Analysis of information measures using generalized type-Ⅰ hybrid censored data

Baria A. Helmy; Amal S. Hassan; Ahmed K. El-Kholy; Rashad A. R. Bantan; Mohammed Elgarhy; Baria A. Helmy; Amal S. Hassan; Ahmed K. El-Kholy; Rashad A. R. Bantan; Mohammed Elgarhy

doi:10.3934/math.20231034

AIMS Mathematics

2023, Volume 8, Issue 9: 20283-20304. doi: 10.3934/math.20231034

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Analysis of information measures using generalized type-Ⅰ hybrid censored data

1.
Department of Mathematics, Al-Azhar University (girls branch), Faculty of Science, Cairo 11651, Egypt
2.
Department of Mathematical Statistics, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza 12613, Egypt
3.
Department of Marine Geology, Faulty of Marine Science, King Abdulaziz University, Jeddah 21551, Saudi Arabia
4.
Mathematics and Computer Science Department, Faculty of Science, Beni-Suef University, Beni-Suef 62521, Egypt

Received: 10 March 2023 Revised: 30 May 2023 Accepted: 12 June 2023 Published: 20 June 2023
MSC : 62F15, 62F30, 94A17

An entropy measure of uncertainty has a complementary dual function called extropy. In the last six years, this measure of randomness has gotten a lot of attention. It cannot, however, be applied to systems that have survived for some time. As a result, the idea of residual extropy was created. To estimate the extropy and residual extropy, Bayesian and non-Bayesian estimators of unknown parameters of the exponentiated gamma distribution are generated. Bayesian estimators are regarded using balanced loss functions like the balanced squared error, balanced linear exponential and balanced general entropy. We use the Lindley method to get the extropy and residual extropy estimates for the exponentiated gamma distribution based on generalized type-Ⅰ hybrid censored data. To test the effectiveness of the proposed methodologies, a simulation experiment was carried out, and the actual data set was studied for illustrative purposes. In summary, the mean squared error values decrease as the number of failures increases, according to the results obtained. The Bayesian estimates of residual extropy under the balanced linear exponential loss function perform well compared to the other estimates. Alternatively, the Bayesian estimates of the extropy perform well under a balanced general entropy loss function in the majority of situations.
- extropy,
- residual extropy,
- exponentiated gamma distribution,
- Bayesian estimation,
- Lindley's method
Citation: Baria A. Helmy, Amal S. Hassan, Ahmed K. El-Kholy, Rashad A. R. Bantan, Mohammed Elgarhy. Analysis of information measures using generalized type-Ⅰ hybrid censored data[J]. AIMS Mathematics, 2023, 8(9): 20283-20304. doi: 10.3934/math.20231034

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Abstract

An entropy measure of uncertainty has a complementary dual function called extropy. In the last six years, this measure of randomness has gotten a lot of attention. It cannot, however, be applied to systems that have survived for some time. As a result, the idea of residual extropy was created. To estimate the extropy and residual extropy, Bayesian and non-Bayesian estimators of unknown parameters of the exponentiated gamma distribution are generated. Bayesian estimators are regarded using balanced loss functions like the balanced squared error, balanced linear exponential and balanced general entropy. We use the Lindley method to get the extropy and residual extropy estimates for the exponentiated gamma distribution based on generalized type-Ⅰ hybrid censored data. To test the effectiveness of the proposed methodologies, a simulation experiment was carried out, and the actual data set was studied for illustrative purposes. In summary, the mean squared error values decrease as the number of failures increases, according to the results obtained. The Bayesian estimates of residual extropy under the balanced linear exponential loss function perform well compared to the other estimates. Alternatively, the Bayesian estimates of the extropy perform well under a balanced general entropy loss function in the majority of situations.

