IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v113y2013icp91-105.html
   My bibliography  Save this article

A necessary test of fit of specific elliptical distributions based on an estimator of Song’s measure

Author

Listed:
  • Batsidis, Apostolos
  • Zografos, Konstantinos

Abstract

In a recent paper, Zografos [K. Zografos, On Mardia’s and Song’s measures of kurtosis in elliptical distributions, J. Multivariate Anal. 99 (2008) 858–879] has obtained general formulas for Song’s measure for the elliptic family of distributions, and he introduced and studied its sample analogue. In this paper, based on the empirical estimator of this measure, we present a test to verify if the data are distributed according to a specific elliptical (spherical) distribution. In this context, the asymptotic distribution of the proposed statistic under the null hypothesis of specific spherical distributions is obtained. The proposed statistic also provides us with a procedure for testing multivariate normality. In order to evaluate the convergence of the proposed statistic to its limiting distribution, under the null hypothesis, a simulation study is performed to analyze the behavior of the percentiles of the proposed statistic in some special cases of spherical distributions. Moreover, a Monte Carlo study is carried out on the performance of the test statistic as a necessary test of fit of specific spherical distributions. In this framework, the type I error rates as well as the power of the test are studied. Finally, a well-known data set is used to illustrate the method developed in this paper.

Suggested Citation

  • Batsidis, Apostolos & Zografos, Konstantinos, 2013. "A necessary test of fit of specific elliptical distributions based on an estimator of Song’s measure," Journal of Multivariate Analysis, Elsevier, vol. 113(C), pages 91-105.
    • Handle: RePEc:eee:jmvana:v:113:y:2013:i:c:p:91-105
    DOI: 10.1016/j.jmva.2011.09.006
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X11001941
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2011.09.006?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. Balakrishnan, N. & Scarpa, Bruno, 2012. "Multivariate measures of skewness for the skew-normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 104(1), pages 73-87, February.
    2. J. P. Royston, 1983. "Some Techniques for Assessing Multivarate Normality Based on the Shapiro‐Wilk W," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 32(2), pages 121-133, June.
    3. Blattberg, Robert C & Gonedes, Nicholas J, 1974. "A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices," The Journal of Business, University of Chicago Press, vol. 47(2), pages 244-280, April.
    4. Huffer, Fred W. & Park, Cheolyong, 2007. "A test for elliptical symmetry," Journal of Multivariate Analysis, Elsevier, vol. 98(2), pages 256-281, February.
    5. Cambanis, Stamatis & Huang, Steel & Simons, Gordon, 1981. "On the theory of elliptically contoured distributions," Journal of Multivariate Analysis, Elsevier, vol. 11(3), pages 368-385, September.
    6. Schott, James R., 2002. "Testing for elliptical symmetry in covariance-matrix-based analyses," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 395-404, December.
    7. Fang, K. T. & Zhu, L. X. & Bentler, P. M., 1993. "A Necessary Test of Goodness of Fit for Sphericity," Journal of Multivariate Analysis, Elsevier, vol. 45(1), pages 34-55, April.
    8. Zografos, K., 2008. "On Mardia's and Song's measures of kurtosis in elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 99(5), pages 858-879, May.
    9. Jiajuan Liang & Kai-Tai Fang & Fred Hickernell, 2008. "Some necessary uniform tests for spherical symmetry," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(3), pages 679-696, September.
    10. Sakhanenko, Lyudmila, 2008. "Testing for ellipsoidal symmetry: A comparison study," Computational Statistics & Data Analysis, Elsevier, vol. 53(2), pages 565-581, December.
    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. Iwashita, Toshiya & Klar, Bernhard, 2014. "The joint distribution of Studentized residuals under elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 203-209.
    2. Sakineh Dehghan & Mohammad Reza Faridrohani & Zahra Barzegar, 2023. "Testing for diagonal symmetry based on center-outward ranking," Statistical Papers, Springer, vol. 64(1), pages 255-283, February.
    3. Liebscher Eckhard & Richter Wolf-Dieter, 2020. "Modelling with star-shaped distributions," Dependence Modeling, De Gruyter, vol. 8(1), pages 45-69, January.
    4. Liebscher Eckhard & Richter Wolf-Dieter, 2020. "Modelling with star-shaped distributions," Dependence Modeling, De Gruyter, vol. 8(1), pages 45-69, January.
    5. Boente, Graciela & Salibián Barrera, Matías & Tyler, David E., 2014. "A characterization of elliptical distributions and some optimality properties of principal components for functional data," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 254-264.

