Home Generalization method of generating the continuous nested distributions
Article
Licensed
Unlicensed Requires Authentication

Generalization method of generating the continuous nested distributions

  • Mian Muhammad Farooq , Muhammad Mohsin , Muhammad Farman , Ali Akgül EMAIL logo and Muhammad Umer Saleem
Published/Copyright: May 20, 2022
Become an author with De Gruyter Brill
Author Information
International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 4

Abstract

In many life time scenarios, life of one component or system nested in other components or systems. To model these complex structures some so called nested models are required rather than conventional models. This paper introduces the generalization of the method of generating continuous distribution proposed by N. Eugene, C. Lee, and F. Famoye, “Beta-normal distribution and its applications,” Commun. Stat. Theor. Methods, vol. 31, no. 4, pp. 497–512, 2002 and A. Alzaatreh, C. Lee, and F. Famoye, “A new method for generating families of continuous distributions,” Metron, vol. 71, no. 1, pp. 63–79, 2013 which nest one model in other to cope with complex systems. Some important characteristics of the proposed family of generalized distribution have been studied. The famous Beta, Kumaraswami and Gamma generated distributions are special cases of our suggested procedure. Some new distributions have also been developed by using the suggested methodology and their important properties have been discussed as well. A variety of real life data sets are used to demonstrate the efficacy of new suggested distributions and illation is made with baseline models.

Keywords: beta-family; distribution function; generalized distribution; Kumaraswami-generated distribution
Corresponding author: Ali Akgül, Department of Mathematics, Arts and Science Faculty, Siirt University, Siirt 56100, Türkiye, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: None declared.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

References

[1] K. Pearson, “Contributions to the mathematical theory of evolution. ii. skew variation in homogeneous material,” Phil. Trans. Roy. Soc. Lond.: Math. Phys. Eng. Sci., vol. 186, pp. 343–414, 1895. https://doi.org/10.1098/rsta.1895.0010.Search in Google Scholar

[2] I. W. Burr, “Cumulative frequency functions,” Ann. Math. Stat., vol. 13, no. 2, pp. 215–232, 1942. https://doi.org/10.1214/aoms/1177731607.Search in Google Scholar

[3] N. L. Johnson, “Systems of frequency curves generated by methods of translation,” Biometrika, vol. 36, nos. 1–2, pp. 149–176, 1949. https://doi.org/10.1093/biomet/36.1-2.149.Search in Google Scholar

[4] J. Tukey, The Practical Relationship Between the Common Transformations of Percentages of Counts and Amounts (Tech. Rep. No. 36), Princeton, NJ, Statistical Techniques Research Group, Princeton University, 1960.Search in Google Scholar

[5] J. S. Ramberg and B. W. Schmeiser, “An approximate method for generating symmetric random variables,” Commun. ACM, vol. 15, no. 11, pp. 987–990, 1972. https://doi.org/10.1145/355606.361888.Search in Google Scholar

[6] J. S. Ramberg and B. W. Schmeiser, “An approximate method for generating asymmetric random variables,” Commun. ACM, vol. 17, no. 2, pp. 78–82, 1974. https://doi.org/10.1145/360827.360840.Search in Google Scholar

[7] J. S. Ramberg, E. J. Dudewicz, P. R. Tadikamalla, and E. F. Mykytka, “A probability distribution and its uses in fitting data,” Technometrics, vol. 21, no. 2, pp. 201–214, 1979. https://doi.org/10.1080/00401706.1979.10489750.Search in Google Scholar

[8] M. Freimer, G. Kollia, G. S. Mudholkar, and C. T. Lin, “A study of the generalized tukey lambda family,” Commun. Stat. Theor. Methods, vol. 17, no. 10, pp. 3547–3567, 1988. https://doi.org/10.1080/03610928808829820.Search in Google Scholar

[9] Z. A. Karian and E. J. Dudewicz, Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized Bootstrap Methods, CRC Press, 2000.10.1201/9781420038040Search in Google Scholar

[10] A. Azzalini, “A class of distributions which includes the normal ones,” Scand. J. Stat., vol. 12, no. 2, pp. 171–178, 1985.Search in Google Scholar

