A novel logarithmic framework for developing probability distributions with applications to item failures and toxic contamination datasets

Aijaz Ahmad; Bassant Elkalzah; Manzoor A. Khanday; R. A Rather; Hatem E. Semary; Mustafa Bayram; Okechukwu J. Obulezi; Aijaz Ahmad; Bassant Elkalzah; Manzoor A. Khanday; R. A Rather; Hatem E. Semary; Mustafa Bayram; Okechukwu J. Obulezi

doi:10.3934/math.2026733

AIMS Mathematics

2026, Volume 11, Issue 6: 17972-18025. doi: 10.3934/math.2026733

Previous Article Next Article

Research article Special Issues

A novel logarithmic framework for developing probability distributions with applications to item failures and toxic contamination datasets

1.
Lincoln University College, Petaling Jaya 47301, Selangor, Malaysia
2.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
3.
Department of Mathematics and Statistics, Lovely Professional University, Punjab, India
4.
Department of Mathematical Science IUST, Awantipora Kashmir, India
5.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi Arabia
6.
Department of Computer Engineering, Biruni University, Istanbul 34010, Turkey
7.
Department of Statistics, Faculty of Physical Sciences, Nnamdi Azikiwe University, P.O. Box 5025, Awka, Nigeria

Received: 16 April 2026 Revised: 09 June 2026 Accepted: 12 June 2026 Published: 18 June 2026
MSC : 62E, 62N, 62F

A new family of distributions, the Log-generator class, is introduced in this work, deriving its basis from the logarithmic function. The study establishes formal expressions for the generator's probability density function. Subsequently, the New Log-Rayleigh distribution (NLRD) is formulated, selecting the Rayleigh distribution as its foundation and concentrating on a particular case within the proposed family. Several distributional properties, including moments, reliability indices, entropies, and order statistics, are derived through systematic mathematical approaches. The finite-sample behavior and effectiveness of the parameters are assessed through simulation experiments, analyzing the bias, mean square error, and mean relative error. To validate its practical utility and highly adaptive right-skewed tail flexibility, the proposed distribution is applied to real-world datasets concerning repairable system failures and groundwater contamination from vinyl chloride, demonstrating its strong capability to model highly skewed empirical profiles. Furthermore, a group acceptance sampling strategy (GASP) for quality assurance is applied to the formulated model.
- logarithmic function,
- Rayleigh distribution,
- moments,
- asymptotic and extremum evaluation,
- Monte Carlo simulation,
- parameter estimation
Citation: Aijaz Ahmad, Bassant Elkalzah, Manzoor A. Khanday, R. A Rather, Hatem E. Semary, Mustafa Bayram, Okechukwu J. Obulezi. A novel logarithmic framework for developing probability distributions with applications to item failures and toxic contamination datasets[J]. AIMS Mathematics, 2026, 11(6): 17972-18025. doi: 10.3934/math.2026733

Related Papers:

Abstract

A new family of distributions, the Log-generator class, is introduced in this work, deriving its basis from the logarithmic function. The study establishes formal expressions for the generator's probability density function. Subsequently, the New Log-Rayleigh distribution (NLRD) is formulated, selecting the Rayleigh distribution as its foundation and concentrating on a particular case within the proposed family. Several distributional properties, including moments, reliability indices, entropies, and order statistics, are derived through systematic mathematical approaches. The finite-sample behavior and effectiveness of the parameters are assessed through simulation experiments, analyzing the bias, mean square error, and mean relative error. To validate its practical utility and highly adaptive right-skewed tail flexibility, the proposed distribution is applied to real-world datasets concerning repairable system failures and groundwater contamination from vinyl chloride, demonstrating its strong capability to model highly skewed empirical profiles. Furthermore, a group acceptance sampling strategy (GASP) for quality assurance is applied to the formulated model.

