Communications for Statistical Applications and Methods

  • 제25권2호
  • /
  • Pages.109-130
  • /
  • 2018
  • /
  • 2287-7843(pISSN)
  • /
  • 2383-4757(eISSN)
한국통계학회 (The Korean Statistical Society)
권호 표지
DOI QR코드

DOI QR Code

Multivariate measures of skewness for the scale mixtures of skew-normal distributions

  • Kim, Hyoung-Moon (Department of Applied Statistics, Konkuk University) ;
  • Zhao, Jun (Department of Applied Statistics, Konkuk University)
  • 투고 : 2017.10.24
  • 심사 : 2018.01.19
  • 발행 : 2018.03.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Several measures of multivariate skewness for scale mixtures of skew-normal distributions are derived. As a special case, those of multivariate skew-t distribution are considered in detail. Furthermore, the similarities, differences, and behavior of these measures are explored for cases of some specific members of the multivariate skew-normal and skew-t distributions using a simulation study. Since some measures are vectors, it is better to take all measures in the same scale when comparing them. In order to attain such a set of comparable indices, the sample version is considered for each of the skewness measures that are taken as test statistics for the hypothesis of t distribution against skew-t distribution. An application is reported for the data set consisting of 71 total glycerol and magnesium contents in Grignolino wine.

키워드

과제정보

연구 과제 주관 기관 : National Research Foundation of Korea (NRF)

참고문헌

  1. Arevalillo JM and Navarro H (2015). A note on the direction maximizing skewness in multivariate skew-t vectors. Statistics and Probability Letters, 96, 328-332. https://doi.org/10.1016/j.spl.2014.10.014
  2. Azzalini A and Capitanio A (2014). The Skew-Normal and Related Families, Cambridge, United Kingdom.
  3. Balakrishnan N, Brito MR, and Quiroz AJ (2007). A vectorial notion of skewness and its use in testing for multivariate symmetry. Communications in Statistics Theory and Methods, 36, 1757-1767. https://doi.org/10.1080/03610920601126225
  4. Balakrishnan N and Scarpa B (2012). Multivariate measures of skewness for the skew-normal distribution. Journal of Multivariate Analysis, 104, 73-87. https://doi.org/10.1016/j.jmva.2011.06.017
  5. Benjamini Y and Krieger AM (2006). Skewness: Concepts and Measures. In Kotz S, Balakrishnan N, Read CB, Vidakovic B (eds), Encyclopedia of Statistical Sciences (2nd ed), John Wiley & Sons, Hoboken, New Jersey.
  6. Branco MD and Dey DK (2001). A general class of multivariate skew elliptical distributions, Journal of Multivariate Analysis, 79, 99-113. https://doi.org/10.1006/jmva.2000.1960
  7. Capitanio A (2012). On the canonical form of scale mixtures of skew-normal distributions. Available on arXiv (http://arxiv.org/abs/1207.0797).
  8. Isogai T (1982). On a measure of multivariate skewness and a test for multivariate normality. Annals of the Institute of Statistical Mathematics, 34A, 531-541.
  9. Kim HM and Kim C (2017). Moments of scale mixtures of skew-normal distributions and their quadratic forms. Communications in Statistics Theory and Methods, 46, 1117-1126. https://doi.org/10.1080/03610926.2015.1011339
  10. Kollo T (2008). Multivariate skewness and kurtosis measures with an application in ICA, Journal of Multivariate Analysis, 99, 2328-2338.
  11. Malkovich JF and Afifi AA (1973). On tests for multivariate normality. Journal of the American Statistical Association, 68, 176-179. https://doi.org/10.1080/01621459.1973.10481358
  12. Mardia KV (1970). Measures of multivariate skewness and kurtosis with applications. Biometrika, 36, 519-530.
  13. Mori TF, Rohatgi VK, and Szekely GJ (1993). On multivariate skewness and kurtosis. Theory of Probability and its Applications, 38, 547-551.
  14. Song KS (2001). Renyi information, loglikelihood and an intrinsic distribution measure. Journal of Statistical Planning and Inference, 93, 51-69. https://doi.org/10.1016/S0378-3758(00)00169-5
  15. Srivastava MS (1984). A measure of skewness and kurtosis and a graphical method for assessing multivariate normality. Statistics & Probability Letters, 2, 263-267 https://doi.org/10.1016/0167-7152(84)90062-2