Kelley’s coefficients of skewness using R

Öz

Kelley developed a robust measure of asymmetry based on quantiles. His proposal was an absolute index which, when divided by the median, results in its relative expression. If the additive complement is standardized with the semi-percentile range, the percentile coefficient of skewness (PCS) is obtained. Additionally, Kelley provided its standard error in case of normal distribution. However, no statistical software currently computes these measures. The aim of this methodological article is to determine their sampling distribution and facilitate their use. Three random samples of 10,000 data points were generated from three symmetric distributions: semicircular (platykurtic), normal (mesokurtic), and logistic (leptokurtic). By bootstrapping, the sampling distribution was obtained for absolute and relative indices, as well as the PCS. The sampling distributions of the absolute index and the PCS conformed to normality, while that of the relative index was leptokurtic with an excessive bootstrap standard error. Furthermore, a script was developed for the R program, adjusted based on these findings, to obtain point and interval estimates of these indices. The script was applied to a random sample as an example. It is concluded that dividing the absolute index by the semi-percentage range is a better standardization option than dividing by the median.

Anahtar Kelimeler

• skewness
• quantile
• bootstrap confidence interval
• asymptotic standard error
• normality

Kelley'nin R ile eğrilik katsayıları

Öz

Kelley, kuantiller temelinde asimetriyi ölçen sağlam bir yöntem geliştirdi. Onun önerisi, medyan ile bölündüğünde göreceli ifadesi elde edilen mutlak bir indeksdi. Ekleyici tamamlayıcı, yarı yüzde aralığı ile standart hale getirildiğinde, yüzde katsayısı eğriliği (PCS) elde edilir. Ayrıca, Kelley, normal dağılım durumunda standart hatayı da sağladı. Ancak, şu anda hiçbir istatistiksel yazılım bu ölçümleri hesaplamamaktadır. Bu metodolojik makalenin amacı, bu ölçümlerin örnekleme dağılımını belirlemek ve kullanımlarını kolaylaştırmaktır. Üç simetrik dağılımdan (yarım daire (platykurtik), normal (mesokurtik) ve lojistik (leptokurtik)) 10.000 veri noktasından oluşan üç rastgele örnek oluşturulmuştur. Bootstrapping yöntemi ile mutlak ve göreceli indekslerin yanı sıra PCS'in örnekleme dağılımı elde edilmiştir. Mutlak indeks ve PCS'in örnekleme dağılımları normalliğe uymuşken, göreceli indeksin dağılımı leptokurtik ve aşırı bootstrapping standart hatası ile görülmüştür. Ayrıca, bu bulgulara dayanarak, bu indekslerin nokta ve aralık tahminlerini elde etmek için R programı için bir betik geliştirilmiştir. Betik, örnek olarak bir rastgele örneğe uygulanmıştır. Sonuç olarak, mutlak indeksi yarı yüzde aralığı ile bölmek, medyan ile bölmekten daha iyi bir standardizasyon seçeneği olduğu sonucuna varılmıştır.

Anahtar Kelimeler

• eğrilik
• kuantiller
• bootstrap güven aralığı
• asimptotik standart hata
• normallik

Kaynakça

1. [1] T. L. Kelley, “A new measure of dispersion,” Quar. Pub. Amer. Statist. Assoc., vol. 17, no. 134, pp. 743-749, June 1921. https://doi.org/10.1080/15225445.1921.10503833
2. [2] T. L. Kelley, Statistical Method, The Macmillan Company, New York, 1923, pp. 75-77. https://doi.org/10.1080/15225445.1921.10503833
3. [3] K. Pearson, “Contributions to the mathematical theory of evolution. I. On the dissection of asymmetrical frequency curves”. Phil. Trans. Roy. Soc. London A, vol. 185, pp. 71−110, January 1894. https://doi.org/10.1098/rsta.1894.0003
4. [4] K. Pearson, “Contributions to the mathematical theory of evolution. II. Skew variation in homogeneous material”, Phil. Trans. Roy. Soc. London A, vol. 186, pp. 343−414, 1895. https://doi.org/10.1098/rsta.1895.0010
5. [5] A. L. Bowley, Elements of Statistics, P. S. King and Son, London, 1901.
6. [6] U. G. Yule, An introduction to the Theory of Statistics. Charles Griffin and Company Limited, London, 1912. https://doi.org/10.1037/13786-000
7. [7] R. A. Fisher, “The moments of the distribution for normal samples of measures of departure from normality”. Proc Roy. Soc. London A, vol. 130, no. 812, pp. 16–28, December 1930. https://doi.org/10.1098/rspa.1930.0185
8. [8] D. Stout, “A question of statistical inference: E. G. Boring, T. L. Kelley, and the probable error”. Am. J. Psychol., vol. 102, no. 4, pp. 549–562, April 1989. https://doi.org/10.2307/1423307