References

[1]	N. Eugene, C. Lee, F. Famoye, Beta-normal distribution and its applications, Commun. Stat. Theory Methods, 31 (2002), 497–512. https://doi.org/10.1081/STA-120003130 doi: 10.1081/STA-120003130
[2]	K. Zografos, N. Balakrishnan, On families of beta-and generalized gamma-generated distributions and associated inference, Stat. Methodol., 6 (2009), 344–362. https://doi.org/10.1016/j.stamet.2008.12.003 doi: 10.1016/j.stamet.2008.12.003
[3]	M. Alizadeh, G. M. Cordeiro, L. G. B. Pinho, I. Ghosh, The Gompertz-G family of distributions, J. Stat. Theory Pract., 11 (2017), 179–207. https://doi.org/10.1080/15598608.2016.1267668 doi: 10.1080/15598608.2016.1267668
[4]	G. O. Orji, H. O. Etaga, E. M. Almetwally, C. P. Igbokwe, O. C. Aguwa, O. J. Obulezi, A new odd reparameterized exponential transformed-X family of distributions with applications to public health data, Innov. Stat. Prob., 1 (2025), 88–118. https://doi.org/10.64389/isp.2025.01107 doi: 10.64389/isp.2025.01107
[5]	A. M. Gemeay, N. A. Noori, K. Alakkari, M. A. Khaleel, F. A. M. Ali, G. T. Gellow, et al., A new modified arctan model: statistical properties, estimation methods, simulation study, and real-life application, Mod. J. Stat., 2 (2026), 112–137. https://doi.org/10.64389/mjs.2026.02139 doi: 10.64389/mjs.2026.02139
[6]	I. P. Reuben, A. D. Obinna, D. I. John, B. B. Bitrus, A novel alpha power Gumbel-X family of distributions with exponential baseline, Mod. J. Stat., 2 (2026), 180–202. https://doi.org/10.64389/mjs.2026.02157 doi: 10.64389/mjs.2026.02157
[7]	A. A. Khalaf, M. A. Khaleel, E. Hussam, G. T. Gellow, A. T. Hammad, A. M. Gemeay, Bayesian and non-Bayesian approaches for estimating the extended exponential distribution: applications to COVID-19 and carbon fibers, Innov. Stat. Prob., 1 (2025), 60–94. https://doi.org/10.64389/isp.2025.01236 doi: 10.64389/isp.2025.01236
[8]	E. Brito, G. M. Cordeiro, H. M. Yousof, M. Alizadeh, G. O. Silva, The Topp-Leone odd log-logistic family of distributions, J. Stat. Comput. Simul., 87 (2017), 3040–3058. https://doi.org/10.1080/00949655.2017.1351972 doi: 10.1080/00949655.2017.1351972
[9]	D. Kumar, U. Singh, S. K. Singh, A new distribution using sine function: its application to bladder cancer patients data, J. Stat. Appl. Probab., 4 (2015), 417–427. https://doi.org/10.12785/jsap/040309 doi: 10.12785/jsap/040309
[10]	A. A. Al-Babtain, I. Elbatal, C. Chesneau, M. Elgarhy, Sine Topp-Leone-G family of distributions: theory and applications, Open Phys., 18 (2020), 575–577. https://doi.org/10.1515/phys-2020-0180 doi: 10.1515/phys-2020-0180
[11]	R. A. R. Bantan, C. Chesneau, F. Jamal, M. Elgarhy, On the analysis of new COVID-19 cases in Pakistan using an exponentiated version of the M family of distributions, Mathematics, 8 (2020), 953. https://doi.org/10.3390/math8060953 doi: 10.3390/math8060953
[12]	Z. Ahmad, M. Elgarhy, G. Hamedani, N. S. Butt, Odd generalized N-H generated family of distributions with application to exponential model, Pak. J. Stat. Oper. Res., 16 (2020), 53–71. https://doi.org/10.18187/pjsor.v16i1.2295 doi: 10.18187/pjsor.v16i1.2295
[13]	A. S. Al-Moisheer, I. Elbatal, W. Almutiry, M. Elgarhy, Odd inverse power generalized Weibull generated family of distributions: properties and applications, Math. Probl. Eng., 2021 (2021), 5082192. https://doi.org/10.1155/2021/5082192 doi: 10.1155/2021/5082192
