamed ¹,
Muqrin A. Almuqrin ^{2
,
,}

1.
Mathematics Department, Faculty of Education, Ain Shams University, Cairo 11341, Egypt
2.
Department of Mathematics, College of Science in Zulfi, Majmaah University, Majmaah 11952, Saudi Arabia

Received: 26 November 2024 Revised: 31 December 2024 Accepted: 09 January 2025 Published: 20 January 2025
MSC : 60E15, 62G10, 94A17

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.
- entropy,
- hazard rate function,
- stochastic order,
- order statistics,
- upper record values,
- symmetry testing
Citation: Mohamed Said Mohamed, Muqrin A. Almuqrin. Properties of fractional generalized entropy in ordered variables and symmetry testing[J]. AIMS Mathematics, 2025, 10(1): 1116-1141. doi: 10.3934/math.2025053

Related Papers:

Abstract

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.

References

[1]	M. AbaOud, M. A. Almuqrin, The weighted inverse Weibull distribution: Heavy-tailed characteristics, Monte Carlo simulation with medical application, Alex. Eng. J., 102 (2024), 99–107. https://doi.org/10.1016/j.aej.2024.05.056 doi: 10.1016/j.aej.2024.05.056
[2]	J. Ahmadi, Characterization results for symmetric continuous distributions based on the properties of k-records and spacings, Stat. Probabil. Lett., 162 (2020), 108764. https://doi.org/10.1016/j.spl.2020.108764 doi: 10.1016/j.spl.2020.108764
[3]	C. D. Aliprantis, O. Burkinshaw, Principles of real analysis, London: Edward Arnold, 1981.
[4]	M. A. Almuqrin, A new flexible distribution with applications to engineering data, Alex. Eng. J., 69 (2023), 371–382. https://doi.org/10.1016/j.aej.2023.01.046 doi: 10.1016/j.aej.2023.01.046
[5]	M. A. Almuqrin, Next-generation statistical methodology: Advances health science research, Alex. Eng. J., 108 (2024), 459–475. https://doi.org/10.1016/j.aej.2024.07.097 doi: 10.1016/j.aej.2024.07.097
[6]	G. Alomani, M. Kayid, Stochastic properties of fractional generalized cumulative residual entropy and its extensions, Entropy, 24 (2022), 1041. https://doi.org/10.3390/e24081041 doi: 10.3390/e24081041
[7]	A. Baklizi, A conditional distribution runs test for symmetry, J. Nonparametr. Stat., 15 (2003), 713–718. https://doi.org/10.1080/10485250310001634737 doi: 10.1080/10485250310001634737
[8]	A. Baklizi, Testing symmetry using a trimmed longest run statistic, Aust. N. Z. J. Stat., 49 (2007), 339–347. https://doi.org/10.1111/j.1467-842X.2007.00485.x doi: 10.1111/j.1467-842X.2007.00485.x
[9]	A. Baklizi, Improving the power of the hybrid test, Int. J. Contemp. Math. Sciences, 3 (2008), 497–499.
[10]	N. Balakrishnan, A. Selvitella, Symmetry of a distribution via symmetry of order statistics, Stat. Probabil. Lett., 129 (2017), 367–372. https://doi.org/10.1016/j.spl.2017.06.023 doi: 10.1016/j.spl.2017.06.023
[11]	F. Belzunce, R. E. Lillo, J. M. Ruiz, M. Shaked, Stochastic comparisons of nonhomogeneous processes, Probab. Eng. Inform. Sc., 15 (2001), 199–224. https://doi.org/10.1017/S0269964801152058 doi: 10.1017/S0269964801152058
[12]	V. Bozin, B. Milosevic, Y. Y. Nikitin, M. Obradovic, New characterization-based symmetry tests, Bull. Malays. Math. Sci. Soc., 43 (2020), 297–320. https://doi.org/10.1007/s40840-018-0680-3 doi: 10.1007/s40840-018-0680-3
[13]	W. H. Cheng, N. Balakrishnan, A modified sign test for symmetry, Commun. Stat.-Simul. C., 33 (2004), 703–709. https://doi.org/10.1081/SAC-200033302 doi: 10.1081/SAC-200033302
[14]	J. Corzo, G. Babativa, A modified runs test for symmetry, J. Stat. Comput. Sim., 83 (2013), 984–991. https://doi.org/10.1080/00949655.2011.647026 doi: 10.1080/00949655.2011.647026
