A Dual Measure of Fuzzy Entropy: Fuzzy Extropy

[1] V. Arya, S. Kumar, Knowledge measure and entropy: A complementary concept in fuzzy theory, Granular Computing,
6(3) (2021), 631-643. https://doi.org/10.1007/s41066-020-00221-7

[2] N. Balakrishnan, F. Buono, M. Longobardi, On weighted extropies, Communications in Statistics Theory and
Methods, 51(18) (2022), 6250-6267. https://doi.org/10.1080/03610926.2020.1860222

[3] N. Balakrishnan, F. Buono, M. Longobardi, On Tsallis extropy with an application to pattern recognition, Statistics
and Probability Letters, 180 (2022), 109241. https://doi.org/10.1016/j.spl.2021.109241

[4] F. Buono, M. Longobardi, A dual measure of uncertainty: The deng extropy, Entropy, 22 (2020), 582. https:
//doi.org/10.3390/e22050582

[5] L. Chen, Y. Deng, Entropy of random permutation set, Communications in Statistics-Theory and Methods, 53(11)
(2024), 4127-4146. https://doi.org/10.1080/03610926.2023.2173975

[6] A. Deluca, S. Tennini, A definition of nonprobabilistic entropy in the setting of fuzzy sets theory, Information and
Control, 20 (1972), 301-312. https://doi.org/10.1016/B978-1-4832-1450-4.50020-1

[7] A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping, In Classic Works of the Dempster-
Shafer Theory of Belief Functions, (2008), 57-72. https://doi.org/10.1007/978-3-540-44792-4_3.

[8] Y. Deng, Deng entropy, Chaos, Solitons and Fractals, 91 (2016), 549-53. https://doi.org/10.1016/j.chaos.
2016.07.014

[9] Y. Deng, Information volume of mass function, International Journal of Computers Communications and Control,
15(6) (2020), 3983. https://doi.org/10.15837/ijccc.2020.6.3983

[10] D. Dubois, H. Prade, A class of fuzzy measures based on triangular norms, International Journal of General
Systems, 8 (1982), 43-61. https://doi.org/10.1080/03081078208934833

[11] D. Dubois, H. Prade, Th´eorie des Possibilit´es, Paris, France: Masson, (1985). https://hal.science/
hal-04058264

[12] N. Ebrahimi, How to measure uncertainty in the residual lifetime distribution, Sankhy´a: The Indian Journal of
Statistics, 58 (1996), 48-56. https://www.jstor.org/stable/i25051076

[13] O. Kamari, F. Buono, On extropy of past lifetime distribution, Ricerche di Matematica, 70 (2021), 505-515.
https://doi.org/10.1007/s11587-020-00488-7

[14] A. Kaufmann, Introduction to the theory of fuzzy subsets-fundamental theoretical elements, Academic, New York,
1 (1975). https://archive.org/details/introductiontoth0000kauf

[15] M. R. Kazemi, S. Tahmasebi, F. Buono, M. Longobardi, Fractional deng entropy and extropy and some applications,
Entropy, 23(5) (2021), 623. https://doi.org/10.3390/e23050623

[16] A. S. Krishnan, S. M. Sunoj, N. U. Nair, Some reliability properties of extropy for residual and past lifetime
random variables, Journal of the Korean Statistical Society, 49 (2020), 457-474. https://doi.org/10.1007/
s42952-019-00023-x

[17] F. Lad, G. Sanfilippo, G. Agr´o, Extropy: Complementary dual of entropy, Statistical Science, 30(1) (2015), 40-58.
https://doi.org/10.1214/14-STS430

[18] P. Li, B. Liu, Entropy of credibility distributions for fuzzy variables, IEEE Transactions on Fuzzy Systems, 16
(2008), 123-129. https://doi.org/10.1109/TFUZZ.2007.894975

[19] B. Liu, Y. K. Liu, Expected value of fuzzy variable and fuzzy expected value models, IEEE Transactions on Fuzzy
Systems, 10(4) (2002), 445-450. https://doi.org/10.1109/TFUZZ.2002.800692

[20] J. Liu, F. Xiao, Information volume of mass function based on extropy, Soft Computing, 26(5) (2022), 2409-2418.
https://doi.org/10.1007/s00500-021-06410-z

[21] J. Liu, F. Xiao, R´enyi extropy, Communications in Statistics-Theory and Methods, 52(16) (2023), 5836-5847.
https://www.tandfonline.com/toc/lsta20/52/16

[22] S. Nahmias, Fuzzy variables, Fuzzy Sets and Systems, 1 (1978), 97-110. https://doi.org/10.1016/
0165-0114(78)90011-8

[23] E. Y. Ozkan, Inequalities for approximation of new defined fuzzy post-quantum Bernstein polynomials via intervalvalued fuzzy numbers, Symmetry, 14 (2022), 696. https://doi.org/10.3390/sym14040696

