Kelley'nin R ile eğrilik katsayıları

Yıl 2024, Sayı: 10, 50 - 75

Jose Moral De La Rubia

https://doi.org/10.52693/jsas.1510230

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

Etik Beyan

Yazar herhangi bir çıkar çatışması olmadığını beyan eder. Bu, simüle edilmiş verilerle yapılan bir çalışmadır, bu nedenle insan veya insan dışı varlıkların manipülasyonunu içermez.

Destekleyen Kurum

Bu araştırma herhangi bir dış finansman almamıştır.

Proje Numarası

0000 The manuscript is not derived from any funded project.

Teşekkür

Yazar, yardımcı yorumları için hakemlere ve editöre teşekkürlerini sunar.

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

Etik Beyan

The author declares no conflict of interest This is a study with simulated data, so it does not involve the manipulation of human or non-human beings.

Destekleyen Kurum

This research did not receive any external funding

Proje Numarası

0000 The manuscript is not derived from any funded project.

Teşekkür

The author expresses gratitude the reviewers and editor for their helpful comments.

Kaynakça

• [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] T. L. Kelley, Statistical Method, The Macmillan Company, New York, 1923, pp. 75-77. https://doi.org/10.1080/15225445.1921.10503833
• [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] 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] A. L. Bowley, Elements of Statistics, P. S. King and Son, London, 1901.
• [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] 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] 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] R. Ihaka, R: past and future history, 2022. https://cran.r-project.org/doc/html/interface98-paper/paper.html
• [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] 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] 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] 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] 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] 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] S. C. Gupta, and V. K. Kapoor, Fundamentals of Mathematical Statistics (12 th ed.), New Delhi: Sultan Chand & Sons, 2020.
• [17] L. M. Chihara, and T. C. Hesterberg, Mathematical Statistics with Resampling and R. New York: John Wiley & Sons, 2022.
• [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] 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] 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] J. W. Tukey, Exploratory Data Analysis, Addison-Wesley, Reading, MA, 1977.
• [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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] R. Kimberk, “Beta distribution of long memory sequences,” arXiv:2404.05736, March 2024. https://doi.org/10.48550/arXiv.2404.05736
• [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] 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] 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]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] 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] 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] 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] 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] 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] 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] F. Caeiro, and A. Mateus, Randtests: Testing Randomness in R, 2024. https://cran.r-project.org/web/packages/randtests/index.html
• [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] 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] 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] 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] 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] 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] J. Gross, and U. Ligges, Package ‘nortest’, 2022. https://cran.r project.org/web/packages/nortest/nortest.pdf
• [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] 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] 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] 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] 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] 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] 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] 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] 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] 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] 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] R Development Core Team. The R Manuals, 2024. https://cran.r-project.org/manuals.html
• [69] A. Canty, B. Ripley, and A. R. Brazzale, Package ‘boot’, 2024. https://cran.r-project.org/web/packages/boot/boot.pdf
• [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] D. Lakens, “Sample size justification,” Collabra: Psychology, vol. 8, no. 1, article 33267, January 2022. https://doi.org/10.1525/collabra.33267
• [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] 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] 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] T. L. Kelley, Fundamentals of Statistics, Cambridge, MA: Harvard University Press, 1947.