Parametrik Yedi Sayılı Özet Temelli Geliştirilmiş Bir Normallik Testi

Abstract

2022 yılında, parametrik yedi sayılı özet temelli bir normallik testi önerilmiştir. Test istatistiği, karelenmiş standartlaştırılmış çeyreklerin toplamından oluşmakta olup, başlangıçta çeyrekler arasındaki korelasyon göz önünde bulundurulmadan yedi serbestlik dereceli ki-kare dağılımı ile yaklaşık olarak değerlendirilmiştir. Amaç, bu korelasyonları dikkate alarak testi geliştirmektir. Bu doğrultuda iki alternatif önerilmiştir: (1) Q-istatistiğinin örnekleme dağılımının bootstrap yöntemiyle tahmin edilmesi (QB) ve (2) normallik altında çeyreklerin korelasyon matrisini içeren bir karelik formun kullanılması (QT). Üç varyant (Q, QB, QT), doğruluk oranı (başarı oranı) ve istatistiksel güç açısından Shapiro–Wilk W-testi ile karşılaştırılmıştır. On iki sürekli dağılımdan, otuz bir farklı örneklem büyüklüğünde toplam 372 rastgele örnek üretilmiştir. Doğru sınıflandırmalar Cochran’ın Q testiyle karşılaştırılmış, güç ise tekrarlı ölçümler ANOVA’sı ile değerlendirilmiştir. QB, anlamlı şekilde en yüksek doğruluğa sahip olmuş ve Q ile QT’ye kıyasla en yüksek ortalama gücü göstermiştir. Doğruluk açısından Shapiro–Wilk W-testi ile eşdeğer olsa da, bu test ortalama güç bakımından üç Q varyantının tamamından üstün performans sergilemiştir. Sonuç olarak, QB, normallik testi için yedi sayılı özetin uygun bir çıkarımsal genişletmesi olarak değerlendirilmektedir.

Keywords

• normallik testi
• normal dağılım
• kare biçimler
• bootstrap
• genelleştirilmiş ki-kare dağılımı
• çıkarımsal istatistikler

Refined Normality Test Based on the Parametric Seven-Number Summary

Abstract

In 2022, a normality test based on the parametric seven-number summary was proposed. Its test statistic is the sum of squared standardized quantiles and was initially approximated by a chi-square distribution with seven degrees of freedom, without accounting for correlation among quantiles. Objective: To improve the test by incorporating these correlations. Two alternatives were proposed: (1) estimating the sampling distribution of the Q-statistic via bootstrap (QB), and (2) using a quadratic form with a correlation matrix of quantiles under normality (QT). Methods: The three variants (Q, QB, QT) were compared with the Shapiro-Wilk W-test in terms of accuracy (hit ratio) and statistical power. A total of 372 random samples were generated across 31 sample sizes from twelve continuous distributions. Correct classifications were compared using Cochran’s Q test, and power was assessed via repeated-measures ANOVA. Results: QB was significantly the most accurate and showed the highest average power compared to Q and QT. Its accuracy was equivalent to that of the Shapiro–Wilk W-test, although the latter outperformed all three Q variants in average power. Conclusions: QB is a suitable inferential extension of the seven-number summary for testing normality.

Keywords

• testing normality
• normal distribution
• quadratic forms
• bootstrap
• generalized chi-square distribution
• inference statistics

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