Extropy-based characterization and nonparametric testing of symmetry in continuous distributions via consecutive systems

Tahani Alshathri; Mohamed Kayid; Tahani Alshathri; Mohamed Kayid

doi:10.3934/math.2026438

AIMS Mathematics

2026, Volume 11, Issue 4: 10638-10667. doi: 10.3934/math.2026438

Previous Article Next Article

Research article Special Issues

Extropy-based characterization and nonparametric testing of symmetry in continuous distributions via consecutive systems

Tahani Alshathri ^,,
Mohamed Kayid

Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 13 December 2025 Revised: 28 March 2026 Accepted: 03 April 2026 Published: 20 April 2026
MSC : 62G10, 94A17, 62E10, 62N05

This paper develops a unified information-theoretic framework for the characterization of continuous symmetric distributions based on extropy and cumulative extropy measures. The proposed approach is formulated through the structural properties of consecutive (n−r)-out-of-$ n $:G and (n−r)-out-of-$ n $:F systems, which naturally arise in reliability and lifetime analysis. We establish necessary and sufficient conditions under which the equality of differential extropy, cumulative residual extropy, and cumulative past extropy uniquely characterizes distributional symmetry for all $ n\ge 2r $. These results contribute to statistical distribution theory by providing new system-based information characterizations of symmetry. Building on the theoretical developments, we introduce a new nonparametric test for symmetry constructed from differences between estimated extropy measures associated with consecutive systems. The proposed test is shown to be consistent, and extensive Monte Carlo simulations demonstrate that it achieves competitive and often superior power compared with several existing symmetry tests, particularly in detecting mild and near-symmetric departures. Applications to real datasets further confirm the practical effectiveness and interpretability of the proposed methodology. The proposed framework naturally admits extensions in multivariate settings and to other information measures.
- symmetric distributions,
- extropy,
- cumulative extropy,
- consecutive reliability systems,
- nonparametric symmetry testing,
- distribution characterization
Citation: Tahani Alshathri, Mohamed Kayid. Extropy-based characterization and nonparametric testing of symmetry in continuous distributions via consecutive systems[J]. AIMS Mathematics, 2026, 11(4): 10638-10667. doi: 10.3934/math.2026438

Related Papers:

Abstract

This paper develops a unified information-theoretic framework for the characterization of continuous symmetric distributions based on extropy and cumulative extropy measures. The proposed approach is formulated through the structural properties of consecutive (n−r)-out-of-$ n $:G and (n−r)-out-of-$ n $:F systems, which naturally arise in reliability and lifetime analysis. We establish necessary and sufficient conditions under which the equality of differential extropy, cumulative residual extropy, and cumulative past extropy uniquely characterizes distributional symmetry for all $ n\ge 2r $. These results contribute to statistical distribution theory by providing new system-based information characterizations of symmetry. Building on the theoretical developments, we introduce a new nonparametric test for symmetry constructed from differences between estimated extropy measures associated with consecutive systems. The proposed test is shown to be consistent, and extensive Monte Carlo simulations demonstrate that it achieves competitive and often superior power compared with several existing symmetry tests, particularly in detecting mild and near-symmetric departures. Applications to real datasets further confirm the practical effectiveness and interpretability of the proposed methodology. The proposed framework naturally admits extensions in multivariate settings and to other information measures.