[15]	X. J. Dai, C. Z. Niu, X. Guo, Testing for central symmetry and inference of the unknown center, Comput. Stat. Data An., 127 (2018), 15–31. https://doi.org/10.1016/j.csda.2018.05.007 doi: 10.1016/j.csda.2018.05.007
[16]	A. Di Crescenzo, M. Longobardi, On cumulative entropies, J. Stat. Plan. Infer., 139 (2009), 4072–4087. https://doi.org/10.1016/j.jspi.2009.05.038
[17]	A. Di Crescenzo, S. Kayal, A. Meoli, Fractional generalized cumulative entropy and its dynamic version, Commun. Nonlinear Sci., 102 (2021), 105899. https://doi.org/10.1016/j.cnsns.2021.105899 doi: 10.1016/j.cnsns.2021.105899
[18]	G. W. Cobb, The problem of the Nile: conditional solution to a change point problem, Biometrika, 65 (1978), 243–251. https://doi.org/10.1093/biomet/65.2.243 doi: 10.1093/biomet/65.2.243
[19]	M. Fashandi, J. Ahmadi, Characterizations of symmetric distributions based on Renyi entropy, Stat. Probabil. Lett., 82 (2012), 798–804. https://doi.org/10.1016/j.spl.2012.01.004 doi: 10.1016/j.spl.2012.01.004
[20]	J. D. Gibbons, S. Chakraborti, Nonparametric statistical inference, New York: Dekker, 1992.
[21]	C. Goffman, G. R. Pedrick, First course in functional analysis, Englewood Cliffs: Prentice-Hall, 1965.
[22]	P. Crzegorzewski, R. Wieczorkowski, Entropy-based goodness-of-fit test for exponentiality, Commun. Stat.-Theor. M., 28 (1999), 1183–1202. https://doi.org/10.1080/03610929908832351 doi: 10.1080/03610929908832351
[23]	N. Gupta, S. K. Chaudhary, Some characterizations of continuous symmetric distributions based on extropy of record values, Stat. Papers, 65 (2024), 291–308. https://doi.org/10.1007/s00362-022-01392-y doi: 10.1007/s00362-022-01392-y
[24]	I. A. Husseiny, H. M. Barakat, M. Nagy, A. H. Mansi, Analyzing symmetric distributions by utilizing extropy measures based on order statistics, J. Radiat. Res. Appl. Sc., 17 (2024), 101100. https://doi.org/10.1016/j.jrras.2024.101100 doi: 10.1016/j.jrras.2024.101100
[25]	J. Jose, E. I. A. Sathar, Symmetry being tested through simultaneous application of upper and lower k-records in extropy, J. Stat. Comput. Sim., 92 (2022), 830–846. https://doi.org/10.1080/00949655.2021.1975283 doi: 10.1080/00949655.2021.1975283
[26]	J. Jozefczyk, Data driven score tests for univariate symmetry based on nonsmooth functions, Probab. Math. Stat., 32 (2012), 301–322.
[27]	J. T. Machado, Fractional order generalized information, Entropy, 16 (2014), 2350–2361. https://doi.org/10.3390/e16042350 doi: 10.3390/e16042350
[28]	M. Mahdizadeh, E. Zamanzade, Estimation of a symmetric distribution function in multistage ranked set sampling, Stat. Papers, 61 (2020), 851–867. https://doi.org/10.1007/s00362-017-0965-x doi: 10.1007/s00362-017-0965-x
[29]	T. P. McWilliams, A distribution-free test for symmetry based on a runs statistic, J. Am. Stat. Assoc., 85 (1990), 1130–1133. https://doi.org/10.2307/2289611 doi: 10.2307/2289611
[30]	R. Modarres, J. L. Gastwirth, A modified runs test for symmetry, Stat. Probabil. Lett., 31 (1996), 107–112. https://doi.org/10.1016/S0167-7152(96)00020-X doi: 10.1016/S0167-7152(96)00020-X
[31]	M. S. Mohamed, On concomitants of ordered random variables under general forms of Morgenstern family, Filomat, 33 (2019), 2771–2780. https://doi.org/10.2298/FIL1909771M doi: 10.2298/FIL1909771M
[32]	M. S. Mohamed, A measure of inaccuracy in concomitants of ordered random variables under Farlie-Gumbel-Morgenstern family, Filomat, 33 (2019), 4931–4942. https://doi.org/10.2298/FIL1915931M doi: 10.2298/FIL1915931M
[33]	M. S. Mohamed, On cumulative residual Tsallis entropy and its dynamic version of concomitants of generalized order statistics, Comm. Statist. Theory Methods, 51 (2022), 2534–2551. https://doi.org/10.1080/03610926.2020.1777306 doi: 10.1080/03610926.2020.1777306