References

[1]	B. Epstein, Truncated life tests in the exponential case, Ann. Math. Stat., 25 (1954), 555–564. http://dx.doi.org/10.1214/aoms/1177728723 doi: 10.1214/aoms/1177728723
[2]	A. Childs, B. Chandrasekar, N. Balakrishnan, D. Kundu, Exact likelihood inference based on Type-Ⅰ and Type-Ⅱ hybrid censored samples from the exponential distribution, Ann. Inst. Stat. Math., 55 (2003), 319–330. https://doi.org/10.1007/BF02530502 doi: 10.1007/BF02530502
[3]	B. Chandrasekar, A. Childs, N. Balakrishnan, Exact likelihood inference for the exponential distribution under generalized Type-Ⅰ and Type-Ⅱ hybrid censoring, Nav. Res. Logist., 51 (2004), 994–1000. https://doi.org/10.1002/nav.20038 doi: 10.1002/nav.20038
[4]	C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech., 27 (1948), 379–432. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x doi: 10.1002/j.1538-7305.1948.tb01338.x
[5]	Y. Cho, H. Sun, K. Lee, An estimation of the entropy for a Rayleigh distribution based on doubly-generalized Type-Ⅱ hybrid censored samples, Entropy, 16 (2014), 3655–3669. https://doi.org/10.3390/e16073655 doi: 10.3390/e16073655
[6]	S. Liu, W. Gui, Estimating the entropy for Lomax distribution based on generalized progressively hybrid censoring, Symmetry, 11 (2019), 1219. https://doi.org/10.3390/sym11101219 doi: 10.3390/sym11101219
[7]	A. S. Hassan, A. N. Zaky, Estimation of entropy for inverse Weibull distribution under multiple censored data, J. Taibah Univ. Sci., 13 (2019), 331–337. https://doi.org/10.1080/16583655.2019.1576493 doi: 10.1080/16583655.2019.1576493
[8]	J. Yu, W. Gui, Y. Shan, Statistical inference on the Shannon entropy of inverse Weibull distribution under the progressive first-failure censoring, Entropy, 21 (2019), 1209. https://doi.org/10.3390/e21121209 doi: 10.3390/e21121209
[9]	A. A. H. Ahmadini, A. S. Hassan, A. N. Zaki, S. S. Alshqaq, Bayesian inference of dynamic cumulative residual entropy from Pareto Ⅱ distribution with application to COVID-19, AIMS Math., 6 (2020), 2196–2216. https://doi.org/10.3934/math.2021133 doi: 10.3934/math.2021133
[10]	A. A. Al-Babtain, A. S. Hassan, A. N. Zaky, I. Elbatal, M. Elgarhy, Dynamic cumulative residual Renyi entropy for Lomax distribution: Bayesian and non-Bayesian methods, AIMS Math., 6 (2021), 3889–3914. https://doi.org/10.3934/math.2021231 doi: 10.3934/math.2021231
[11]	A. S. Hassan, A. N. Zaky, Entropy Bayesian estimation for Lomax distribution based on record, Thailand Stat., 19 (2021), 96–115.
[12]	A. I. Al-Omari, A. S. Hassan, H. F. Nagy, A. R. Al-Anzi, L. Alzoubi, Entropy Bayesian analysis for the generalized inverse exponential distribution based on URRSS, Comput. Mater. Contin., 69 (2021), 3795–3811. https://doi.org/10.32604/cmc.2021.019061 doi: 10.32604/cmc.2021.019061
[13]	A. M. Almarashi, A. Algarni, A. S. Hassan, A. N. Zaky, M. Elgarhy, Bayesian analysis of dynamic cumulative residual entropy for Lindley distribution, Entropy, 23 (2021), 1256. https://doi.org/10.3390/e23101256 doi: 10.3390/e23101256
[14]	A. S. Hassan, E. A. Elsherpieny, R. E. Mohamed, Estimation of information measures for power-function distribution in the presence of outliers and their applications, Int. J. Inf. Commun. Technol., 21 (2022), 1–25. https://doi.org/10.32890/jict2022.21.1.1 doi: 10.32890/jict2022.21.1.1
[15]	B. A. Helmy, A. S. Hassan, A. K. El-Kholy, Analysis of uncertainty measure using unified hybrid censored data with applications, J. Taibah Univ. Sci., 15 (2022), 1130–1143. https://doi.org/10.1080/16583655.2021.2022901 doi: 10.1080/16583655.2021.2022901
[16]	A. S. Hassan, E. A. Elsherpieny, R. E. Mohamed, Classical and Bayesian estimation of entropy for Pareto distribution in presence of outliers with application, Sankhya A, 85 (2023), 707–740. https://doi.org/10.1007/s13171-021-00274-z doi: 10.1007/s13171-021-00274-z
[17]	D. Ellerman, An introduction to logical entropy and its relation to Shannon entropy, Int. J. Semant. Comput., 7 (2013), 121–145. https://doi.org/10.48550/arXiv.2112.01966 doi: 10.48550/arXiv.2112.01966
[18]	D. Markechová, B. Riečan, Logical entropy of fuzzy dynamical systems, Entropy, 18 (2016), 157. https://doi.org/10.3390/e18040157 doi: 10.3390/e18040157