9. [9] R. Ihaka, R: past and future history, 2022. https://cran.r-project.org/doc/html/interface98-paper/paper.html
10. [10] D. R. Bickel, “Robust estimators of the mode and skewness of continuous data,” Comput. Stat. Data Anal., vol. 39, no. 2, pp. 153−163, April 2002. https://doi.org/10.1016/S0167-9473(01)00057-3
11. [11] G. Altinay, A simple class of measures of skewness. Munich Personal RePEc Archive, Paper No. 72353, pp. 1−13, September 2016. https://mpra.ub.uni-muenchen.de/72353/
12. [12] A. Singh, L. Gewali, and J. Khatiwada, “New measures of skewness of a probability distribution,” Open J. Stat., vol. 9, no. 3, pp. 601−621, October 2019. http://dx.doi.org/10.4236/ojs.2019.95039
13. [13] A. Eberl, and B. Klar, “Asymptotic distributions and performance of empirical skewness measures,” Comput. Stat. Data Anal., vol. 146, article 106939, June 2020. https://doi.org/10.1016/j.csda.2020.106939
14. [14] R Development Core Team, Quantile {stats}. R Documentation. Sample Quantiles, 2024. https://stat.ethz.ch/R-manual/R-devel/library/stats/html/quantile.html
15. [15] K. Stapor, “Descriptive and inferential statistics,” in Introduction to Probabilistic and Statistical Methods with Examples in R, vol 176, Intelligent Systems Reference Library, Cham, Switzerland: Springer, 2020, pp. 63–131. https://doi.org/10.1007/978-3-030-45799-0_2
16. [16] S. C. Gupta, and V. K. Kapoor, Fundamentals of Mathematical Statistics (12 th ed.), New Delhi: Sultan Chand & Sons, 2020.
17. [17] L. M. Chihara, and T. C. Hesterberg, Mathematical Statistics with Resampling and R. New York: John Wiley & Sons, 2022.
18. [18] A. Linden, "CENTILE2: Stata module to enhance centile command and provide additional definitions for computing sample quantiles," Statistical Software Components S459262, Boston College Department of Economics, 2023.
19. [19] D. I.Sukhoplyuev, and A. N. Nazarov, Methods of descriptive statistics in telemetry tasks, in Proceedings of the 2024 Systems of Signals Generating and Processing in the Field of on-Board Communications, Moscow, Russian Federation, (article 10496798), New Orleans, LA: Institute of Electrical and Electronics Engineers (IEEE), 2024. https://doi.org/10.1109/IEEECONF60226.2024.10496798
20. [20] R. J. Hyndman, and Y. Fan, “Sample quantiles in statistical packages,” Am. Stat., vol. 50, no. 4, pp. 361-365, November 1996. https://doi.org/10.2307/2684934
21. [21] J. W. Tukey, Exploratory Data Analysis, Addison-Wesley, Reading, MA, 1977.
22. [22] A. Roques, and A. Zhao, “Association rules discovery of deviant events in multivariate time series: an analysis and implementation of the sax-arm algorithm”, Image Processing On Line, vol. 12, pp. 604-624, December 2022. https://doi.org/10.5201/ipol.2022.437
23. [23] P. M. Dixon, “The bootstrap and the jackknife: describing the precision of ecological indices,” in S. Scheiner, Ed., Design and Analysis of Ecological Experiments, London: Chapman and Hall/CRC, 2020, pp. 290-318. https://doi.org/10.1201/9781003059813
24. [24] R. B. D’Agostino, “Transformation to normality of the null distribution of g1,” Biometrika, vol. 57, no. 3, pp. 679–681, December 1970. https://doi.org/10.1093/biomet/57.3.679
25. [25] B. Efron, and B. Narasimhan, “The automatic construction of bootstrap confidence intervals,” J. Comput. Graph. Stat., vol. 29, no. 3, pp. 608–619, March 2020. https://doi.org/10.1080/10618600.2020.1714633
