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Generalized beta-generated distributions

Author

Listed:
  • Alexander, Carol
  • Cordeiro, Gauss M.
  • Ortega, Edwin M.M.
  • Sarabia, José María

Abstract

This article introduces generalized beta-generated (GBG) distributions. Sub-models include all classical beta-generated, Kumaraswamy-generated and exponentiated distributions. They are maximum entropy distributions under three intuitive conditions, which show that the classical beta generator skewness parameters only control tail entropy and an additional shape parameter is needed to add entropy to the centre of the parent distribution. This parameter controls skewness without necessarily differentiating tail weights. The GBG class also has tractable properties: we present various expansions for moments, generating function and quantiles. The model parameters are estimated by maximum likelihood and the usefulness of the new class is illustrated by means of some real data sets.

Suggested Citation

  • Alexander, Carol & Cordeiro, Gauss M. & Ortega, Edwin M.M. & Sarabia, José María, 2012. "Generalized beta-generated distributions," Computational Statistics & Data Analysis, Elsevier, vol. 56(6), pages 1880-1897.
    • Handle: RePEc:eee:csdana:v:56:y:2012:i:6:p:1880-1897
    DOI: 10.1016/j.csda.2011.11.015
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    1. James B. McDonald, 2008. "Some Generalized Functions for the Size Distribution of Income," Economic Studies in Inequality, Social Exclusion, and Well-Being, in: Duangkamon Chotikapanich (ed.), Modeling Income Distributions and Lorenz Curves, chapter 3, pages 37-55, Springer.
    2. M. C. Jones & M. J. Faddy, 2003. "A skew extension of the t‐distribution, with applications," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 65(1), pages 159-174, February.
    3. Brys, Guy & Hubert, Mia & Struyf, Anja, 2006. "Robust measures of tail weight," Computational Statistics & Data Analysis, Elsevier, vol. 50(3), pages 733-759, February.
    4. Zhu, Dongming & Galbraith, John W., 2010. "A generalized asymmetric Student-t distribution with application to financial econometrics," Journal of Econometrics, Elsevier, vol. 157(2), pages 297-305, August.
    5. Nadarajah, Saralees & Gupta, Arjun K., 2007. "A generalized gamma distribution with application to drought data," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 74(1), pages 1-7.
    6. Nadarajah, Saralees, 2008. "Explicit expressions for moments of order statistics," Statistics & Probability Letters, Elsevier, vol. 78(2), pages 196-205, February.
    7. Amit Choudhury, 2005. "A Simple Derivation of Moments of the Exponentiated Weibull Distribution," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 62(1), pages 17-22, September.
    8. Nadarajah, Saralees & Kotz, Samuel, 2006. "The beta exponential distribution," Reliability Engineering and System Safety, Elsevier, vol. 91(6), pages 689-697.
    9. M. C. Jones & P. V. Larsen, 2004. "Multivariate distributions with support above the diagonal," Biometrika, Biometrika Trust, vol. 91(4), pages 975-986, December.
    10. McDonald, James B. & Xu, Yexiao J., 1995. "A generalization of the beta distribution with applications," Journal of Econometrics, Elsevier, vol. 69(2), pages 427-428, October.
    11. 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.
    12. Hansen, Bruce E, 1994. "Autoregressive Conditional Density Estimation," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 35(3), pages 705-730, August.
    13. M. Jones, 2004. "Families of distributions arising from distributions of order statistics," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 13(1), pages 1-43, June.
    14. Kjersti Aas & Ingrid Hobaek Haff, 2006. "The Generalized Hyperbolic Skew Student's t-Distribution," Journal of Financial Econometrics, Oxford University Press, vol. 4(2), pages 275-309.
    15. Arnold, Barry C. & Castillo, Enrique & Sarabia, Jose Maria, 2006. "Families of Multivariate Distributions Involving the Rosenblatt Construction," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 1652-1662, December.
    16. Stasinopoulos, D. Mikis & Rigby, Robert A., 2007. "Generalized Additive Models for Location Scale and Shape (GAMLSS) in R," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 23(i07).
    17. Ebrahimi, Nader & Maasoumi, Esfandiar & Soofi, Ehsan S., 1999. "Ordering univariate distributions by entropy and variance," Journal of Econometrics, Elsevier, vol. 90(2), pages 317-336, June.
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    More about this item

    Keywords

    Entropy; Exponentiated; Kumaraswamy; Kurtosis; McDonald; Minimax; Skewness;
    All these keywords.

    JEL classification:

    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions
    • G1 - Financial Economics - - General Financial Markets

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