IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v10y2022i14p2370-d856928.html
   My bibliography  Save this article

Modelling Bimodal Data Using a Multivariate Triangular-Linked Distribution

Author

Listed:
  • Daan de Waal

    (Department of Statistics, University of Pretoria, Pretoria 0002, South Africa
    Department of Mathematical Statistics and Actuarial Science, University of Free State, Bloemfontein 9301, South Africa
    These authors contributed equally to this work.)

  • Tristan Harris

    (Department of Statistics, University of Pretoria, Pretoria 0002, South Africa
    These authors contributed equally to this work.)

  • Alta de Waal

    (Department of Statistics, University of Pretoria, Pretoria 0002, South Africa
    Centre for Artificial Intelligence Research, Cape Town, South Africa
    These authors contributed equally to this work.)

  • Jocelyn Mazarura

    (Department of Statistics, University of Pretoria, Pretoria 0002, South Africa
    These authors contributed equally to this work.)

Abstract

Bimodal distributions have rarely been studied although they appear frequently in datasets. We develop a novel bimodal distribution based on the triangular distribution and then expand it to the multivariate case using a Gaussian copula. To determine the goodness of fit of the univariate model, we use the Kolmogorov–Smirnov (KS) and Cramér–von Mises (CVM) tests. The contributions of this work are that a simplistic yet robust distribution was developed to deal with bimodality in data, a multivariate distribution was developed as a generalisation of this univariate distribution using a Gaussian copula, a comparison between parametric and semi-parametric approaches to modelling bimodality is given, and an R package called btld is developed from the workings of this paper.