26. [26] G. Rousselet, C. R. Pernet, and R. R. Wilcox, “An introduction to the bootstrap: a versatile method to make inferences by using data-driven simulations,” Meta-Psychology, 7, artícle 2019.2058, December 2023 https://doi.org/10.15626/MP.2019.2058
27. [27] G. Sánchez-Barajas, and A. Gómez-Navarro, Estadística General Aplicada con Excel [General Statistics Applied with Excel]. Universidad Autónoma Nacional de México, Ciudad de México, 2017.
28. [28] D. V. Hinkley, “On power transformations to symmetry,” Biometrika, vol. 62, no. 1, pp. 101−111, April 1975. https://doi.org/10.2307/2334491
29. [29] R. Chattamvelli, and R. Shanmugam, “Skewness,” in Descriptive Statistics for Scientists and Engineers. Applications in R, Cham, Switzerland: Springer, 2023, pp. 91-110. https://doi.org/10.1007/978-3-031-32330-0_4
30. [30] S. Luo and D. Villar, The skewness of the price change distribution: A new touchstone for sticky price models. Journal of Money, Credit and Banking, vol. 53, no. 1, pp. 41-72, September 2021. https://doi.org/10.1111/jmcb.12700
31. [31] P. L. Mamidi, N. D. Arigela, K.N.V.R. Lakshmi, and A Srilakshmi, “Skewness corrected control charts: a new probability model,” Obstetrics and Gynaecology Forum, vol. 34, no. 3s, pp. 775–779, May 2024. Retrieved from https://obstetricsandgynaecologyforum.com/index.php/ogf/article/view/352
32. [32] F. Najafi, S. Naderpour, M. Moradinazar, M. Khoramdad, A. Vahedian-Azimi, T. Jamialahmadi, and A. Sahebkar, “Percentiles for anthropometric measures in 11–18 years-old students of 73 developing countries,” Diabetes Metab. Syndr., vol. 14, no. 6, pp. 1957-1962, Nov-Dec 2020. https://doi.org/10.1016/j.dsx.2020.10.002
33. [33] M. Iseringhausen, I. Petrella, and K. Theodoridis, “Aggregate skewness and the business cycle,” Rev. Econ. Stat., 1-37, October 2023. https://doi.org/10.1162/rest_a_01390
34. [34] K. Harmenberg, The labor-market origins of cyclical skewness, Technical Report, University of Copenhagen, February 2021. Available at https://www.karlharmenberg.com/papers/skewness_harmenberg.pdf
35. [35] R. A. Groeneveld, and G. Meeden, “Measuring skewness and kurtosis,” J. Roy. Stat. Soc. Ser. D Statistician, vol. 33, no. 4, pp. 391–399, December 1984. https://doi.org/10.2307/2987742
36. [36] S. Salgado, F. Guvenen, and N. Bloom, “Skewed business cycles,” National Bureau of Economic Research, Working Paper 26565, December 2019. https://doi.org/10.3386/w26565
37. [37] S. Chowdhury, Monte Carlo Methods Utilizing Mathematica®: Applications in Inverse Transform and Acceptance-Rejection Sampling, Berlin: Springer Nature, 2023. https://doi.org/10.1007/978-3-031-23294-7
38. [38] R. Kimberk, “Beta distribution of long memory sequences,” arXiv:2404.05736, March 2024. https://doi.org/10.48550/arXiv.2404.05736
39. [39] F. Grubbs, “Procedures for detecting outlying observations in samples,” Technometrics, vol. 11, no. 1, pp. 1-21, Febrary 1969. https://doi.org/10.1080/00401706.1969.10490657
40. [40] F. J. Anscombe, and W. J. Glynn, “Distribution of kurtosis statistics b2 for normal samples,” Biometrika, vol. 70, no. 1, pp. 227-234, April 1983. https://doi.org/10.1093/BIOMET/70.1.227
41. [41] T. W. Anderson, and D. A. Darling, “Asymptotic theory of certain goodness-of-fit criteria based on stochastic processes,” Ann. Math. Stat., vol. 23, no. 2, pp. 193-212, June 1952. http://dx.doi.org/10.1214/aoms/1177729437
