Estimation of the coefﬁcients of variation for inverse power Lomax distribution

Samah M. Ahmed; Abdelfattah Mustafa; Samah M. Ahmed; Abdelfattah Mustafa

doi:10.3934/math.20241595

AIMS Mathematics

2024, Volume 9, Issue 12: 33423-33441. doi: 10.3934/math.20241595

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Estimation of the coefﬁcients of variation for inverse power Lomax distribution

Samah M. Ahmed ¹,
Abdelfattah Mustafa ^{2,3
,
,}

1.
Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt
2.
Mathematics Department, Faculty of Science, Islamic University of Madinah, Madinah 42351, KSA
3.
Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 09 August 2024 Revised: 05 November 2024 Accepted: 12 November 2024 Published: 25 November 2024
MSC : 62F10, 62F15, 62N01

One useful descriptive metric for measuring variability in applied statistics is the coefficient of variation (CV) of a distribution. However, it is uncommon to report conclusions about the CV of non-normal distributions. This study develops a method for estimating the CV for the inverse power Lomax (IPL) distribution using adaptive Type-Ⅱ progressive censored data. The experiment is a well-liked plan for gathering data, particularly for a very dependable product. The point and interval estimate of CV are formulated under the classical approach (maximum likelihood and bootstrap) and the Bayesian approach with respect to the symmetric loss function. For the unknown parameters, the joint prior density is calculated using the Bayesian technique as a product of three independent gamma densities. Additionally, it is recommended to use the Markov Chain Monte Carlo (MCMC) method to calculate the Bayes estimate and generate posterior distributions. A simulation study and a numerical example are given to assess the performance of the maximum likelihood and Bayes estimations.
- Gibbs and Metropolis sampler,
- inverse power Lomax distribution,
- adaptive Type-Ⅱ progressive censoring scheme,
- coefficient of variation,
- Bayesian approach
Citation: Samah M. Ahmed, Abdelfattah Mustafa. Estimation of the coefﬁcients of variation for inverse power Lomax distribution[J]. AIMS Mathematics, 2024, 9(12): 33423-33441. doi: 10.3934/math.20241595

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Abstract

One useful descriptive metric for measuring variability in applied statistics is the coefficient of variation (CV) of a distribution. However, it is uncommon to report conclusions about the CV of non-normal distributions. This study develops a method for estimating the CV for the inverse power Lomax (IPL) distribution using adaptive Type-Ⅱ progressive censored data. The experiment is a well-liked plan for gathering data, particularly for a very dependable product. The point and interval estimate of CV are formulated under the classical approach (maximum likelihood and bootstrap) and the Bayesian approach with respect to the symmetric loss function. For the unknown parameters, the joint prior density is calculated using the Bayesian technique as a product of three independent gamma densities. Additionally, it is recommended to use the Markov Chain Monte Carlo (MCMC) method to calculate the Bayes estimate and generate posterior distributions. A simulation study and a numerical example are given to assess the performance of the maximum likelihood and Bayes estimations.