Suggested Citation

  • Daan de Waal & Tristan Harris & Alta de Waal & Jocelyn Mazarura, 2022. "Modelling Bimodal Data Using a Multivariate Triangular-Linked Distribution," Mathematics, MDPI, vol. 10(14), pages 1-20, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:14:p:2370-:d:856928
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/10/14/2370/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/10/14/2370/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Carlo De Michele & Francesco Accatino, 2014. "Tree Cover Bimodality in Savannas and Forests Emerging from the Switching between Two Fire Dynamics," PLOS ONE, Public Library of Science, vol. 9(3), pages 1-7, March.
    2. Hien D. Nguyen & Geoffrey McLachlan, 2019. "On approximations via convolution-defined mixture models," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 48(16), pages 3945-3955, August.
    3. Yanyuan Ma & Marc G. Genton, 2004. "Flexible Class of Skew‐Symmetric Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 31(3), pages 459-468, September.
    4. Barry Arnold & Robert Beaver & A. Azzalini & N. Balakrishnan & A. Bhaumik & D. Dey & C. Cuadras & J. Sarabia & Barry Arnold & Robert Beaver, 2002. "Skewed multivariate models related to hidden truncation and/or selective reporting," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 11(1), pages 7-54, June.
    5. Benaglia, Tatiana & Chauveau, Didier & Hunter, David R. & Young, Derek S., 2009. "mixtools: An R Package for Analyzing Mixture Models," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 32(i06).
    6. Altemir da Silva Braga & Gauss M. Cordeiro & Edwin M. M. Ortega, 2018. "A new skew-bimodal distribution with applications," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(12), pages 2950-2968, June.
    7. Adelchi Azzalini & Antonella Capitanio, 2003. "Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t‐distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(2), pages 367-389, May.
    8. Pragya Sur & Galit Shmueli & Smarajit Bose & Paromita Dubey, 2015. "Modeling Bimodal Discrete Data Using Conway-Maxwell-Poisson Mixture Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(3), pages 352-365, July.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Guillermo Martínez-Flórez & Diego I. Gallardo & Osvaldo Venegas & Heleno Bolfarine & Héctor W. Gómez, 2021. "Flexible Power-Normal Models with Applications," Mathematics, MDPI, vol. 9(24), pages 1-15, December.
    2. Ley, Christophe & Paindaveine, Davy, 2010. "On the singularity of multivariate skew-symmetric models," Journal of Multivariate Analysis, Elsevier, vol. 101(6), pages 1434-1444, July.
    3. Juan Duarte & Guillermo Martínez-Flórez & Diego Ignacio Gallardo & Osvaldo Venegas & Héctor W. Gómez, 2023. "A Bimodal Extension of the Epsilon-Skew-Normal Model," Mathematics, MDPI, vol. 11(3), pages 1-18, January.
    4. Isaac E. Cortés & Osvaldo Venegas & Héctor W. Gómez, 2022. "A Symmetric/Asymmetric Bimodal Extension Based on the Logistic Distribution: Properties, Simulation and Applications," Mathematics, MDPI, vol. 10(12), pages 1-17, June.
    5. Panagiotelis, Anastasios & Smith, Michael, 2010. "Bayesian skew selection for multivariate models," Computational Statistics & Data Analysis, Elsevier, vol. 54(7), pages 1824-1839, July.
    6. Ali Genç, 2013. "A skew extension of the slash distribution via beta-normal distribution," Statistical Papers, Springer, vol. 54(2), pages 427-442, May.
    7. Cornelis J. Potgieter & Marc G. Genton, 2013. "Characteristic Function-based Semiparametric Inference for Skew-symmetric Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(3), pages 471-490, September.
    8. Samuel Kotz & Donatella Vicari, 2005. "Survey of developments in the theory of continuous skewed distributions," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 225-261.
    9. Zinoviy Landsman & Udi Makov & Tomer Shushi, 2017. "Extended Generalized Skew-Elliptical Distributions and their Moments," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 79(1), pages 76-100, February.
    10. David Elal-Olivero & Juan F. Olivares-Pacheco & Osvaldo Venegas & Heleno Bolfarine & Héctor W. Gómez, 2020. "On Properties of the Bimodal Skew-Normal Distribution and an Application," Mathematics, MDPI, vol. 8(5), pages 1-16, May.
    11. Julio Mulero & Miguel A. Sordo & Marilia C. de Souza & Alfonso Suárez‐LLorens, 2017. "Two stochastic dominance criteria based on tail comparisons," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 33(6), pages 575-589, November.
    12. Hok Shing Kwong & Saralees Nadarajah, 2022. "A New Robust Class of Skew Elliptical Distributions," Methodology and Computing in Applied Probability, Springer, vol. 24(3), pages 1669-1691, September.
    13. Jamalizadeh, A. & Mehrali, Y. & Balakrishnan, N., 2009. "Recurrence relations for bivariate t and extended skew-t distributions and an application to order statistics from bivariate t," Computational Statistics & Data Analysis, Elsevier, vol. 53(12), pages 4018-4027, October.
    14. M. C. Jones, 2015. "On Families of Distributions with Shape Parameters," International Statistical Review, International Statistical Institute, vol. 83(2), pages 175-192, August.
    15. S. Cabras & M. E. Castellanos, 2009. "Default Bayesian goodness-of-fit tests for the skew-normal model," Journal of Applied Statistics, Taylor & Francis Journals, vol. 36(2), pages 223-232.
    16. Christophe Ley, 2014. "Flexible Modelling in Statistics: Past, present and Future," Working Papers ECARES ECARES 2014-42, ULB -- Universite Libre de Bruxelles.
    17. Sreenivasa Rao Jammalamadaka & Emanuele Taufer & Gyorgy H. Terdik, 2021. "On Multivariate Skewness and Kurtosis," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 607-644, August.
    18. Emilio Gómez-Déniz & Barry C. Arnold & José M. Sarabia & Héctor W. Gómez, 2021. "Properties and Applications of a New Family of Skew Distributions," Mathematics, MDPI, vol. 9(1), pages 1-18, January.
    19. Azzalini, Adelchi, 2022. "An overview on the progeny of the skew-normal family— A personal perspective," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    20. Kim, Hea-Jung, 2008. "A class of weighted multivariate normal distributions and its properties," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1758-1771, September.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:10:y:2022:i:14:p:2370-:d:856928. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.