42. [42]S. S. Shapiro, and R. S. Francia, “An approximate analysis of variance test for normality,” J. Am. Stat. Assoc., vol. 67, no. 337, pp. 215–216, April 1972. https://doi.org/10.1080/01621459.1972.10481232
43. [43] J. P. Royston, “A toolkit for testing for non-normality in complete and censured samples,” J. Roy. Stat. Soc., Ser. D-Statistician, vol. 42, no. 1, pp. 37-43, March 1993. https://doi.org/10.2307/2348109
44. [44] R. B. D’Agostino, A. Berlanger, and R. B. Jr. D’Agostino, “A suggestion for using powerful and informative tests of normality,” Am. Stat., vol. 44, no. 4, pp. 316-321, November 1990. https://doi.org/10.2307/2684359
45. [45] P. Kvam, B. Vidakovic, and S. J. Kim, Nonparametric Statistics with Applications to Science and Engineering with R, 2nd ed., Hoboken, NJ: John Wiley & Sons, 2022. https://doi.org/10.1002/9781119268178
46. [46] A. Blöchlinger, Gauss versus Cauchy: a comparative study on risk, in T. Hüttche, Ed., Finance in Crises. Contributions to Finance and Accounting, Cham, Switzerland: Springer, 2023, pp. 177–198. https://doi.org/10.1007/978-3-031-48071-3_12
47. [47] S. Nadarajah, and T. Hitchen, “Estimation of models for stock returns,” Computational Economics, March 2024. https://doi.org/10.1007/s10614-024-10580-x
48. [48] B. Coker, C. Rudin, and G. King, “A theory of statistical inference for ensuring the robustness of scientific results,” Management Science, vol. 67, no. 10, pp. 6174-6197, October 2021. https://doi.org/10.1287/mnsc.2020.3818
49. [49] F. Caeiro, and A. Mateus, Randtests: Testing Randomness in R, 2024. https://cran.r-project.org/web/packages/randtests/index.html
50. [50] S. Kwak, “Are only p-values less than 0.05 significant? A p-value greater than 0.05 is also significant!,” J. Lipid. Atheroscler., vol. 12, no. 2, pp. 89–95, May 2023. https://doi.org/10.12997/jla.2023.12.2.89
51. [51] G. Di Leo, and F. Sardanelli, “Statistical significance: p value, 0.05 threshold, and applications to radiomics—reasons for a conservative approach,” Eur. Radiol. Exp., vol. 4, article 18, March 2020. https://doi.org/10.1186/s41747-020-0145-y
52. [52] J. H. Kim, and I. Choi, “Choosing the Level of Significance: A Decision-theoretic Approach,” Abacus, vol. 57, no. 1, pp. 27-71, March 2021. https://doi.org/10.1111/abac.12172
53. [53] K. J. Nicholson, M. Sherman, S. N. Divi, D. R. Bowles, and A. R. Vaccaro, “The Role of Family-wise Error Rate in Determining Statistical Significance,” Clin. Spine Surg., vol. 35, no. 5, pp. 222-223, June 2022. https://doi.org/10.1097/BSD.0000000000001287
54. [54] C. Avram, and M. Mărușteri, “Normality assessment, few paradigms and use cases,” Rev. Romana Med. Lab., vol. 30, no. 3, pp. 251-260, July 2022. https://doi.org/10.2478/rrlm-2022-0030
55. [55] L. Komsta, and F. Novomestky, Moments: moments, cumulants, skewness, kurtosis and related tests, 2022. https://cran.r-project.org/web/packages/moments/index.html
56. [56] J. Gross, and U. Ligges, Package ‘nortest’, 2022. https://cran.r project.org/web/packages/nortest/nortest.pdf
57. [57] H. W. Lilliefors, “On the Kolmogorov-Smirnov test for normality with mean and variance unknow,” J. Am. Stat. Assoc., vol. 62, no. 318, pp. 399-402, August 1967. https://doi.org/10.2307/2283970
58. [58] S. Demir, “Comparison of normality tests in terms of sample sizes under different skewness and Kurtosis coefficients,” Int. J. Assess. Tool. Educ., vol. 9, no. 2, pp. 397-409, May 2022. https://doi.org/10.21449/ijate.1101295