[19]	S. Boffa, D. Ciucci, Logical entropy and aggregation of fuzzy orthopartitions, Fuzzy Set. Syst., 455 (2023), 77–101. https://doi.org/10.1016/j.fss.2022.07.014 doi: 10.1016/j.fss.2022.07.014
[20]	F. Lad, G. Sanfilippo, G. Agro, Extropy: Complementary dual of entropy, Stat. Sci., 30 (2015), 40–58. https://doi.org/10.48550/arXiv.1109.6440 doi: 10.48550/arXiv.1109.6440
[21]	T. Gneiting, A. E. Raftery, Strictly proper scoring rules, prediction and estimation, J. Am. Stat. Assoc., 102 (2007), 359–378. https://doi.org/10.1198/016214506000001437 doi: 10.1198/016214506000001437
[22]	S. Furuichi, F. C. Mitroi, Mathematical inequalities for some divergences, Physica A, 391 (2012), 388–400. https://doi.org/10.48550/arXiv.1104.5603 doi: 10.48550/arXiv.1104.5603
[23]	G. Qiu, The extropy of order statistics and record values, Stat. Probab. Lett., 120 (2017), 52–60. https://doi.org/10.1016/j.spl.2016.09.016 doi: 10.1016/j.spl.2016.09.016
[24]	G. Qiu, K. Jia, The residual extropy of order statistics, Stat. Probab. Lett., 133 (2018), 15–22. https://doi.org/10.1016/j.spl.2017.09.014 doi: 10.1016/j.spl.2017.09.014
[25]	M. Z. Raqab, G. Qiu, On extropy properties of ranked set sampling, Am. J. Theor. Appl. Stat., 53 (2019), 210–226. https://doi.org/10.1080/02331888.2018.1533963 doi: 10.1080/02331888.2018.1533963
[26]	H. A. Noughabi, J. Jarrahiferiz, On the estimation of extropy, J. Nonparametr. Stat., 31 (2019), 88–99. https://doi.org/10.1080/10485252.2018.1533133 doi: 10.1080/10485252.2018.1533133
[27]	R. Hazeb, M. Z. Raqab, H. A. Bayoud, Non-parametric estimation of the extropy and the entropy measures based on progressive type-Ⅱ censored data with testing uniformity, J. Stat. Comput. Simul., 91 (2021), 1–33. https://doi.org/10.1080/00949655.2021.1888953 doi: 10.1080/00949655.2021.1888953
[28]	A. S. Hassan, E. Elsherpieny, R. Mohamed, Cumulative residual extropy for Pareto distribution in the presence of outliers: Bayesian and non-Bayesian methods, Stat. Optim. Inf. Comput., 10 (2022), 1095–1109. https://doi.org/10.19139/soic-2310-5070-1200 doi: 10.19139/soic-2310-5070-1200
[29]	R. C. Gupta, P. L. Gupta, R. D. Gupta, Modeling failure time data by Lehman alternatives, Commun. Stat. Theor. M., 27 (1998), 887–904. https://doi.org/10.1080/03610929808832134 doi: 10.1080/03610929808832134
[30]	A. I. Shawky, R. A. Bakoban, Bayesian and non-Bayesian estimations on the exponentiated gamma distribution, Appl. Math. Sci., 2 (2008), 2521–2530.
[31]	A. I. Shawky, R. A. Bakoban, Order statistics from exponentiated gamma distribution and associated inference, Int. J. Contemp. Math. Sci., 4 (2009), 71–91.
[32]	N. Feroze, M. Aslam, Bayesian analysis of exponentiated gamma distribution under type Ⅱ censored samples, Sci. J. Pure. Appl. Sci., 49 (2012), 30–39.
[33]	U. Singh, S. K. Singh, A. S. Yadav, Bayesian estimation for exponentiated gamma distribution under progressive type-Ⅱ censoring using different approximation techniques, Data Sci. J., 13 (2015), 551–568. https://doi.org/10.6339/JDS.201507_13(3).0008 doi: 10.6339/JDS.201507_13(3).0008
[34]	M. A. W. Mahmoud, L. S. Diab, M. G. M. Ghazal, A. H. Baria, Bayesian prediction of exponentiated gamma distribution based on unified hybrid censored data, J. Stat. : Adv. Theory Appl., 22 (2019), 21–43. https://doi.org/10.18642/jsata_7100122102 doi: 10.18642/jsata_7100122102
[35]	M. A. W. Mahmoud, L. S. Diab, M. G. M. Ghazal, A. H. Baria, On study of exponentiated gamma distribution based on unified hybrid censored data, Al-Azhar Bull. Sci., 30 (2019), 13–27. https://doi.org/10.21608/absb.2019.86749 doi: 10.21608/absb.2019.86749
[36]	A. Zellner, Bayesian and non-Bayesian estimation using balanced loss functions, Springer, New York, 1994. https://doi.org/10.1007/978-1-4612-2618-5_28
[37]	D. V. Lindley, Approximate Bayesian methods, Trab. Estad. Invest. Oper., 31 (1980), 223–245. https://doi.org/10.1007/BF02888353 doi: 10.1007/BF02888353
[38]	M. G. M. Ghazal, H. M. Hasaballah, Exponentiated Rayleigh distribution: A Bayes study using MCMC approach based on unified hybrid censored data, J. Adv. Math., 12 (2017), 6863–6880. https://doi.org/10.24297/jam.v12i12.4599 doi: 10.24297/jam.v12i12.4599

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