[14]	S. A. Alyami, M. G. Babu, I. Elbatal, N. Alotaibi, M. Elgarhy, Type Ⅱ half-logistic odd Fréchet class of distributions: statistical theory and applications, Symmetry, 14 (2022), 1222. https://doi.org/10.3390/sym14061222 doi: 10.3390/sym14061222
[15]	C. Chesneau, H. S. Bakouch, T. Hussain, A new class of probability distributions via cosine and sine functions with applications, Commun. Stat.-Simul. Comput., 48 (2019), 2287–2300. https://doi.org/10.1080/03610918.2018.1440303 doi: 10.1080/03610918.2018.1440303
[16]	A. Alzaatreh, C. Lee, F. Famoye, A new method for generating families of continuous distributions, Metron, 71 (2013), 63–79. https://doi.org/10.1007/s40300-013-0007-y doi: 10.1007/s40300-013-0007-y
[17]	L. Souza, W. R. O. Junior, C. C. R. Brito, C. Chesneau, T. A. E. Ferreira, L. Soares, General properties of cos-G class of distributions with applications, Eurasian Bull. Math., 2 (2019), 63–79.
[18]	Z. Mahmood, C. Chesneau, M. H. Tahir, A new sine-G family of distributions: properties and applications, Bull. Comput. Appl. Math., 7 (2019), 53–81.
[19]	F. Jamal, C. Chesneau, A new family of polyno-expo-trigonometric distributions with applications, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 22 (2020), 1950027. https://doi.org/10.1142/S0219025719500279 doi: 10.1142/S0219025719500279
[20]	H. Anwar, I. H. Dar, M. A. Lone, A novel family of generating distributions based on trigonometric functions with an application to exponential distribution, J. Sci. Res., 65 (2021), 173–179. https://doi.org/10.37398/JSR.2021.650519 doi: 10.37398/JSR.2021.650519
[21]	E. A. El-Sherpieny, E. M. Almetwally, H. Z. Muhammed, Bivariate Weibull-G family based on copula function: properties, Bayesian and non-Bayesian estimation and applications, Stat. Optim. Inf. Comput., 10 (2022), 678–709. https://doi.org/10.19139/soic-2310-5070-1129 doi: 10.19139/soic-2310-5070-1129
[22]	H. Z. Muhammed, E. M. Almetwally, Bayesian and non-Bayesian estimation for the bivariate inverse Weibull distribution under progressive Type-Ⅱ censoring, Ann. Data Sci., 10 (2023), 481–512. https://doi.org/10.1007/s40745-020-00316-7 doi: 10.1007/s40745-020-00316-7
[23]	M. N. Raihen, S. Akter, F. Tabassum, F. Jahan, S. Begum, A statistical analysis of excess mortality mean at COVID-19 in 2020-2021, Comput. J. Math. Stat. Sci., 2 (2022), 223–239. Available from: https://api.semanticscholar.org/CorpusID: 263215922.
[24]	I. Elbatal, M. Elgarhy, B. M. Golam Kibria, Alpha power transformed Weibull-G family of distributions: theory and applications, J. Stat. Theory Appl., 20 (2021), 340–354. https://doi.org/10.2991/jsta.d.210222.002 doi: 10.2991/jsta.d.210222.002
[25]	S. M. Alghamdi, M. Shrahili, A. S. Hassan, R. E. Mohamed, I. Elbatal, M. Elgarhy, Analysis of milk production and failure data: using unit exponentiated half logistic power series class of distributions, Symmetry, 15 (2023), 714. https://doi.org/10.3390/sym15030714 doi: 10.3390/sym15030714
[26]	H. Haj Ahmad, E. M. Almetwally, M. Elgarhy, D. A. Ramadan, On unit exponential Pareto distribution for modeling the recovery rate of COVID-19, Processes, 11 (2023), 232. https://doi.org/10.3390/pr11010232 doi: 10.3390/pr11010232