[24] E. Y. Ozkan, Approximation by fuzzy (p, q)-Bernstein-Chlodowsky operators, Sahand Communications in Mathematical Analysis, 19(2) (2022), 113-132. https://doi.org/10.22130/scma.2022.524506.910

[25] E. Y. Ozkan, B. Hazarika, Approximation results by fuzzy Bernstein type rational functions via intervalvalued fuzzy
number, Soft Computing, 27 (2023), 6893-6901. https://doi.org/10.1007/s00500-023-08013-2

[26] S. K. Pal, B. Chakraborty, Fuzzy set theoretic measure for automatic feature evaluation, IEEE Transactions on
Systems, Man, and Cybernetics, SMC, 16(5) (1986), 754-760. http://hdl.handle.net/10263/921

[27] N. R. Pal, S. K. Pal, Object background segmentation using a new definition of entropy, IEE Proceedings E
(Computers and Digital Techniques), 136 (1989), 284-295. https://repository.ias.ac.in/26062/

[28] N. R. Pal, S. K. Pal, Higher order fuzzy entropy and hybrid entropy of a set, Information Sciences, 61 (1992),
211-231. https://doi.org/10.1016/0020-0255(92)90051-9

[29] L. Pan, Y. Deng, A new belief entropy to measure uncertainty of basic probability assignments based on belief
function and plausibility function, Entropy, 20(11) (2018), 842. https://doi.org/10.3390/e20110842

[30] G. Qiu, The extropy of order statistics and record values, Statistics and Probability Letters, 120 (2017), 52-60.
https://doi.org/10.1016/j.spl.2016.09.016

[31] A. R´enyi, On measures of entropy and information, In Proceedings of the Fourth Berkeley Symposium on
Mathematical Statistics and Probability, Contributions to the Theory of Statistics, 1 (1961), 547-561. http:
//l.academicdirect.org/Horticulture/GAs/Refs/Renyi_1961.pdf

[32] G. Shafer, A mathematical theory of evidence, Princeton University Press, 42 (1976). https://www.ams.org/
journals/bull/1977-83-04/S0002-9904-1977-14338-3/S0002-9904-1977-14338-3.pdf

[33] C. E. Shannon, A mathematical theory of communication, The Bell System Technical Journal, 27 (1948), 379-423.
https://doi.org/10.1002/j.1538-7305.1948.tb01338.x

[34] C. E. Shannon, W. Weaver, The mathematical theory of communication, University of Illinois Press, 27 (1949),
379-423, 623-656. https://psycnet.apa.org/record/1950-04584-000

[35] K. S. Song, R´enyi information, Loglikelihood and an Intrinsic Distribution Measure. Journal of Statistical Planning
and Inference, 93(1-2) (2001), 51-69. https://doi.org/10.1016/S0378-3758(00)00169-5

[36] C. Tsallis, Possible generalization of Boltzmann-Gibbs statistics, Journal of Statistical Physics, 52 (1988), 479-487.
http://dx.doi.org/10.1007/BF01016429

[37] M. R. Ubriaco, Entropies based on fractional calculus, Physics Letters A, 373 (2009), 2516-2519. https://doi.
org/10.1016/j.physleta.2009.05.026

[38] N. M. Vaselabadi, S. Tahmasebi, M. R. Kazemi, F. Buono, Results on varextropy measure of random variables,
Entropy, 23(3) (2021), 356. https://doi.org/10.3390/e23030356

[39] Y. Xue, Y. Deng, Tsallis extropy, Communications in Statistics-Theory and Methods, 52(3) (2023), 751-762.
https://doi.org/10.1080/03610926.2021.1921804

[40] P. Yu, Y. Deng, On the scaled R´enyi entropy and application, Communications in Statistics Theory and Methods,
1 (2024). https://doi.org/10.1080/03610926.2024.2301986

[41] L. A. Zadeh, Fuzzy sets, Information and Control, 8(3) (1965), 338-353. https://doi.org/10.1016/
S0019-9958(65)90241-X

[42] L. A. Zadeh, Probability measures of fuzzy events, Journal of Mathematical Analysis and Applications, 2 (1968),
421-427. https://doi.org/10.1016/0022-247X(68)90078-4

[43] L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems, 1 (1978), 3-28. https:
//doi.org/10.1016/0165-0114(78)90029-5

[44] R. Zamini, S. Ghafouri, F. Goodarzi, Varextropy of doubly truncated random variable, arXiv preprintarXiv:
2407.21423, (2024). https://doi.org/10.48550/arXiv.2407.21423

[45] Q. Zhou, Y. Deng, Belief extropy: Measure uncertainty from negation, Communications in Statistics-Theory and
Methods, 52(11) (2023), 3825-3847. https://doi.org/10.1080/03610926.2021.1980049