IDEAS home Printed from https://ideas.repec.org/a/eee/phsmap/v395y2014icp200-208.html
   My bibliography  Save this article

Asymptotic form of the Kullback–Leibler divergence for multivariate asymmetric heavy-tailed distributions

Author

Listed:
  • Contreras-Reyes, Javier E.

Abstract

An asymptotic expression for the Kullback–Leibler (KL) divergence measure of multivariate skew-t distributions (MST) is derived. This novel class of flexible family distributions incorporates a shape and degree of freedom parameters, in order to manipulate the skewness and heavy-tail presence of the data, respectively. The quadratic form expressions of MST models are used to provide asymptotic measures. Additional inequalities for MST entropy and simulation studies are reported. Finally, the expected values of the KL divergence of a sample correlation matrix obtained by Pearson’s correlation coefficient are discussed.

Suggested Citation

  • Contreras-Reyes, Javier E., 2014. "Asymptotic form of the Kullback–Leibler divergence for multivariate asymmetric heavy-tailed distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 395(C), pages 200-208.
    • Handle: RePEc:eee:phsmap:v:395:y:2014:i:c:p:200-208
    DOI: 10.1016/j.physa.2013.10.035
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378437113010078
    Download Restriction: Full text for ScienceDirect subscribers only. Journal offers the option of making the article available online on Science direct for a fee of $3,000
    File URL: https://libkey.io/10.1016/j.physa.2013.10.035?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Reinaldo B. Arellano-Valle & Javier E. Contreras-Reyes & Marc G. Genton, 2013. "Shannon Entropy and Mutual Information for Multivariate Skew-Elliptical Distributions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(1), pages 42-62, March.
    2. M. Tumminello & F. Lillo & R. N. Mantegna, 2007. "Shrinkage and spectral filtering of correlation matrices: a comparison via the Kullback-Leibler distance," Papers 0710.0576, arXiv.org.
    3. Jose, K.K. & Naik, Shanoja R., 2008. "A class of asymmetric pathway distributions and an entropy interpretation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(28), pages 6943-6951.
    4. Eling, Martin, 2012. "Fitting insurance claims to skewed distributions: Are the skew-normal and skew-student good models?," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 239-248.
    5. Kim, Hyoung-Moon & Mallick, Bani K., 2003. "Moments of random vectors with skew t distribution and their quadratic forms," Statistics & Probability Letters, Elsevier, vol. 63(4), pages 417-423, July.
    6. Sandoval, Leonidas & Franca, Italo De Paula, 2012. "Correlation of financial markets in times of crisis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 187-208.
    7. A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
    8. Branco, Márcia D. & Dey, Dipak K., 2001. "A General Class of Multivariate Skew-Elliptical Distributions," Journal of Multivariate Analysis, Elsevier, vol. 79(1), pages 99-113, October.
    9. 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.
    10. Giulio Biroli & Jean-Philippe Bouchaud & Marc Potters, 2007. "The Student ensemble of correlation matrices: eigenvalue spectrum and Kullback-Leibler entropy," Papers 0710.0802, arXiv.org.
    11. Arellano-Valle, Reinaldo B. & Genton, Marc G., 2005. "On fundamental skew distributions," Journal of Multivariate Analysis, Elsevier, vol. 96(1), pages 93-116, September.
    12. Zografos, K. & Nadarajah, S., 2005. "Expressions for Rényi and Shannon entropies for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 71(1), pages 71-84, January.
    13. Tumminello, Michele & Lillo, Fabrizio & Mantegna, Rosario N., 2010. "Correlation, hierarchies, and networks in financial markets," Journal of Economic Behavior & Organization, Elsevier, vol. 75(1), pages 40-58, July.
    14. Seokho Lee & Marc G. Genton & Reinaldo B. Arellano-Valle, 2010. "Perturbation of Numerical Confidential Data via Skew-t Distributions," Management Science, INFORMS, vol. 56(2), pages 318-333, February.
    15. Michele Tumminello & Fabrizio Lillo & Rosario Nunzio Mantegna, 2007. "Kullback-Leibler distance as a measure of the information filtered from multivariate data," Papers 0706.0168, arXiv.org.
    16. Domínguez-Molina, J. Armando & Rocha-Arteaga, Alfonso, 2007. "On the infinite divisibility of some skewed symmetric distributions," Statistics & Probability Letters, Elsevier, vol. 77(6), pages 644-648, March.
    17. Zografos, K., 1999. "On Maximum Entropy Characterization of Pearson's Type II and VII Multivariate Distributions," Journal of Multivariate Analysis, Elsevier, vol. 71(1), pages 67-75, October.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Contreras-Reyes, Javier E., 2021. "Chaotic systems with asymmetric heavy-tailed noise: Application to 3D attractors," Chaos, Solitons & Fractals, Elsevier, vol. 145(C).
    2. Xu, Hai-Yan & Kuo, Shyh-Hao & Li, Guoqi & Legara, Erika Fille T. & Zhao, Daxuan & Monterola, Christopher P., 2016. "Generalized Cross Entropy Method for estimating joint distribution from incomplete information," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 453(C), pages 162-172.
    3. Contreras-Reyes, Javier E., 2022. "Rényi entropy and divergence for VARFIMA processes based on characteristic and impulse response functions," Chaos, Solitons & Fractals, Elsevier, vol. 160(C).
    4. Christian Caamaño-Carrillo & Javier E. Contreras-Reyes, 2022. "A Generalization of the Bivariate Gamma Distribution Based on Generalized Hypergeometric Functions," Mathematics, MDPI, vol. 10(9), pages 1-17, May.
    5. Contreras-Reyes, Javier E., 2015. "Rényi entropy and complexity measure for skew-gaussian distributions and related families," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 433(C), pages 84-91.