[27]	L. Murtaza, H. D. Ishfaq, R. J. Tariq, A new method for generating distributions with an application to Weibull distribution, Reliab. Theory Appl., 17 (2022), 223–239. https://doi.org/10.24412/1932-2321-2022-167-223-239 doi: 10.24412/1932-2321-2022-167-223-239
[28]	O. J. Obulezi, Obulezi distribution: a novel one-parameter distribution for lifetime data modeling, Mod. J. Stat., 2 (2025), 32–74. https://doi.org/10.64389/mjs.2026.02140 doi: 10.64389/mjs.2026.02140
[29]	C. K. Onyekwere, O. C. Aguwa, O. J. Obulezi, An updated Lindley distribution: properties, estimation, acceptance sampling, actuarial risk assessment and applications, Innov. Stat. Prob., 1 (2025), 1–27. https://doi.org/10.64389/isp.2025.01103 doi: 10.64389/isp.2025.01103
[30]	J. Muzamil, A. Aijaz, T. Rajnee, Weibull-Power Rayleigh distribution with applications related to distinct fields of science, Reliab. Theory Appl., 17 (2022), 272–290. https://doi.org/10.24412/1932-2321-2022-268-272-290 doi: 10.24412/1932-2321-2022-268-272-290
[31]	A. Aijaz, S. Qurat ul-Ain, A. Afaq, T. Rajnee, Inverse Weibull-Rayleigh distribution characterisation with applications related to cancer data, Reliab. Theory Appl., 16 (2021), 364–382. https://doi.org/10.24412/1932-2321-2021-465-364-382 doi: 10.24412/1932-2321-2021-465-364-382
[32]	A. Aijaz, J. Muzamil, A. Afaq, A novel approach for constructing distributions with an example of the Rayleigh distribution, Reliab. Theory Appl., 17 (2022), 52–64. https://doi.org/10.24412/1932-2321-2022-167-52-64 doi: 10.24412/1932-2321-2022-167-52-64
[33]	K. Singh, A. K. Mahto, Y. Tripathi, L. Wang, Inference for reliability in a multicomponent stress-strength model for a unit inverse Weibull distribution under type-Ⅱ censoring, Qual. Technol. Quant. Manag., 22 (2025), 147–176. https://doi.org/10.1080/16843703.2023.2177811 doi: 10.1080/16843703.2023.2177811
[34]	M. M. Yousef, E. M. Almetwally, Multi stress-strength reliability based on progressive first failure for Kumaraswamy model: Bayesian and non-Bayesian estimation, Symmetry, 13 (2021), 2120. https://doi.org/10.3390/sym13112120 doi: 10.3390/sym13112120
[35]	F. G. Abd EL-Maksoud, G. R. AL-Dayian, A. A. EL-Helbawy, A new mixture of two components of exponentiated family with applications to real life data sets, Comput. J. Math. Stat. Sci., 3 (2024), 316–340. https://doi.org/10.21608/CJMSS.2024.262548.1038 doi: 10.21608/CJMSS.2024.262548.1038
[36]	A. Aijaz, M. A. Fatimah, A. K. Manzoor, G. A. Abd-Elmougod, G. T. Mekiso, M. A. El-Qurashi, et al., Innovative survival modelling in pandemics with a novel family of distributions: a comparative study of UK and Mexico pandemic data, Sci. Rep., 15 (2025), 39378. https://doi.org/10.1038/s41598-025-26210-x doi: 10.1038/s41598-025-26210-x
[37]	X. Gu, S. Li, B. Zhao, Z. Deng, Nonlinear dynamic analysis of a viscoelastic rotating cantilever beam under stochastic perturbation, J. Sound Vib., 619 (2025), 119428. https://doi.org/10.1016/j.jsv.2025.119428 doi: 10.1016/j.jsv.2025.119428
[38]	X. D. Gu, Y. Y. Zhang, I. Mughal, Z. C. Deng, Stochastic responses of nonlinear inclined cables with an attached damper and random excitations, Nonlinear Dyn., 112 (2024), 15969–15986. https://doi.org/10.1007/s11071-024-09877-1 doi: 10.1007/s11071-024-09877-1
[39]	X. Gu, B. Zhao, Z. Deng, T. Wu, Approximate analytical response of nonlinear functionally graded beams subjected to harmonic and random excitations, Int. J. Non-Linear Mech., 148 (2023), 104269. https://doi.org/10.1016/j.ijnonlinmec.2022.104269 doi: 10.1016/j.ijnonlinmec.2022.104269
