Advertisement

An alternative data analytic approach to measure the univariate and multivariate skewness

  • Ravindra KhattreeEmail author
  • Manoj Bahuguna
Regular Paper

Abstract

We introduce a new measure of univariate skewness of a distribution or data based on quantiles and by using the concepts of even and odd functions. Based on this new measure, we then suggest an approach to define the multivariate skewness for the multivariate distributions and multidimensional data and accordingly suggest a measure for it. Using numerous data sets, we illustrate that Mardia’s measure of multivariate skewness appears to be ambiguous in what it actually measures and show that our measure not only has an intuitive appeal, it also unambiguously quantifies what one would view as the multivariate skewness. Approach presented here is data analytic and can be implemented on a computer. Based on the idea of orthogonal transformation of the data, we also suggest another multivariate measure of skewness which may be simpler to compute.

Keywords

Asymmetry Mardia’s measure of multivariate skewness Quantiles Skewness 

Notes

Acknowledgements

We wish to thank three referees for their helpful suggestions.

References

  1. 1.
    Baringhaus, L., Henze, N.: Limit distributions for measures of multivariate skewness and kurtosis based on projections. J. Multivar. Anal. 38(1), 51–69 (1991)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Benjamini, Y., Krieger, A.M.: Skewness: Concepts and Measures Encyclopedia of Statistical Sciences. Wiley Online Library, Hoboken (2006)Google Scholar
  3. 3.
    Bowley, A.L.: Elements of Statistics, vol. 2. P. S. King, Westminster (1920)zbMATHGoogle Scholar
  4. 4.
    Brown, C.A., Robinson, D.M.: Skewness and kurtosis implied by option prices: a correction. J. Financ. Res. 25(2), 279–282 (2002)Google Scholar
  5. 5.
    Chatterjee, S., Hadi, A.S., Price, B.: Regression Analysis by Example. Wiley, Hoboken (2000)zbMATHGoogle Scholar
  6. 6.
    Chvosta, J., Erdman, D.J., Little, M.: Modeling financial risk factor correlation with the copula procedure. In: SAS Global Forum, pp. 340–2011 (2011)Google Scholar
  7. 7.
    Corrado, C.J., Su, T.: Skewness and kurtosis in S&P 500 index returns implied by option prices. J. Financ. Res. XIX(2), 175–192 (1996)Google Scholar
  8. 8.
    Flurry, B., Riedwyl, H.: Multivariate Statistics: A Practical Approach. Chapman and Hall, London (1988)Google Scholar
  9. 9.
    Groeneveld, R.A.: Skewness, Bowley’s Measures of Encyclopedia of Statistical Sciences. Wiley Online Library, Hoboken (2006)Google Scholar
  10. 10.
    Groeneveld, R.A., Meeden, G.: Measuring skewness and kurtosis. Statistician 33, 391–399 (1984)Google Scholar
  11. 11.
    Harvey, C.R., Siddique, A.: Conditional skewness in asset pricing tests. J. Finance 55(3), 1263–1295 (2000a)Google Scholar
  12. 12.
    Harvey, C.R., Siddique, A.: Time-varying conditional skewness and the market risk premium. Res. Bank. Finance 1(1), 27–60 (2000b)Google Scholar
  13. 13.
    Hinkley, D.V.: On power transformations to symmetry. Biometrika 62(1), 101–111 (1975)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Joanes, D.N., Gill, C.A.: Comparing measures of sample skewness and kurtosis. J. R. Stat. Soc. Ser. D (Statistician) 47(1), 183–189 (1998)Google Scholar
  15. 15.
    Khattree, R., Naik, D.N.: Multivariate Data Reduction and Discrimination with SAS Software. SAS Institute Inc, Cary (2000)Google Scholar
  16. 16.
    Kim, T.H., White, H.: On more robust estimation of skewness and kurtosis. Finance Res. Lett. 1(1), 56–73 (2004)Google Scholar
  17. 17.
    Kirby, M.: Geometric Data Analysis: An Empirical Approach to Dimensionality Reduction and the Study of Patterns. Wiley, Hoboken (2001)zbMATHGoogle Scholar
  18. 18.
    Kraus, A., Litzenberger, R.H.: Skewness preference and the valuation of risk assets. J. Finance 31(4), 1085–1100 (1976)Google Scholar
  19. 19.
    MacGillivray, H.L.: Skewness and asymmetry: measures of ordering. Ann. Stat. 14, 994–1011 (1986)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Malkovich, J.F., Afifi, A.A.: On tests for multivariate normality. J. Am. Stat. Assoc. 68, 176–179 (1973)Google Scholar
  21. 21.
    Mardia, K.V.: Measures of multivariate skewness and kurtosis with applications. Biometrika 57, 519–530 (1970)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mardia, K.V.: Applications of some measures of multivariate skewness and kurtosis in testing normality and robustness studies. Sankhyā Indian J. Stat. Ser. B 36(2), 115–128 (1974)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Mardia, K.V., Foster, K.: Omnibus tests of multinormality based on skewness and kurtosis. Commun. Stat. Theory Methods 12(2), 207–221 (1983)MathSciNetGoogle Scholar
  24. 24.
    Mardia, K.V., Zemroch, P.J.: Algorithm AS 84: measures of multivariate skewness and kurtosis. J. R. Stat. Soc. Ser. C (Appl. Stat.) 24(2), 262–265 (1975)Google Scholar
  25. 25.
    Móri, T., Rohatgi, V.K., Székely, G.J.: On multivariate skewness and kurtosis. Theory Probab. Appl. 38(3), 547–551 (1994)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Naik, D.N., Khattree, R.: Revisiting Olympic track records: some practical considerations in the principal component analysis. Am. Stat. 50, 140–144 (1996)Google Scholar
  27. 27.
    Oja, H.: On location, scale, skewness and kurtosis of univariate distributions. Scand. J. Stat. 8(3), 154–168 (1981)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Oja, H.: Descriptive statistics for multivariate distributions. Stat. Probab. Lett. 6, 327–332 (1983)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Pearson, K.: Contributions to the mathematical theory of evolution. Philos. Trans. R. Soc. Lond. A 185, 71–110 (1894)zbMATHGoogle Scholar
  30. 30.
    Pearson, K.: Contributions to the mathematical theory of evolution II: skew variation in homogeneous material. Philos. Trans. R. Soc. Lond. A 86, 343–414 (1895)Google Scholar
  31. 31.
    Serfling, R.J.: Multivariate Symmetry and Asymmetry Encyclopedia of Statistical Sciences. Wiley Online Library, Hoboken (2006)Google Scholar
  32. 32.
    Siotani, M., Hayakawa, T., Fujikoshi, Y.: Modern Multivariate Statistical Analysis: A Graduate Course and Handbook. American Sciences Press, Columbus (1985)zbMATHGoogle Scholar
  33. 33.
    TC2000 Software-Version 7, Available at tc2000.com (2010)Google Scholar
  34. 34.
    van Zwet, W.R.: Convex Transformations of Random Variables, Mathematical Centre Tract, vol. 7. Mathematisch Centrum, Amsterdam (1964)Google Scholar
  35. 35.
    Von Hippel, P.: Skewness International Encyclopedia of Statistical Science. Springer, New York (2011)Google Scholar
  36. 36.
    Yule, G.U.: An Introduction to the Theory of Statistics. C. Griffin Limited, London (1919)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Statistics and Center for Data Science and Big Data AnalyticsOakland UniversityRochesterUSA
  2. 2.Department of Mathematics and StatisticsOakland UniversityRochesterUSA

Personalised recommendations