[11] R. D. Gupta and D. Kundu, “Exponentiated exponential family: an alternative to gamma and Weibull distributions,” Biom. J., vol. 43, no. 1, pp. 117–130, 2001. https://doi.org/10.1002/1521-4036(200102)43:1<117::aid-bimj117>3.0.co;2-r.10.1002/1521-4036(200102)43:1<117::AID-BIMJ117>3.0.CO;2-RSearch in Google Scholar

[12] N. Eugene, C. Lee, and F. Famoye, “Beta-normal distribution and its applications,” Commun. Stat. Theor. Methods, vol. 31, no. 4, pp. 497–512, 2002. https://doi.org/10.1081/sta-120003130.Search in Google Scholar

[13] M. Jones, “Families of distributions arising from distributions of order statistics,” Test, vol. 13, no. 1, pp. 1–43, 2004. https://doi.org/10.1007/bf02602999.Search in Google Scholar

[14] S. Nadarajah and S. Kotz, “The beta Gumbel distribution,” Math. Probl Eng., vol. 2004, no. 4, pp. 323–332, 2004. https://doi.org/10.1155/s1024123x04403068.Search in Google Scholar

[15] S. Nadarajah and S. Kotz, “The beta exponential distribution,” Reliab. Eng. Syst. Saf., vol. 91, no. 6, pp. 689–697, 2006. https://doi.org/10.1016/j.ress.2005.05.008.Search in Google Scholar

[16] F. Famoye, C. Lee, and O. Olumolade, “The beta-Weibull distribution,” J. Stat. Theory App., vol. 4, no. 2, pp. 121–136, 2005.Search in Google Scholar

[17] L. Kong, C. Lee, and J. Sepanski, “On the properties of beta-gamma distribution,” J. Mod. Appl. Stat. Methods, vol. 6, no. 1, p. 18, 2007. https://doi.org/10.22237/jmasm/1177993020.Search in Google Scholar

[18] A. Akinsete, F. Famoye, and C. Lee, “The beta-pareto distribution,” Statistics, vol. 42, no. 6, pp. 547–563, 2008. https://doi.org/10.1080/02331880801983876.Search in Google Scholar

[19] M. Jones, “Kumaraswamy’s distribution: a beta-type distribution with some tractability advantages,” Stat. Methodol., vol. 6, no. 1, pp. 70–81, 2009. https://doi.org/10.1016/j.stamet.2008.04.001.Search in Google Scholar

[20] G. M. Cordeiro and M. de Castro, “A new family of generalized distributions,” J. Stat. Comput. Simulat., vol. 81, no. 7, pp. 883–898, 2011. https://doi.org/10.1080/00949650903530745.Search in Google Scholar

[21] P. Kumaraswamy, “A generalized probability density function for double-bounded random processes,” J. Hydrol., vol. 46, no. 1, pp. 79–88, 1980. https://doi.org/10.1016/0022-1694(80)90036-0.Search in Google Scholar

[22] G. M. Cordeiro, E. M. Ortega, and S. Nadarajah, “The Kumaraswamy Weibull distribution with application to failure data,” J. Franklin Inst., vol. 347, no. 8, pp. 1399–1429, 2010. https://doi.org/10.1016/j.jfranklin.2010.06.010.Search in Google Scholar

[23] M. A. de Pascoa, E. M. Ortega, and G. M. Cordeiro, “The Kumaraswamy generalized gamma distribution with application in survival analysis,” Stat. Methodol., vol. 8, no. 5, pp. 411–433, 2011. https://doi.org/10.1016/j.stamet.2011.04.001.Search in Google Scholar

[24] G. M. Cordeiro, R. R. Pescim, and E. M. Ortega, “The Kumaraswamy generalized half-normal distribution for skewed positive data,” J. Data Sci., vol. 10, no. 2, pp. 195–224, 2012.10.6339/JDS.201204_10(2).0003Search in Google Scholar

[25] J. T. A. S. Ferreira and M. F. J. Steel, “A constructive representation of univariate skewed distributions,” J. Am. Stat. Assoc., vol. 101, no. 474, pp. 823–829, 2006. https://doi.org/10.1198/016214505000001212.Search in Google Scholar