    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. Albisetti, Isaia & Balabdaoui, Fadoua & Holzmann, Hajo, 2020. "Testing for spherical and elliptical symmetry," Journal of Multivariate Analysis, Elsevier, vol. 180(C).
    2. Jiajuan Liang & Kai Wang Ng & Guoliang Tian, 2019. "A class of uniform tests for goodness-of-fit of the multivariate $$L_p$$ L p -norm spherical distributions and the $$l_p$$ l p -norm symmetric distributions," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 71(1), pages 137-162, February.
    3. Sladana Babic & Laetitia Gelbgras & Marc Hallin & Christophe Ley, 2019. "Optimal tests for elliptical symmetry: specified and unspecified location," Working Papers ECARES 2019-26, ULB -- Universite Libre de Bruxelles.
    4. Frank Schuhmacher & Hendrik Kohrs & Benjamin R. Auer, 2021. "Justifying Mean-Variance Portfolio Selection when Asset Returns Are Skewed," Management Science, INFORMS, vol. 67(12), pages 7812-7824, December.
    5. Manzotti, A. & Pérez, Francisco J. & Quiroz, Adolfo J., 2002. "A Statistic for Testing the Null Hypothesis of Elliptical Symmetry," Journal of Multivariate Analysis, Elsevier, vol. 81(2), pages 274-285, May.
    6. Valdez, Emiliano A. & Dhaene, Jan & Maj, Mateusz & Vanduffel, Steven, 2009. "Bounds and approximations for sums of dependent log-elliptical random variables," Insurance: Mathematics and Economics, Elsevier, vol. 44(3), pages 385-397, June.
    7. Ali Genç, 2013. "Moments of truncated normal/independent distributions," Statistical Papers, Springer, vol. 54(3), pages 741-764, August.
    8. Yin, Chuancun & Balakrishnan, Narayanaswamy, 2024. "Stochastic representations and probabilistic characteristics of multivariate skew-elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 199(C).
    9. Norbert Henze & Celeste Mayer, 2020. "More good news on the HKM test for multivariate reflected symmetry about an unknown centre," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(3), pages 741-770, June.
    10. Francq, C. & Jiménez-Gamero, M.D. & Meintanis, S.G., 2017. "Tests for conditional ellipticity in multivariate GARCH models," Journal of Econometrics, Elsevier, vol. 196(2), pages 305-319.
    11. Sakineh Dehghan & Mohammad Reza Faridrohani & Zahra Barzegar, 2023. "Testing for diagonal symmetry based on center-outward ranking," Statistical Papers, Springer, vol. 64(1), pages 255-283, February.
    12. Baishuai Zuo & Narayanaswamy Balakrishnan & Chuancun Yin, 2023. "An analysis of multivariate measures of skewness and kurtosis of skew-elliptical distributions," Papers 2311.18176, arXiv.org.
    13. Fotopoulos, Stergios B. & Jandhyala, Venkata K., 2004. "Bessel inequalities with applications to conditional log returns under GIG scale mixtures of normal vectors," Statistics & Probability Letters, Elsevier, vol. 66(2), pages 117-125, January.
    14. Pere, Jaakko & Ilmonen, Pauliina & Viitasaari, Lauri, 2024. "On extreme quantile region estimation under heavy-tailed elliptical distributions," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    15. Niu, Lu & Liu, Xiumin & Zhao, Junlong, 2020. "Robust estimator of the correlation matrix with sparse Kronecker structure for a high-dimensional matrix-variate," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    16. Faliva, Mario & Quatto, Piero & Zoia, Maria Grazia, 2018. "Gram–Charlier-like expansions of power-raised hyperbolic secant laws," Statistics & Probability Letters, Elsevier, vol. 137(C), pages 229-234.
    17. Boente, Graciela & Salibián Barrera, Matías & Tyler, David E., 2014. "A characterization of elliptical distributions and some optimality properties of principal components for functional data," Journal of Multivariate Analysis, Elsevier, vol. 131(C), pages 254-264.
    18. Jensen, Mark J. & Maheu, John M., 2010. "Bayesian semiparametric stochastic volatility modeling," Journal of Econometrics, Elsevier, vol. 157(2), pages 306-316, August.
    19. Schott, James R., 2002. "Testing for elliptical symmetry in covariance-matrix-based analyses," Statistics & Probability Letters, Elsevier, vol. 60(4), pages 395-404, December.
    20. Geluk, J.L. & De Vries, C.G., 2006. "Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities," Insurance: Mathematics and Economics, Elsevier, vol. 38(1), pages 39-56, February.

    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:jmvana:v:113:y:2013:i:c:p:91-105. 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.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    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.