IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v11y2023i16p3587-d1220465.html
   My bibliography  Save this article

The Log-Bimodal Asymmetric Generalized Gaussian Model with Application to Positive Data

Author

Listed:
  • Guillermo Martínez-Flórez

    (Departamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Montería 230002, Colombia
    These authors contributed equally to this work.)

  • Roger Tovar-Falón

    (Departamento de Matemáticas y Estadística, Facultad de Ciencias Básicas, Universidad de Córdoba, Montería 230002, Colombia
    These authors contributed equally to this work.)

  • Heleno Bolfarine

    (Departamento de Estatística, Instituto de Matemática e Estatística (IME), Universidade de São Paulo, São Paulo 1010, Brazil)

Abstract

One of the most widely known probability distributions used to explain the probabilistic behavior of positive data is the log-normal (LN). Although the LN distribution is capable of adjusting data types, it is not always fully true that the model manages to adequately model the behavior of the response of interest since in some cases, the degree of skewness and/or kurtosis of the data are greater or less than those that the LN distribution can capture. Another peculiarity of the LN distribution is that it only fits unimodal positive data, which constitutes a limitation when dealing with data that present more than one mode (bimodality). On the other hand, the log-normal model only fits unimodal positive data and in reality there are multiple applications where the behavior of materials is bimodal. To fill this gap, this paper introduces a new probability distribution that is capable of fitting unimodal or bimodal positive data with a high or low degree of skewness and/or kurtosis. The new distribution is a generalization of the LN distribution. For the new proposal, its main properties are studied and the process of estimation of the parameters involved in the model is carried out from a classical perspective using the maximum likelihood method. An important feature of this distribution is the non-singularity of the Fisher information matrix, which guarantees the use of asymptotic theory to study the properties of the parameter estimators. A Monte Carlo type simulation study is carried out to evaluate the properties of the estimators and finally, an illustration is presented with a set of data related to the concentration of nickel in soil samples, allowing to show that the proposed distribution fits extremely well in certain situations.

