Measuring skewness: We do not assume much

Document Type : Article

Authors

1 Department of Statistics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan

2 Department of Mathematical and Physical Sciences, Newcastle University 2308, Australia

3 Department of Mathematics, Statistics and Physics, Qatar University, Doha, Qatar

Abstract

Skewness plays a vital role in different engineering phenomena so it is desired to measure this characteristic accurately. Several measures to quantify the extent of skewness in distributions have been developed over the course of history but each measure has some serious limitations. Therefore, in this article, we propose a new skewness measuring functional, based on distribution function evaluated at mean with minimal assumptions and limitations. Four well recognized properties for an appropriate measure of skewness are verified and demonstrated for the new measure. Comparisons with the conventional moment-based measure are carried out by employing both functionals over range of distributions available in literature. Furthermore, the robustness of the proposed measure against unusual data points is explored through the application of influence function. The Mathematical findings are verified through meticulous simulation studies and further verified by real data sets coming from diverse fields of inquiries. It is witnessed that the suggested measure passes all the checks with distinction while comparing to the classical moment-based measure. Based on computational simplicity, applicability in more general environment and preservation of c-ordering of distribution, it may be considered as an attractive addition to the family of skewness measures.

Keywords

References

References:
1. Pearson, K. "Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material", Philosophical Transactions of the Royal Society of London, 186(Part I), pp. 343-424 (1895).
2. Kendall, M. and Stuart, A., The Advanced Theory of Statistics, 4th Ed., 1, Distribution Theory, London: Griffin (1977).
3. Firat, M., Koc, A.C., Dikbas, F., et al. "Identification of homogeneous regions and regional frequency analysis for Turkey", Scientia Iranica, Transactions A, Civil Engineering, 21(5), pp. 1492-1502 (2014).
4. Harvey, A. and Sucarrat, G. "EGARCH models with fat tails, skewness and leverage", Computational Statistics & Data Analysis, 76, pp. 320-338 (2014).
5. Amaya, D., Christoffersen, P., Jacobs, K., et al. "Does  realized skewness predict the cross-section of equity returns?", Journal of Financial Economics, 118(1), pp. 135-167 (2015).
6. Astebro, T., Mata, J., and Santos-Pinto, L. "Skewness seeking: risk loving, optimism or overweighting of small probabilities?", Theory and Decision, 78(2), pp. 189-208 (2015).
7. Ho, A.D. and Yu, C.C. "Descriptive statistics for modern test score distributions: Skewness, kurtosis, discreteness, and ceiling effects", Educational and Psychological Measurement, 75(3), pp. 365-388 (2015).
8. Decker, R.A., Haltiwanger, J., Jarmin, R.S., et al. "Where has all the skewness gone? The decline in highgrowth (young) firms in the US", European Economic Review, 86, pp. 4-23 (2016).
9. Colacito, R., Ghysels, E., Meng, J., et al. "Skewness in expected macro fundamentals and the predictability of equity returns: Evidence and theory", The Review of Financial Studies, 29(8), pp. 2069-2109 (2016).
10. Ensthaler, L., Nottmeyer, O., Weizsacker, G., et al. "Hidden skewness: On the difficulty of multiplicative compounding under random shocks", Management Science, 64(4), pp. 1693-1706 (2017).
11. Amjadzadeh, M. and Ansari-Asl, K. "An innovative emotion assessment using physiological signals based on the combination mechanism", Scientia Iranica, 24(6), pp. 3157-3170 (2017).
12. Taylor, S. and Fang, M. "Unbiased weighted variance and skewness estimators for overlapping returns", Swiss Journal of Economics and Statistics, 154(1), pp. 1-8 (2018).
13. Zwet, W.R., Convex Transformations of Random Variables, Mathematisch Centrum, Amsterdam (1964).
14. Goldstein, J.L., Schrott, H.G., Hazzard, W.R., et al. "Hyperlipidemia in coronary heart disease II. Genetic analysis of lipid levels in 176 families and delineation of a new inherited disorder, combined hyperlipidemia", The Journal of Clinical Investigation, 52(7), pp. 1544- 1568 (1973).
15. Mardia, K.V., Families of Bivariate Distributions, 27, Lubrecht & Cramer Ltd (1970).
16. Aucremanne, L., Brys, G., Hubert, M., et al. "A study of Belgian inflation, relative prices and nominal rigidities using new robust measures of skewness and tail weight", Statistics for Industry and Technology, pp. 13-25 (2004).
17. Brys, G., Hubert, M., and Struyf, A. "A robust measure of skewness", Journal of Computational and Graphical Statistics, 13(4), pp. 996-1017 (2004).
18. Doane, D.P. and Seward, L.E. "Measuring skewness: a forgotten statistic?", Journal of Statistics Education, 19(2), pp. 1-18 (2011).
19. Hosking, J. "Moments or L moments? An example comparing two measures of distributional shape", The American Statistician, 46(3), pp. 186-189 (1992).
20. Li, X., Qin, Z., and Kar, S. "Mean-variance-skewness model for portfolio selection with fuzzy returns", European Journal of Operational Research, 202(1), pp. 239-247 (2010).
21. Arnold, B.C. and Groeneveld, R.A. "Measuring skewness with respect to the mode", The American Statistician, 49(1), pp. 34-38 (1995).
22. Brys, G., Hubert, M., and Struyf, A. "A comparison of some new measures of skewness", Developments in Robust Statistics (ICORS 2001), 14, pp. 98-113 (2003).
23. Kim, T.H. and White, H. "On more robust estimation of skewness and kurtosis", Finance Research Letters, 1(1), pp. 56-73 (2004).
24. Holgersson, H. "A modified skewness measure for testing asymmetry", Communications in Statistics- Simulation and Computation, 39(2), pp. 335-346 (2010).
25. Tajuddin, I. "On the use of medcouple as a measure of skewness", Pakistan Journal of Statistics, 28(1), pp. 69-80 (2012).
26. Altinay, G. "A simple class of measures of skewness", MPRA Paper No. 72353, posted 4 July 2016, https://mpra.ub.uni-muenchen.de/72353/(2016).
27. Oja, H. "On location, scale, skewness and kurtosis of univariate distributions", Scandinavian Journal of Statistics, 8(3), pp. 154-168 (1981).
Scientia Iranica
Volume 28, Issue 6 - Serial Number 6
Transactions on Industrial Engineering (E)
November and December 2021
Pages 3525-3537

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