Kelley’s coefficients of skewness using R

Abstract

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.

Keywords

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

Kelley'nin R ile eğrilik katsayıları

Abstract

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.

Keywords

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

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