Rényi entropy of past lifetime from lower $ k $-record values

Mansour Shrahili; Mohamed Kayid; Mansour Shrahili; Mohamed Kayid

doi:10.3934/math.20241189

AIMS Mathematics

2024, Volume 9, Issue 9: 24401-24417. doi: 10.3934/math.20241189

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Research article

Rényi entropy of past lifetime from lower $ k $-record values

Mansour Shrahili ^,,
Mohamed Kayid

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

Received: 12 July 2024 Revised: 12 August 2024 Accepted: 13 August 2024 Published: 19 August 2024
MSC : 94A08, 62P99

This paper explored the concept of past Rényi entropy within the context of $ k $-record values. We began by introducing a representation of the past Rényi entropy for the $ n $-th lower $ k $-record values, sampled from any continuous distribution function $ F, $ concerning the past Rényi entropy of the $ n $-th lower $ k $-record values sampled from a uniform distribution. Then, we delved into the examination of the monotonicity properties of the past Rényi entropy of $ k $-record values. Specifically, we focused on the aging properties of the component lifetimes and investigated how they impacted the monotonicity of the past Rényi entropy. Additionally, we derived an expression for the $ n $-th lower $ k $-records in terms of the past Rényi entropy, specifically when the first lower $ k $-record was less than a specified threshold level, and then investigated several properties of the given formula.
- $ k $-record values,
- Rényi entropy,
- past Rényi entropy,
- stochastic orders
Citation: Mansour Shrahili, Mohamed Kayid. Rényi entropy of past lifetime from lower $ k $-record values[J]. AIMS Mathematics, 2024, 9(9): 24401-24417. doi: 10.3934/math.20241189

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Abstract

This paper explored the concept of past Rényi entropy within the context of $ k $-record values. We began by introducing a representation of the past Rényi entropy for the $ n $-th lower $ k $-record values, sampled from any continuous distribution function $ F, $ concerning the past Rényi entropy of the $ n $-th lower $ k $-record values sampled from a uniform distribution. Then, we delved into the examination of the monotonicity properties of the past Rényi entropy of $ k $-record values. Specifically, we focused on the aging properties of the component lifetimes and investigated how they impacted the monotonicity of the past Rényi entropy. Additionally, we derived an expression for the $ n $-th lower $ k $-records in terms of the past Rényi entropy, specifically when the first lower $ k $-record was less than a specified threshold level, and then investigated several properties of the given formula.