References

[1]	D. S. Bhoj, M. Ahsanullah, Testing equality of coefficients of variation of two populations, Biometrical J., 35 (1993), 355–359. https://doi.org/10.1002/bimj.4710350311 doi: 10.1002/bimj.4710350311
[2]	W. Reh, B. Scheffler, Significance tests and confidence intervals for coefficient of variation, Comput. Stat. Data An., 22 (1996), 449–453. https://doi.org/10.1016/0167-9473(96)83707-8 doi: 10.1016/0167-9473(96)83707-8
[3]	K.-I. Ahn, On the use of coefficient of variation for uncertainty analysis in fault tree analysis, Reliab. Eng. Syst. Safe., 47 (1995), 229–230. https://doi.org/10.1016/0951-8320(94)00061-R doi: 10.1016/0951-8320(94)00061-R
[4]	A. J. Hamer, J. R. Strachan, M. M. Black, C. Ibbotson, R. A. Elson, A new method of comparative bone strength measurement, Journal of Medical Engineering & Technology, 19 (1995), 1–5. https://doi.org/10.3109/03091909509030263 doi: 10.3109/03091909509030263
[5]	J. Gong, Y. Li, Relationship between the estimated Weibull modulus and the coefficient of variation of the measured strength for ceramics, J. Amer. Ceram. Soc., 82 (1999), 449–452. https://doi.org/10.1111/j.1551-2916.1999.tb20084.x doi: 10.1111/j.1551-2916.1999.tb20084.x
[6]	W. K. Pang, W. T.-Y. Bosco, M. D. Troutt, H. H. Shui, A simulation-based approach to the study of coefficient of variation of dividend yields, Eur. J. Oper. Res., 189 (2008), 559–569. https://doi.org/10.1016/j.ejor.2007.05.032 doi: 10.1016/j.ejor.2007.05.032
[7]	W. K. Pang, P.-K. Leung, W.-K. Huang, W. Liu, On interval estimation of the coefficient of variation for the three-parameter Weibull, lognormal and gamma distribution: a simulation based approach, Eur. J. Oper. Res., 164 (2005), 367–377. https://doi.org/10.1016/j.ejor.2003.04.005 doi: 10.1016/j.ejor.2003.04.005
[8]	M. M. M. El-Din, M. M. Amein, A. M. A. El-Raheem, H. E. El-Attar, E. H. Hafez, Estimation of the coefﬁcient of variation for Lindley distribution based on progressive first failure censored data, Journal of Statistics Applications & Probability, 8 (2019), 83–90. http://doi.org/10.18576/jsap/080202 doi: 10.18576/jsap/080202
[9]	K. S. Lomax, Business failures: another example of the analysis of failure data, J. Amer. Stat. Assoc., 49 (1954), 847–852. https://doi.org/10.2307/2281544 doi: 10.2307/2281544
[10]	A. B. Atkinson, A. J. Harrison, Distribution of personal wealth in Britain, Cambridge: Cambridge University Press, 1978.
[11]	O. Holland, A. Golaup, A. H. Aghvami, Traffic characteristics of aggregated module downloads for mobile terminal reconfiguration, IEE Proceedings-Communications, 153 (2006), 683–690.
[12]	A. Corbellini, L. Crosato, P. Ganugi, M. Mazzoli, Fitting Pareto Ⅱ distributions on firm size: statistical methodology and economic puzzles, In: Advances in data analysis, Boston: Birkhäuser, 2010,321–328. https://doi.org/10.1007/978-0-8176-4799-5_26
[13]	A. S. Hassan, A. S. Al-Ghamdi, Optimum step stress accelerated life testing for Lomax distribution, Journal of Applied Sciences Research, 5 (2009), 2153–2164.
[14]	A. S. Hassan, S. M. Assar, A. Shelbaia, Optimum step-stress accelerated life test plan for Lomax distribution with an adaptive Type-Ⅱ progressive hybrid censoring, Journal of Advances in Mathematics and Computer Science, 13 (2016), 1–19. https://doi.org/10.9734/BJMCS/2016/21964 doi: 10.9734/BJMCS/2016/21964
[15]	C. Kleiber, S. Kotz, Statistical size distributions in economics and actuarial sciences, Hoboken, New Jersey: John Wiley & Sons, Inc., 2003. https://doi.org/10.1002/0471457175
[16]	A. S. Hassan, M. Abd-Allah, On the inverse power Lomax distribution, Ann. Data Sci., 6 (2019), 259–278. https://doi.org/10.1007/s40745-018-0183-y doi: 10.1007/s40745-018-0183-y
[17]	N. Balakrishnan, R. Aggarwala, Progressive censoring: theory, methods and applications, Boston: Birkhäuser, 2000. https://doi.org/10.1007/978-1-4612-1334-5
[18]	H. K. T. Ng, D. Kundu, P. S. Chan, Statistical analysis of exponential lifetimes under an adaptive Type-Ⅱ progressive censoring scheme, Nav. Res. Log., 56 (2009), 687–698. https://doi.org/10.1002/nav.20371 doi: 10.1002/nav.20371
[19]	M. Nassar, O. E. Abo-Kasem, Estimation of the inverse Weibull parameters under adaptive Type-Ⅱ progressive hybrid censoring scheme, J. Comput. Appl. Math., 315 (2017), 228–239. https://doi.org/10.1016/j.cam.2016.11.012 doi: 10.1016/j.cam.2016.11.012
[20]	S. F. Ateya, H. S. Mohammed, Statistical inferences based on an adaptive progressive type-Ⅱ censoring from exponentiated exponential distribution, Journal of the Egyptian Mathematical Society, 25 (2017), 393–399. http://doi.org/10.1016/j.joems.2017.06.001 doi: 10.1016/j.joems.2017.06.001