[26] A. Alzaatreh, C. Lee, and F. Famoye, “A new method for generating families of continuous distributions,” Metron, vol. 71, no. 1, pp. 63–79, 2013. https://doi.org/10.1007/s40300-013-0007-y.Search in Google Scholar

[27] G. M. Cordeiro, E. M. Ortega, and D. C. da Cunha, “The exponentiated generalized class of distributions,” J. Data Sci., vol. 11, no. 1, pp. 1–27, 2013.10.6339/JDS.201301_11(1).0001Search in Google Scholar

[28] M. Mahdy and B. Ahmed, “Skew-generalized inverse Weibull distribution and its properties,” Pak. J. Stat. Oper. Res., vol. 32, no. 5, pp. 329–348, 2016.Search in Google Scholar

[29] M. Rasekhi, G. Hamedani, and R. Chinipardaz, “A flexible extension of skew generalized normal distribution,” Metron, vol. 75, no. 1, pp. 87–107, 2017. https://doi.org/10.1007/s40300-017-0106-2.Search in Google Scholar

[30] I. Ghosh and A. Alzaatreh, “A new class of generalized logistic distribution,” Commun. Stat. Theor. Methods, vol. 47, no. 9, pp. 2043–2055, 2018. https://doi.org/10.1080/03610926.2013.835420.Search in Google Scholar

[31] G. M. Cordeiro, A. Z. Afify, E. M. Ortega, A. K. Suzuki, and M. E. Mead, “The odd lomax generator of distributions: properties, estimation and applications,” J. Comput. Appl. Math., vol. 347, pp. 222–237, 2019. https://doi.org/10.1016/j.cam.2018.08.008.Search in Google Scholar

[32] E. Mahmoudi, H. Jafari, and R. Meshkat, “Alpha-skew generalized normal distribution and its applications,” Appl. Appl. Math., vol. 14, no. 2, pp. 784–804, 2019.10.29252/jsri.14.2.219Search in Google Scholar

[33] M. A. Aljarrah, F. Famoye, and C. Lee, “A new generalized normal distribution: properties and applications,” Commun. Stat. Theor. Methods, vol. 48, no. 18, pp. 4474–4491, 2019. https://doi.org/10.1080/03610926.2018.1483509.Search in Google Scholar

[34] M. A. Aljarrah, F. Famoye, and C. Lee, “Generalized logistic distribution and its regression model,” J. Stat. Distrib. Appl., vol. 7, no. 1, pp. 1–21, 2020. https://doi.org/10.1186/s40488-020-00107-8.Search in Google Scholar

[35] J. H. Guardiola, “The spherical-Dirichlet distribution,” J. Stat. Distrib. Appl., vol. 7, no. 1, pp. 1–14, 2020. https://doi.org/10.1186/s40488-020-00106-9.Search in Google Scholar

[36] M. Ijaz, W. K. Mashwani, A. Göktaş, and Y. A. Unvan, “A novel alpha power transformed exponential distribution with real-life applications,” J. Appl. Stat., vol. 48, no. 11, pp. 1–16, 2021.10.1080/02664763.2020.1870673Search in Google Scholar PubMed PubMed Central

[37] R Core Team, R: A Language and Environment for Statistical Computing [Computer Software Manual], Vienna, Austria, R Team, 2016. Available at: http://www.R-project.org/.Search in Google Scholar

[38] T. Ryan, B. Joiner, and B. Ryan, Minitab Student Handbook, Hoboken, New Jersey, Canada, John Wiley & Sons, Inc., 1976. Available at: https://books.google.com.pk/books?id=hRH0piMuseMC.Search in Google Scholar

[39] D. Murthy, M. Xie, and R. Jiang, Weibull Models, Wiley, New York, 2004. Available at: https://books.google.com.pk/books?id=1c5B6w9RZHYC.Search in Google Scholar

[40] P. Manavalan and W. C. JohnsonJr, “Variable selection method improves the prediction of protein secondary structure from circular dichroism spectra,” Anal. Biochem., vol. 167, no. 1, pp. 76–85, 1987. https://doi.org/10.1016/0003-2697(87)90135-7.Search in Google Scholar PubMed

