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Multivariate measures of skewness for the skew-normal distribution


  • Balakrishnan, N.
  • Scarpa, Bruno


The main objective of this work is to calculate and compare different measures of multivariate skewness for the skew-normal family of distributions. For this purpose, we consider the Mardia (1970) [10], Malkovich and Afifi (1973) [9], Isogai (1982) [17], Srivastava (1984) [15], Song (2001) [14], Móri et al. (1993) [11], Balakrishnan et al. (2007) [3] and Kollo (2008) [7] measures of skewness. The exact expressions of all measures of skewness, except for Song's, are derived for the family of skew-normal distributions, while Song's measure of shape is approximated by the use of delta method. The behavior of these measures, their similarities and differences, possible interpretations, and their practical use in testing for multivariate normal are studied by evaluating their power in the case of some specific members of the multivariate skew-normal family of distributions.

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  • 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.
  • Handle: RePEc:eee:jmvana:v:104:y:2012:i:1:p:73-87

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    References listed on IDEAS

    1. Kollo, Tõnu, 2008. "Multivariate skewness and kurtosis measures with an application in ICA," Journal of Multivariate Analysis, Elsevier, vol. 99(10), pages 2328-2338, November.
    2. Genton, Marc G. & He, Li & Liu, Xiangwei, 2001. "Moments of skew-normal random vectors and their quadratic forms," Statistics & Probability Letters, Elsevier, vol. 51(4), pages 319-325, February.
    3. Srivastava, M. S., 1984. "A measure of skewness and kurtosis and a graphical method for assessing multivariate normality," Statistics & Probability Letters, Elsevier, vol. 2(5), pages 263-267, October.
    4. 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.
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    Cited by:

    1. 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.
    2. repec:eee:stapro:v:134:y:2018:i:c:p:1-4 is not listed on IDEAS
    3. repec:eee:ejores:v:263:y:2017:i:2:p:510-523 is not listed on IDEAS
    4. Mehdi Amiri & Ahad Jamalizadeh & Mina Towhidi, 2015. "Inference and further probabilistic properties of the $$ SUN_{n,2}$$ S U N n , 2 -distribution," Statistical Papers, Springer, vol. 56(4), pages 1071-1098, November.
    5. repec:eee:csdana:v:120:y:2018:i:c:p:42-57 is not listed on IDEAS
    6. Balakrishnan, N. & Capitanio, A. & Scarpa, B., 2014. "A test for multivariate skew-normality based on its canonical form," Journal of Multivariate Analysis, Elsevier, vol. 128(C), pages 19-32.


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