References

[1]	M. Ahsanullah, Record values-theory and applications, University Press of America, 2004.
[2]	B. C. Arnold, N. Balakrishnan, H. N. Nagaraja, Records, John Wiley & Sons, 768 (2011).
[3]	V. B. Nevzorov, Records, Theor. Probab. Appl., 32 (1988), 201–228. https://doi.org/10.1137/1132032 doi: 10.1137/1132032
[4]	W. Dziubdziela, B. Kopociński, Limiting properties of the k-th record values, Appl. Math., 2 (1976), 187–190. https://doi.org/10.4064/am-15-2-187-190 doi: 10.4064/am-15-2-187-190
[5]	H. A. David, H. N. Nagaraja, Order statistics, John Wiley & Sons, 2004. https://doi.org/10.1080/02331889208802367
[6]	B. C. Arnold, N. Balakrishnan, H. N. Nagaraja, A first course in order statistics, SIAM, 2008.
[7]	M. Abramowitz, I. A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, US Government printing office, 55 (1948).
[8]	G. Arfken, The incomplete gamma function and related functions, mathematical methods for physicists, Academic Press San Diego, 15 (1985).
[9]	C. E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J., 27 (1948), 379–423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x doi: 10.1002/j.1538-7305.1948.tb01338.x
[10]	A. Rényi, On measures of entropy and information, In: Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, Berkeley, California, USA, 1 (1961).
[11]	N. Ebrahimi, E. S. Soofi, R. Soyer, Information measures in perspective, Int. Stat. Rev., 78 (2010), 383–412. https://doi.org/10.1111/j.1751-5823.2010.00105.x doi: 10.1111/j.1751-5823.2010.00105.x
[12]	I. Csiszár, Generalized cutoff rates and Rényi's information measures, IEEE T. Inform. Theory, 41 (1995), 26–34.
[13]	H. Neemuchwala, A. Hero, P. Carson, Image matching using alpha-entropy measures and entropic graphs, Signal Proces., 85 (2005), 277–296. https://doi.org/10.1016/j.sigpro.2004.10.002 doi: 10.1016/j.sigpro.2004.10.002
[14]	I. Molina, D. Morales, Renyi statistics for testing hypotheses in mixed linear regression models, J. Stat. Plan. Infer., 137 (2007), 87–102.
[15]	S. Vinga, J. S. Almeida, Rényi continuous entropy of dna sequences, J. Theor. Biol., 231 (2004), 377–388.
[16]	M. Basseville, Distance measures for signal processing and pattern recognition, Signal Proc., 18 (1989), 349–369. https://doi.org/10.1016/0165-1684(89)90079-0 doi: 10.1016/0165-1684(89)90079-0
[17]	R. D. Gupta, A. K. Nanda, $\alpha$-and $\beta$-entropies and relative entropies of distributions, J. Stat. Theory Appl., 1 (2002), 177–190.
[18]	M. Asadi, N. Ebrahimi, E. S. Soofi, Dynamic generalized information measures, Stat. Probab. Lett., 71 (2005), 85–98. https://doi.org/10.1016/j.spl.2004.10.033 doi: 10.1016/j.spl.2004.10.033
[19]	A. K. Nanda, P. Paul, Some results on generalized residual entropy, Inform. Sci., 176 (2006), 27–47. https://doi.org/10.1016/j.ins.2004.10.008 doi: 10.1016/j.ins.2004.10.008
[20]	M. Mesfioui, M. Kayid, M. Shrahili, Renyi entropy of the residual lifetime of a reliability system at the system level, Axioms, 12 (2023), 320. https://doi.org/10.3390/axioms12040320 doi: 10.3390/axioms12040320
[21]	A. Di Crescenzo, M. Longobardi, Entropy-based measure of uncertainty in past lifetime distributions, J. Appl. Probab., 39 (2002), 434–440. https://doi.org/10.1239/jap/1025131441 doi: 10.1239/jap/1025131441
[22]	N. U. Nair, S. M. Sunoj, Some aspects of reversed hazard rate and past entropy, Commun. Stat.-Theor. M., 32 (2021), 2106–2116.
[23]	R. C. Gupta, H. C. Taneja, R. Thapliyal, Stochastic comparisons of residual entropy of order statistics and some characterization results, J. Stat. Theor. Appl., 13 (2014), 27–37. https://doi.org/10.2991/jsta.2014.13.1.3 doi: 10.2991/jsta.2014.13.1.3
[24]	K. N. Chandler, The distribution and frequency of record values, J. Roy. Stat. Soc. B, 14 (1952), 220–228. https://doi.org/10.1111/j.2517-6161.1952.tb00115.x doi: 10.1111/j.2517-6161.1952.tb00115.x