Received: 2021-06-09
Revised: 2021-10-16
Accepted: 2022-04-18
Published Online: 2022-05-20

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Image-based 3D reconstruction precision using a camera mounted on a robot arm
  4. Switched-line network with digital phase shifter
  5. M-lump waves and their interactions with multi-soliton solutions for the (3 + 1)-dimensional Jimbo–Miwa equation
  6. Optimal control for a class of fractional order neutral evolution equations
  7. Perceptual evaluation for Zhangpu paper-cut patterns by using improved GWO-BP neural network
  8. Two new iterative schemes to approximate the fixed points for mappings
  9. Ulam’s type stability of impulsive delay integrodifferential equations in Banach spaces
  10. Generalization method of generating the continuous nested distributions
  11. Wellposedness of impulsive functional abstract second-order differential equations with state-dependent delay
  12. Numerical study of heat and mass transfer on the pulsatile flow of blood under atherosclerotic condition
  13. Dynamic propagation behaviors of pure mode I crack under stress wave loading by caustics
  14. Numerical simulation of buoyancy-induced heat transfer and entropy generation in 3D C-shaped cavity filled with CNT–Al2O3/water hybrid nanofluid
  15. On coupled system of nonlinear Ψ-Hilfer hybrid fractional differential equations
  16. Hellinger–Reissner variational principle for a class of specified stress problems
  17. Viscous dissipation effect on steady natural convection Couette flow with convective boundary condition
  18. Fredholm determinants and Z n -mKdV/Z n -sinh-Gordon hierarchies
  19. New soliton waves and modulation instability analysis for a metamaterials model via the integration schemes
  20. A modified high-order symmetrical WENO scheme for hyperbolic conservation laws
  21. Cryptanalysis of various images based on neural networks with leakage and time varying delays
  22. Spectral collocation method approach to thermal stability of MHD reactive squeezed fluid flow through a channel
  23. Higher order Traub–Steffensen type methods and their convergence analysis in Banach spaces
https://doi.org/10.1515/ijnsns-2021-0231
Keywords for this article
beta-family; distribution function; generalized distribution; Kumaraswami-generated distribution

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. Image-based 3D reconstruction precision using a camera mounted on a robot arm
  4. Switched-line network with digital phase shifter
  5. M-lump waves and their interactions with multi-soliton solutions for the (3 + 1)-dimensional Jimbo–Miwa equation
  6. Optimal control for a class of fractional order neutral evolution equations
  7. Perceptual evaluation for Zhangpu paper-cut patterns by using improved GWO-BP neural network
  8. Two new iterative schemes to approximate the fixed points for mappings
  9. Ulam’s type stability of impulsive delay integrodifferential equations in Banach spaces
  10. Generalization method of generating the continuous nested distributions
  11. Wellposedness of impulsive functional abstract second-order differential equations with state-dependent delay
  12. Numerical study of heat and mass transfer on the pulsatile flow of blood under atherosclerotic condition
  13. Dynamic propagation behaviors of pure mode I crack under stress wave loading by caustics
  14. Numerical simulation of buoyancy-induced heat transfer and entropy generation in 3D C-shaped cavity filled with CNT–Al2O3/water hybrid nanofluid
  15. On coupled system of nonlinear Ψ-Hilfer hybrid fractional differential equations
  16. Hellinger–Reissner variational principle for a class of specified stress problems
  17. Viscous dissipation effect on steady natural convection Couette flow with convective boundary condition
  18. Fredholm determinants and Z n -mKdV/Z n -sinh-Gordon hierarchies
  19. New soliton waves and modulation instability analysis for a metamaterials model via the integration schemes
  20. A modified high-order symmetrical WENO scheme for hyperbolic conservation laws
  21. Cryptanalysis of various images based on neural networks with leakage and time varying delays
  22. Spectral collocation method approach to thermal stability of MHD reactive squeezed fluid flow through a channel
  23. Higher order Traub–Steffensen type methods and their convergence analysis in Banach spaces
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 29.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ijnsns-2021-0231/html
Scroll to top button