References

[1]	C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., 27 (1948), 623–656. https://doi.org/10.1002/j.1538-7305.1948.tb00917.x doi: 10.1002/j.1538-7305.1948.tb00917.x
[2]	F. Lad, G. Sanfilippo, G. Agro, Extropy: complementary dual of entropy, Statist. Sci., 30 (2015), 40–58. https://doi.org/10.1214/14-sts430 doi: 10.1214/14-sts430
[3]	P. Xiong, W. Zhuang, G. 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
[4]	M. Z. Raqab, G. Qiu, On extropy properties of ranked set sampling, Statistics, 53 (2019), 210–226. https://doi.org/10.1080/02331888.2018.1533963 doi: 10.1080/02331888.2018.1533963
[5]	J. Jose, E. I. A. Sathar, Symmetry being tested through simultaneous application of upper and lower k-records in extropy, J. Stat. Comput. Simul., 92 (2022), 830–846. https://doi.org/10.1080/00949655.2021.1975283 doi: 10.1080/00949655.2021.1975283
[6]	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
[7]	S. M. A. Jahanshahi, H. Zarei, A. H. Khammar, On cumulative residual extropy, Probab. Eng. Inf. Sci., 34 (2020), 605–625. https://doi.org/10.1017/s0269964819000196 doi: 10.1017/s0269964819000196
[8]	K. H. Jung, H. Kim, Linear consecutive-k-out-of-n: F system reliability with common-mode forced outages, Reliab. Eng. Syst. Safety, 41 (1993), 49–55. https://doi.org/10.1016/0951-8320(93)90017-s doi: 10.1016/0951-8320(93)90017-s
[9]	J. Shen, M. J. Zuo, Optimal design of series consecutive-k-out-of-n: G systems, Reliab. Eng. Syst. Safety, 45 (1994), 277–283. https://doi.org/10.1016/0951-8320(94)90144-9 doi: 10.1016/0951-8320(94)90144-9
[10]	W. Kuo, M. J. Zuo, Optimal reliability modeling: principles and applications, John Wiley and Sons, New York, 2003
[11]	S. Eryilmaz, Reliability properties of consecutive k-out-of-n systems of arbitrarily dependent components, Reliab. Eng. Syst. Safety, 94 (2009), 350–356. https://doi.org/10.1016/j.ress.2008.03.027 doi: 10.1016/j.ress.2008.03.027
[12]	J. Honerkamp, Statistical physics: an advanced approach with applications, Springer, New York, 2012. https://doi.org/10.1007/978-3-642-28684-1
[13]	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
[14]	M. Fashandi, J. Ahmadi, Characterizations of symmetric distributions based on Rényi entropy, Stat. Probab. Lett., 82 (2012), 798–804. https://doi.org/10.1016/j.spl.2012.01.004 doi: 10.1016/j.spl.2012.01.004
[15]	N. Balakrishnan, A. Selvitella, Symmetry of a distribution via symmetry of order statistics, Stat. Probab. Lett., 129 (2017), 367–372. https://doi.org/10.1016/j.spl.2017.06.023 doi: 10.1016/j.spl.2017.06.023
[16]	J. Ahmadi, M. Fashandi, Characterization of symmetric distributions based on some information measure properties of order statistics, Phys. A, 517 (2019), 141–152. https://doi.org/10.1016/j.physa.2018.11.009 doi: 10.1016/j.physa.2018.11.009
[17]	M. Kayid, M. Shrahili, Information properties of consecutive systems using fractional generalized cumulative residual entropy, Fractal Fract., 8 (2024), 568. https://doi.org/10.3390/fractalfract8100568 doi: 10.3390/fractalfract8100568
[18]	M. Kayid, M. A. Alshehri, Shannon differential entropy properties of consecutive k-out-of-n: G systems, Oper. Res. Lett., 57 (2024), 107190. https://doi.org/10.1016/j.orl.2024.107190 doi: 10.1016/j.orl.2024.107190
[19]	U. Kamps, Characterizations of distributions by recurrence relations and identities for moments of order statistics, In: N. Balakrishnan, C. R. Rao, Handbook of statistics, Elsevier, 16 (1998), 291–311. https://doi.org/10.1016/s0169-7161(98)16012-1
[20]	J. S. Hwang, G. D. Lin, On a generalized moment problem Ⅱ, Proc. Amer. Math. Soc., 91 (1984), 577–580. https://doi.org/10.2307/2044804 doi: 10.2307/2044804
[21]	F. Alrewely, M. Kayid, Extropy analysis in consecutive r-out-of-n: G systems with applications in reliability and exponentiality testing, AIMS Math., 10 (2025), 6040–6068. https://doi.org/10.3934/math.2025276 doi: 10.3934/math.2025276