    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. Azzalini, Adelchi, 2022. "An overview on the progeny of the skew-normal family— A personal perspective," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    2. 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.
    3. Reinaldo B. Arellano-Valle & Marc G. Genton, 2010. "Multivariate extended skew-t distributions and related families," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 201-234.
    4. Yin, Chuancun & Balakrishnan, Narayanaswamy, 2024. "Stochastic representations and probabilistic characteristics of multivariate skew-elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    5. Kim, Hyoung-Moon & Genton, Marc G., 2011. "Characteristic functions of scale mixtures of multivariate skew-normal distributions," Journal of Multivariate Analysis, Elsevier, vol. 102(7), pages 1105-1117, August.
    6. Arellano-Valle, Reinaldo B. & Azzalini, Adelchi, 2013. "The centred parameterization and related quantities of the skew-t distribution," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 73-90.
    7. Jorge M. Arevalillo & Hilario Navarro, 2021. "Skewness-Kurtosis Model-Based Projection Pursuit with Application to Summarizing Gene Expression Data," Mathematics, MDPI, vol. 9(9), pages 1-18, April.
    8. 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.
    9. Kahrari, F. & Rezaei, M. & Yousefzadeh, F. & Arellano-Valle, R.B., 2016. "On the multivariate skew-normal-Cauchy distribution," Statistics & Probability Letters, Elsevier, vol. 117(C), pages 80-88.
    10. Kim, Hyoung-Moon, 2008. "A note on scale mixtures of skew normal distribution," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1694-1701, September.
    11. Reinaldo B. Arellano-Valle, 2010. "On the information matrix of the multivariate skew-t model," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(3), pages 371-386.
    12. Shushi, Tomer, 2018. "A proof for the existence of multivariate singular generalized skew-elliptical density functions," Statistics & Probability Letters, Elsevier, vol. 141(C), pages 50-55.
    13. Hossein Negarestani & Ahad Jamalizadeh & Sobhan Shafiei & Narayanaswamy Balakrishnan, 2019. "Mean mixtures of normal distributions: properties, inference and application," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 82(4), pages 501-528, May.
    14. Sandoval, Leonidas & Franca, Italo De Paula, 2012. "Correlation of financial markets in times of crisis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(1), pages 187-208.
    15. Fang, B.Q., 2008. "Noncentral matrix quadratic forms of the skew elliptical variables," Journal of Multivariate Analysis, Elsevier, vol. 99(6), pages 1105-1127, July.
    16. Fang, B.Q., 2006. "Sample mean, covariance and T2 statistic of the skew elliptical model," Journal of Multivariate Analysis, Elsevier, vol. 97(7), pages 1675-1690, August.
    17. Eling, Martin, 2014. "Fitting asset returns to skewed distributions: Are the skew-normal and skew-student good models?," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 45-56.
    18. Batiz-Zuk, Enrique & Christodoulakis, George & Poon, Ser-Huang, 2015. "Credit contagion in the presence of non-normal shocks," International Review of Financial Analysis, Elsevier, vol. 37(C), pages 129-139.
    19. Douadia Bougherara & Laurent Piet, 2018. "On the role of probability weighting on WTP for crop insurance with and without yield skewness," Working Papers hal-02790605, HAL.
    20. Sharon Lee & Geoffrey McLachlan, 2013. "On mixtures of skew normal and skew $$t$$ -distributions," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 7(3), pages 241-266, 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:eee:phsmap:v:395:y:2014:i:c:p:200-208. 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: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/physica-a-statistical-mechpplications/ .

    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.