[25]	N. Glick, Breaking records and breaking boards, Am. Math. Mon., 85 (1978), 2–26. https://doi.org/10.1080/00029890.1978.11994501 doi: 10.1080/00029890.1978.11994501
[26]	U. Kamps, A concept of generalized order statistics, J. Stat. Plan. Infer., 48 (1995), 1–23. https://doi.org/10.1016/0378-3758(94)00147-N doi: 10.1016/0378-3758(94)00147-N
[27]	S. Zarezadeh, M. Asadi, Results on residual Rényi entropy of order statistics and record values, Inform. Sci., 180 (2010), 4195–4206. https://doi.org/10.1016/j.ins.2010.06.019 doi: 10.1016/j.ins.2010.06.019
[28]	A. Habibi, N. R. Arghami, J. Ahmadi, Statistical evidence in experiments and in record values, Commun. Stat.-Theor. M., 35 (2006), 1971–1983. https://doi.org/10.1080/03610920600762780 doi: 10.1080/03610920600762780
[29]	M. Abbasnezhad, N. R. Arghami, Potential statistical evidence in experiments and Renyi information, JIRSS-J. Iran. Stat. So., 5 (2006), 39–52.
[30]	S. Baratpour, J. Ahmadi, N. R. Arghami, Entropy properties of record statistics, Stat. Pap., 48 (2007), 197–213. https://doi.org/10.1007/s00362-006-0326-7 doi: 10.1007/s00362-006-0326-7
[31]	J. Jose, E. I. A. Sathar, Rényi entropy of k-records: Properties and applications, REVSTAT-Stat. J., 20 (2022), 481–500.
[32]	P. S. Asha, M. Chacko, Residual Rényi entropy of k-record values, Commun. Stat.-Theor. M., 45 (2016), 4874–4885. https://doi.org/10.1080/03610926.2014.932806 doi: 10.1080/03610926.2014.932806
[33]	M. Shrahili, M. Kayid, Residual Tsallis entropy and record values: Some new insights, Symmetry, 15 (2023), 2040. https://doi.org/10.3390/sym15112040 doi: 10.3390/sym15112040
[34]	J. Ahmadi, N. Balakrishnan, Preservation of some reliability properties by certain record statistics, Statistics, 39 (2005), 347–354. https://doi.org/10.1080/02331880500178752 doi: 10.1080/02331880500178752
[35]	Y. Wang, P. Zhao, A note on DRHR preservation property of generalized order statistics, Commun. Stat.-Theor. M., 39 (2010), 815–822. https://doi.org/10.1080/03610920902796072 doi: 10.1080/03610920902796072
[36]	Z. Zamani, M. Madadi, Quantile-based entropy function in past lifetime for order statistics and its properties, Filomat, 37 (2023), 3321–3334. https://doi.org/10.2298/FIL2310321Z doi: 10.2298/FIL2310321Z
[37]	M. Kayid, M. Shrahili, Rényi entropy for past lifetime distributions with application in inactive coherent systems, Symmetry, 15 (2023), 1310.
[38]	M. Mahmoudi, M. Asadi, On the monotone behavior of time dependent entropy of order $\alpha$, JIRSS-J. Iran. Stat. So., 9 (2010), 65–83.
[39]	S. C. Kochar, Some partial ordering results on record values, Commun. Stat.-Theor. M., 19 (1990), 299–306. https://doi.org/10.1080/03610929008830201 doi: 10.1080/03610929008830201
[40]	M. Z. Raqab, W. A. Amin, A note on reliability properties of k-record statistics, Metrika, 46 (1997), 245–251. https://doi.org/10.1007/BF02717177 doi: 10.1007/BF02717177
[41]	M. J. Raqab, W. A. Amin, Some ordering results on order statistics and record values, IAPQR Transactions, 21 (1996), 1–8.
[42]	B. E. Khaledi, Some new results on stochastic orderings between generalized order statistics, J. Iran. Stat. So., 4 (2022), 35–49.
[43]	B. E. Khaledi, R. Shojaei, On stochastic orderings between residual record values, Stat. Probab. Lett., 77 (2007), 1467–1472. https://doi.org/10.1016/j.spl.2007.03.033 doi: 10.1016/j.spl.2007.03.033
[44]	C. Kundu, A. K. Nanda, T. Hu, A note on reversed hazard rate of order statistics and record values, J. Stat. Plan. Infer., 139 (2009), 1257–1265. https://doi.org/10.1016/j.jspi.2008.08.002 doi: 10.1016/j.jspi.2008.08.002
[45]	M. Tavangar, M. Asadi, Some results on conditional expectations of lower record values, Statistics, 45 (2011), 237–255. https://doi.org/10.1080/02331880903348481 doi: 10.1080/02331880903348481

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