[22]	G. Alomani, F. Alrewely, M. Kayid, Information-theoretic reliability analysis of linear consecutive r-out-of-n: F systems and uniformity testing, Entropy, 27 (2025), 590. https://doi.org/10.3390/e27060590 doi: 10.3390/e27060590
[23]	M. A. Kon, Probability distributions in quantum statistical mechanics, Springer, New York, 1985. https://doi.org/10.1007/BFb0077154
[24]	J. Naudts, Generalised thermostatistics, Springer, New York, 2011. https://doi.org/10.1007/978-0-85729-355-8
[25]	S. Sirca, Probability for physicists, Springer, New York, 2016. https://doi.org/10.1007/978-3-319-31611-6.
[26]	J. Ahmadi, Characterization results for symmetric continuous distributions based on the properties of k-records and spacings, Stat. Probab. Lett., 162 (2020), 108764. https://doi.org/10.1016/j.spl.2020.108764 doi: 10.1016/j.spl.2020.108764
[27]	O. Vasicek, A test for normality based on sample entropy, J. R. Stat. Soc., Ser. 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
[28]	M. Li, W. Chen, 3×3 optimal ranked set sampling design with cycles and best linear invariant estimators of the parameters for normal distribution, Stat. Probab. Lett., 224 (2025), 110455. https://doi.org/10.1016/j.spl.2025.110455 doi: 10.1016/j.spl.2025.110455
[29]	H. A. Noughabi, N. R. Arghami, Testing exponentiality based on characterizations of the exponential distribution, J. Stat. Comput. Simul., 81 (2011), 1641–1651. https://doi.org/10.1080/00949655.2010.498373 doi: 10.1080/00949655.2010.498373
[30]	N. Ebrahimi, K. Pflughoeft, E. S. Soofi, Two measures of sample entropy, Stat. Probab. Lett., 20 (1994), 225–234. https://doi.org/10.1016/0167-7152(94)90046-9 doi: 10.1016/0167-7152(94)90046-9
[31]	M. Amiri, B. E. Khaledi, A new test for symmetry against right skewness, J. Stat. Comput. Simul., 86 (2016), 1479–1496. https://doi.org/10.1080/00949655.2015.1071374 doi: 10.1080/00949655.2015.1071374
[32]	T. P. McWilliams, A distribution-free test for symmetry based on a runs statistic, J. Amer. Stat. Assoc., 85 (1990), 1130–1133. https://doi.org/10.1080/01621459.1990.10474985 doi: 10.1080/01621459.1990.10474985
[33]	A. Baklizi, A conditional distribution-free runs test for symmetry, J. Nonparametr. Stat., 15 (2003), 713–718. https://doi.org/10.1080/10485250310001634737 doi: 10.1080/10485250310001634737
[34]	J. D. Gibbons, S. Chakraborti, Nonparametric statistical inference, Marcel Dekker, New York, 1992.
[35]	I. H. Tajuddin, Distribution-free test for symmetry based on Wilcoxon two-sample test, J. Appl. Stat., 21 (1994), 409–415. https://doi.org/10.1080/757584017 doi: 10.1080/757584017
[36]	W. H. Cheng, N. Balakrishnan, A modified sign test for symmetry, Commun. Stat.—Simul. Comput., 33 (2004), 703–709. https://doi.org/10.1081/sac-200033302 doi: 10.1081/sac-200033302
[37]	R. Modarres, J. L. Gastwirth, Hybrid test for the hypothesis of symmetry, J. Appl. Stat., 25 (1998), 777–783. https://doi.org/10.1080/02664769822765 doi: 10.1080/02664769822765
[38]	A. Baklizi, Testing symmetry using a trimmed longest run statistic, Aust. NZ 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
[39]	A. Baklizi, Improving the power of the hybrid test of symmetry, Int. J. Contemp. Math. Sci., 3 (2008), 497–499.
[40]	J. Corzo, G. Babativa, A modified runs test for symmetry, J. Stat. Comput. Simul., 83 (2013), 984–991. https://doi.org/10.1080/00949655.2011.647026 doi: 10.1080/00949655.2011.647026
[41]	G. Qiu, K. Jia, Extropy estimators with applications in testing uniformity, J. Nonparametr. Stat., 30 (2018), 182–196. https://doi.org/10.1080/10485252.2017.1404063 doi: 10.1080/10485252.2017.1404063
[42]	J. F. Lawless, Statistical models and methods for lifetime data, John Wiley and Sons, New York, 2011.
[43]	G. W. Cobb, The problem of the Nile: Conditional solution to a changepoint problem, Biometrika, 65 (1978), 243–251. https://doi.org/10.2307/2335202 doi: 10.2307/2335202

Reader Comments

Your name:*

Email:*
© 2026 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)