[40]	P. Beckmann, Rayleigh distribution and its generalizations, J. Res. Natl. Bur. Stand. D, 68D (1964), 927–932. https://doi.org/10.6028/jres.068d.092 doi: 10.6028/jres.068d.092
[41]	C. D. Lai, M. Xie, D. N. P. Murthy, A modified Weibull distribution, IEEE Trans. Reliab., 52 (2003), 33–37. https://doi.org/10.1109/TR.2002.805788 doi: 10.1109/TR.2002.805788
[42]	M. S. Khan, R. King, I. L. Hudson, Transmuted Weibull distribution: properties and estimation, Commun. Stat. Theory Methods, 46 (2017), 5394–5418. https://doi.org/10.1080/03610926.2015.1100744 doi: 10.1080/03610926.2015.1100744
[43]	F. Ashkar, S. Mahdi, Fitting the log-logistic distribution by generalized moments, J. Hydrol., 328 (2006), 694–703. https://doi.org/10.1016/j.jhydrol.2006.01.014 doi: 10.1016/j.jhydrol.2006.01.014
[44]	T. H. M. Abouelmagd, S. Al-Mualim, M. Elgarhy, A. Z. Afify, M. Ahmad, Properties of the four-parameter Weibull distribution and its applications, Pak. J. Statist., 33 (2017), 449–466.
[45]	N. Alsadat, A. S. Hassan, M. Elgarhy, C. Chesneau, R. E. Mohamed, An efficient stress-strength reliability estimate of the unit Gompertz distribution using ranked set sampling, Symmetry, 15 (2023), 1121. https://doi.org/10.3390/sym15051121 doi: 10.3390/sym15051121
[46]	M. Elgarhy, Garhy distribution with different estimation methods and applications to engineering and medical data, Rev. Int. Métodos Numér. Cálc. Diseño Ing., 42 (2026). https://doi.org/10.23967/j.rimni.2026.078324
[47]	S. Al-Marzouki, F. Jamal, C. Chesneau, M. Elgarhy, Type Ⅱ Topp Leone power Lomax distribution with applications, Mathematics, 8 (2019), 4. https://doi.org/10.3390/math8010004 doi: 10.3390/math8010004
[48]	N. M. Alotaibi, I. Elbatal, M. Shrahili, A. S. Al-Moisheer, M. Elgarhy, E. M. Almetwally, Statistical inference for the Kavya-Manoharan Kumaraswamy model under ranked set sampling with applications, Symmetry, 15 (2023), 587. https://doi.org/10.3390/sym15030587 doi: 10.3390/sym15030587
[49]	E. M. Almetwally, M. A. Meraou, Application of environmental data with new extension of Nadarajah-Haghighi distribution, Comput. J. Math. Stat. Sci., 1 (2022), 26–41. https://doi.org/10.21608/CJMSS.2022.271186 doi: 10.21608/CJMSS.2022.271186
[50]	A. M. Basheer, E. M. Almetwally, H. M. Okasha, Marshall-Olkin alpha power inverse Weibull distribution: non Bayesian and Bayesian estimations, J. Stat. Appl. Probab., 10 (2021), 327–345. https://doi.org/10.18576/jsap/100205
[51]	H. Jiang, M. Xie, L. C. Tang, On the odd Weibull distribution, Proc. Inst. Mech. Eng. O, 222 (2008), 583–594. https://doi.org/10.1243/1748006XJRR168 doi: 10.1243/1748006XJRR168
[52]	A. S. Hassan, S. G. Nassr, The inverse Weibull generator of distributions: properties and applications, J. Data Sci., 16 (2018), 723–742. https://doi.org/10.6339/JDS.201810_16(4).00004 doi: 10.6339/JDS.201810_16(4).00004
[53]	D. N. P. Murthy, M. Xie, R. Jiang, Weibull models, Hoboken, NJ: John Wiley & Sons, 2004. https://doi.org/10.1002/047147326X
[54]	D. K. Bhaumik, K. Kapur, R. D. Gibbons, Testing parameters of a gamma distribution for small samples, Technometrics, 51 (2009), 326–334. https://doi.org/10.1198/tech.2009.07038 doi: 10.1198/tech.2009.07038

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)