Suggested Citation

  • Guillermo Martínez-Flórez & Roger Tovar-Falón & Heleno Bolfarine, 2023. "The Log-Bimodal Asymmetric Generalized Gaussian Model with Application to Positive Data," Mathematics, MDPI, vol. 11(16), pages 1-14, August.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:16:p:3587-:d:1220465
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/11/16/3587/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/11/16/3587/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Adelchi Azzalini & Thomas Del Cappello & Samuel Kotz, 2002. "Log-Skew-Normal and Log-Skew-t Distributions as Models for Family Income Data," Journal of Income Distribution, Ad libros publications inc., vol. 11(3-4), pages 2-2, September.
    2. Hamparsum Bozdogan, 1987. "Model selection and Akaike's Information Criterion (AIC): The general theory and its analytical extensions," Psychometrika, Springer;The Psychometric Society, vol. 52(3), pages 345-370, September.
    3. Rameshwar Gupta & Ramesh Gupta, 2008. "Analyzing skewed data by power normal model," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 17(1), pages 197-210, May.
    4. Monica Chiogna, 1998. "Some results on the scalar Skew-normal distribution," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 7(1), pages 1-13, April.
    5. Heleno Bolfarine & Guillermo Martínez-Flórez & Hugo S. Salinas, 2018. "Bimodal symmetric-asymmetric power-normal families," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 47(2), pages 259-276, January.
    6. Arthur Pewsey & Héctor Gómez & Heleno Bolfarine, 2012. "Likelihood-based inference for power distributions," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(4), pages 775-789, December.
    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. Roger Tovar-Falón & Guillermo Martínez-Flórez & Heleno Bolfarine, 2022. "Modelling Asymmetric Data by Using the Log-Gamma-Normal Regression Model," Mathematics, MDPI, vol. 10(7), pages 1-16, April.
    2. 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.
    3. R. N. Rattihalli, 2023. "A Class of Multivariate Power Skew Symmetric Distributions: Properties and Inference for the Power-Parameter," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 85(2), pages 1356-1393, August.
    4. Guillermo Martínez-Flórez & Heleno Bolfarine & Héctor Gómez, 2015. "Doubly censored power-normal regression models with inflation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 24(2), pages 265-286, June.
    5. 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.
    6. Guillermo Martínez-Flórez & Hector W. Gomez & Roger Tovar-Falón, 2021. "Modeling Proportion Data with Inflation by Using a Power-Skew-Normal/Logit Mixture Model," Mathematics, MDPI, vol. 9(16), pages 1-20, August.
    7. Guillermo Martínez-Flórez & Heleno Bolfarine & Héctor W. Gómez, 2017. "The Log-Linear Birnbaum-Saunders Power Model," Methodology and Computing in Applied Probability, Springer, vol. 19(3), pages 913-933, September.
    8. Ioana Gutu & Daniela Tatiana Agheorghiesei & Alexandru Tugui, 2023. "Assessment of a Workforce Sustainability Tool through Leadership and Digitalization," IJERPH, MDPI, vol. 20(2), pages 1-30, January.
    9. Daniela Andreini & Diego Rinallo & Giuseppe Pedeliento & Mara Bergamaschi, 2017. "Brands and Religion in the Secularized Marketplace and Workplace: Insights from the Case of an Italian Hospital Renamed After a Roman Catholic Pope," Journal of Business Ethics, Springer, vol. 141(3), pages 529-550, March.
    10. S. A. Abu Bakar & Saralees Nadarajah & Z. A. Absl Kamarul Adzhar, 2018. "Loss modeling using Burr mixtures," Empirical Economics, Springer, vol. 54(4), pages 1503-1516, June.
    11. Byrd, T. A. & Marshall, T. E., 1997. "Relating information technology investment to organizational performance: a causal model analysis," Omega, Elsevier, vol. 25(1), pages 43-56, February.
    12. García, V.J. & Gómez-Déniz, E. & Vázquez-Polo, F.J., 2010. "A new skew generalization of the normal distribution: Properties and applications," Computational Statistics & Data Analysis, Elsevier, vol. 54(8), pages 2021-2034, August.
    13. Herbert Hoijtink & Meinte Vollema, 2003. "Contemporary Extensions of the Rasch Model," Quality & Quantity: International Journal of Methodology, Springer, vol. 37(3), pages 263-276, August.
    14. Jaewoong Yun, 2023. "Strategies for Improving the Sustainability of Fare-Free Policy for the Elderly through Preferences by Travel Modes," Sustainability, MDPI, vol. 15(20), pages 1-14, October.
    15. Malerba, Martino E. & Connolly, Sean R. & Heimann, Kirsten, 2015. "An experimentally validated nitrate–ammonium–phytoplankton model including effects of starvation length and ammonium inhibition on nitrate uptake," Ecological Modelling, Elsevier, vol. 317(C), pages 30-40.
    16. Aline Riboli Marasca & Maurício Scopel Hoffmann & Anelise Reis Gaya & Denise Ruschel Bandeira, 2021. "Subjective Well-Being and Psychopathology Symptoms: Mental Health Profiles and their Relations with Academic Achievement in Brazilian Children," Child Indicators Research, Springer;The International Society of Child Indicators (ISCI), vol. 14(3), pages 1121-1137, June.
    17. Friederike Paetz, 2016. "Persönlichkeitsmerkmale als Segmentierungsvariablen: Eine empirische Studie [Personality traits for market segmentation: An empirical study]," Schmalenbach Journal of Business Research, Springer, vol. 68(3), pages 279-306, August.
    18. Emre Demirkaya & Yang Feng & Pallavi Basu & Jinchi Lv, 2022. "Large-scale model selection in misspecified generalized linear models [Information theory and an extension of the maximum likelihood principle]," Biometrika, Biometrika Trust, vol. 109(1), pages 123-136.
    19. Rosbergen, Edward & Wedel, Michel & Pieters, Rik, 1997. "Analyzing visual attention tot repeated print advertising using scanpath theory," Research Report 97B32, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    20. Eduardo Correia & Rodrigo Calili & José Francisco Pessanha & Maria Fatima Almeida, 2023. "Definition of Regulatory Targets for Electricity Non-Technical Losses: Proposition of an Automatic Model-Selection Technique for Panel Data Regressions," Energies, MDPI, vol. 16(6), pages 1-22, March.

    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:11:y:2023:i:16:p:3587-:d:1220465. 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.