[34]	J. Navarro, Y. del Aguila, M. Asadi, Some new results on the cumulative residual entropy, J. Stat. Plan. Infer., 140 (2010), 310–322. https://doi.org/10.1016/j.jspi.2009.07.015 doi: 10.1016/j.jspi.2009.07.015
[35]	H. A. Noughabi, Tests of symmetry based on the sample entropy of order statistics and power comparison, Sankhya B, 77 (2015), 240–255. https://doi.org/10.1007/s13571-015-0103-5 doi: 10.1007/s13571-015-0103-5
[36]	H. A. Noughabi, J. Jarrahiferiz, Extropy of order statistics applied to testing symmetry, Commun. Stat.-Simul. C., 51 (2022), 3389–3399. https://doi.org/10.1080/03610918.2020.1714660 doi: 10.1080/03610918.2020.1714660
[37]	S. Park, A goodness-of-fit test for normality based on the sample entropy of order statistics, Stat. Probabil. Lett., 44 (1999), 359–363. https://doi.org/10.1016/S0167-7152(99)00027-9 doi: 10.1016/S0167-7152(99)00027-9
[38]	G. Psarrakos, J. Navarro, Generalized cumulative residual entropy and record values, Metrika, 76 (2013), 623–640. https://doi.org/10.1007/s00184-012-0408-6 doi: 10.1007/s00184-012-0408-6
[39]	G. Psarrakos, A. Toomaj, On the generalized cumulative residual entropy with applications in actuarial science, J. Comput. Appl. Math., 309 (2017), 186–199. https://doi.org/10.1016/j.cam.2016.06.037 doi: 10.1016/j.cam.2016.06.037
[40]	M. Rao, Y. Chen, B. C. Vemuri, F. Wang, Cumulative residual entropy: a new measure of information, IEEE T. Inform. Theory, 50 (2004), 1220–1228. https://doi.org/10.1109/TIT.2004.828057 doi: 10.1109/TIT.2004.828057
[41]	H. H. Sakr, M. S. Mohamed, Sharma-Taneja-Mittal entropy and its application of obesity in Saudi Arabia, Mathematics, 12 (2024), 2639. https://doi.org/10.3390/math12172639 doi: 10.3390/math12172639
[42]	M. Shaked, J. G. Shanthikumar, Stochastic orders and their applications, San Diego: Academic Press, 1994.
[43]	C. E. Shannon, A mathematical theory of communication, Bell System Technical Journal, 27 (1948), 379–423. http://doi.org/10.1002/j.1538-7305.1948.tb01338.x doi: 10.1002/j.1538-7305.1948.tb01338.x
[44]	I. H. Tajuddin, Distribution-free test for symmetry based on the Wilcoxon two-sample test, J. Appl. Stat., 21 (1994), 409–415. https://doi.org/10.1080/757584017 doi: 10.1080/757584017
[45]	A. Toomaj, A. Di Crescenzo, Generalized entropies, variance and applications, Entropy, 22 (2020), 709. https://doi.org/10.3390/e22060709
[46]	M. R. Ubriaco, Entropies based on fractional calculus, Phys. Lett. A, 373 (2009), 2516–2519. https://doi.org/10.1016/j.physleta.2009.05.026 doi: 10.1016/j.physleta.2009.05.026
[47]	O. Vasicek, A test for normality based on sample entropy, J. R. Stat. Soc. B, 38 (1976), 54–59. https://doi.org/10.1111/j.2517-6161.1976.tb01566.x doi: 10.1111/j.2517-6161.1976.tb01566.x
[48]	H. Xiong, P. J. Shang, Y. L. Zhang, Fractional cumulative residual entropy, Commun. Nonlinear Sci., 78 (2019), 104879. https://doi.org/10.1016/j.cnsns.2019.104879 doi: 10.1016/j.cnsns.2019.104879
[49]	P. H. Xiong, W. W. Zhuang, G. X. Qiu, Testing symmetry based on the extropy of record values, J. Nonparametr. Stat., 33 (2021), 134–155. https://doi.org/10.1080/10485252.2021.1914338 doi: 10.1080/10485252.2021.1914338
[50]	S. Yin, Y. D. Zhao, A. Hussain, K. Ullah, Comprehensive evaluation of rural regional integrated clean energy systems considering multi-subject interest coordination with pythagorean fuzzy information, Eng. Appl. Artif. Intel., 138 (2024), 109342. https://doi.org/10.1016/j.engappai.2024.109342 doi: 10.1016/j.engappai.2024.109342

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)