[21]	M. M. M. Mohie El-Din, M. M. Amein, A. R. Shafay, S. Mohamed, Estimation of generalized exponential distribution based on an adaptive progressively Type-Ⅱ censored sample, J. Stat. Comput. Sim., 87 (2017), 1292–1304. https://doi.org/10.1080/00949655.2016.1261863 doi: 10.1080/00949655.2016.1261863
[22]	S. Liu, W. Gui, Estimating the parameters of the two-parameter Rayleigh distribution based on adaptive Type Ⅱ progressive hybrid censored data with competing risks, Mathematics, 8 (2020), 1783. https://doi.org/10.3390/math8101783 doi: 10.3390/math8101783
[23]	A. Elshahhat, M. Nassar, Bayesian survival analysis for adaptive Type-Ⅱ progressive hybrid censored Hjorth data, Comput. Stat., 36 (2021), 1965–1990. https://doi.org/10.1007/s00180-021-01065-8 doi: 10.1007/s00180-021-01065-8
[24]	A. Kohansal, H. S. Bakouch, Estimation procedures for Kumaraswamy distribution parameters under adaptive type-Ⅱ hybrid progressive censoring, Commun. Stat.–Simul. Comput., 50 (2021), 4059–4078. https://doi.org/10.1080/03610918.2019.1639734 doi: 10.1080/03610918.2019.1639734
[25]	R. Alotaibi, M. Nassar, A. Elshahhat, Computational analysis of XLindley parameters using adaptive Type-Ⅱ progressive hybrid censoring with applications in chemical engineering, Mathematics, 10 (2022), 3355. https://doi.org/10.3390/math10183355 doi: 10.3390/math10183355
[26]	A. Xu, J. Wang, Y. Tang, P. Chen, Efficient online estimation and remaining useful life prediction based on the inverse Gaussian process, Nav. Res. Log., in press. https://doi.org/10.1002/nav.22226
[27]	L. Zhuang, A. Xu, Y. Wang, Y. Tang, Remaining useful life prediction for two-phase degradation model based on reparameterized inverse Gaussian process, Eur. J. Oper. Res., 319 (2024), 877–890. https://doi.org/10.1016/j.ejor.2024.06.032 doi: 10.1016/j.ejor.2024.06.032
[28]	R. S. Kenett, S. Zacks, P. Gedeck, Bayesian reliability estimation and prediction, In: Industrial Statistics, Cham: Birkhäuser, 2023,371–396. https://doi.org/10.1007/978-3-031-28482-3_10
[29]	R. C. Kurchin, Using Bayesian parameter estimation to learn more from data without black boxes, Nat. Rev. Phys., 6 (2024), 152–154. https://doi.org/10.1038/s42254-024-00698-0 doi: 10.1038/s42254-024-00698-0
[30]	H. M. Aljohani, N. M. Alfar, Estimations with step-stress partially accelerated life tests for competing risks Burr XII lifetime model under Type-Ⅱ censored data, Alex. Eng. J., 59 (2020), 1171–1180. https://doi.org/10.1016/j.aej.2020.01.022 doi: 10.1016/j.aej.2020.01.022
[31]	C. P. Robert, G. Casella, Monte Carlo statistical methods, 2 Eds., New York: Springer, 2004. https://doi.org/10.1007/978-1-4757-4145-2
[32]	S. Rezali, R. Tahmasbi, M. Mahmoodi, Estimation of P[Y < X] for generalized Pareto distribution, J. Stat. Plan. Infer., 140 (2010), 480–494. https://doi.org/10.1016/j.jspi.2009.07.024 doi: 10.1016/j.jspi.2009.07.024
[33]	A. A. Soliman, E. A. Ahmed, N. A. Abou-Elheggag, S. M. Ahmed, Step-stress partially accelerated life tests model in estimation of inverse Weibull parameters under progressive Type-Ⅱ censoring, Appl. Math. Inform. Sci., 11 (2017), 1369–1381. http://doi.org/10.18576/amis/110514 doi: 10.18576/amis/110514
[34]	S. M. Ahmed, Constant-stress partially accelerated life testing for Weibull inverted exponential distribution with censored data, Iraqi Journal for Computer Science and Mathematics, 5 (2024), 94–111. https://doi.org/10.52866/ijcsm.2024.05.02.009 doi: 10.52866/ijcsm.2024.05.02.009
[35]	N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, E. Teller, Equations of state calculations by fast computing machines, J. Chem. Phys., 21 (1953), 1087–1091. http://doi.org/10.1063/1.1699114 doi: 10.1063/1.1699114
[36]	W. H. Greene, Econometric analysis, 4 Eds., New York: Prentice Hall, 2000.
[37]	B. Efron, The jackknife, the bootstrap and other resampling plans, Philadelphia, PA: SIAM, 1982.
[38]	M.-H. Chen, Q.-M. Shao, Monte Carlo estimation of Bayesian credible and HPD intervals, J. Comput. Graph. Stat., 8 (1999), 69–92. https://doi.org/10.2307/1390921 doi: 10.2307/1390921
[39]	T. Bjerkedal, Acquisition of resistance in guinea pigs infected with different doses of virulent tubercle bacilli, Amer. J. Epidemiol., 72 (1960), 130–148. https://doi.org/10.1093/oxfordjournals.aje.a120129 doi: 10.1093/oxfordjournals.aje.a120129

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