59. [59] N. Khatun, “Applications of Normality Test in Statistical Analysis,” Open J. Stat., vol. 11, no. 1, pp. 113-122, February 2021. https://doi.org/10.4236/ojs.2021.111006
60. [60] I. Mala, V. Sladek, and D. Bılkova, “Power comparisons of normality tests based on l-moments and classical tests,” Math. Stat., vol. 9, no. 6, pp. 994-1003, November 2021. https://doi.org/10.13189/ms.2021.090615
61. [61] D. K. Wijekularathna, A. B Manage, and S. M. Scariano, “Power analysis of several normality tests: A Monte Carlo simulation study,” Commun. Stat-Simul C Journal, vol. 51, no. 3, pp. 757-773, September 2020. https://doi.org/10.1080/03610918.2019.1658780
62. [62] R. Sahann, T. Müller, and J. Schmidt, “Histogram binning revisited with a focus on human perception,” in Proceeding of the 2021 IEEE Visualization Conference (VIS), New Orleans, LA: Institute of Electrical and Electronics Engineers (IEEE), 2021, pp. 66-70. https://doi.org/10.1109/VIS49827.2021.9623301
63. [63] V. A. Epanechnikov, “Nonparametric estimation of a multidimensional probability density,” Theory of Probability and its Applications, vol. 14, no. 1, pp. 156–161, January 1969. https://doi.org/10.1137/1114019
64. [64] N. Fadillah, P. A. Dariah, A. Anggraeni, N. Cahyani, and L. Handayani, “Comparison of Gaussian and Epancehnikov Kernels,” Tadulako Social Science and Humaniora Journal, vol. 3, no. 1, pp. 13-22, September 2022. https://doi.org/10.22487/sochum.v3i1.15745
65. [65] S. J. Sheather, and M. C. Jones, “A reliable data-based bandwidth selection method for kernel density estimation,” J. Roy. Stat. Soc., Ser. B Stat. Methodol., vol. 53, no. 3, pp. 683–690, January 1991. https://doi.org/10.1111/j.2517-6161.1991.tb01857.x
66. [66] W. N. Venables, and B. D. Ripley, Modern Applied Statistics with S, 4th ed., Cham, Switzerland: Springer, 2002. https://doi.org/10.1007/978-0-387-21706-2
67. [67] E. C. Ogwu, and H. I. A. Ojarikre, “Comparative study of the rule of thumb, unbiased cross validation and the Sheather Jones-direct plug-in approaches of kernel density estimation using real life data,” Int. J. Innov. Res. Sci. Eng. Technol., vol. 11, no. 3, pp. 1800-1809, March 2023.
68. [68] R Development Core Team. The R Manuals, 2024. https://cran.r-project.org/manuals.html
69. [69] A. Canty, B. Ripley, and A. R. Brazzale, Package ‘boot’, 2024. https://cran.r-project.org/web/packages/boot/boot.pdf
70. [70] G. Cavaliere, and I. Georgiev, “Inference under random limit bootstrap measures,” Econometrica, vol. 88, no. 6, pp. 2547-2574, November 2020. https://doi.org/10.3982/ECTA16557
71. [71] D. Lakens, “Sample size justification,” Collabra: Psychology, vol. 8, no. 1, article 33267, January 2022. https://doi.org/10.1525/collabra.33267
72. [72] B. Efron, and B. Narasimhan, Package ‘bcaboot’. Bias corrected bootstrap confidence intervals, 2022. https://cran.r-project.org/web/packages/bcaboot/bcaboot.pdf
73. [73] A. Pekgör, “A novel goodness-of-fit test for Cauchy distribution,” J. Math., vol. 2023, no. 1, article 9200213, March 2023. https://doi.org/10.1155/2023/9200213
74. [74] B. Lawson, L. Leemis, and V. Kudlay, Set.seed: Seeding random variate generators, 2023. https://www.rdocumentation.org/packages/simEd/versions/2.0.1/topics/set.seed
75. [75] T. L. Kelley, Fundamentals of Statistics, Cambridge, MA: Harvard University Press, 1947.