Normality Test Based on a Truncated Mean Characterization

Abstract

This article generalizes a characterization based on a truncated mean to include higher truncated moments, and introduces a new normality goodness-of-fit test based on the truncated mean. The test is a weighted integral of the squared distance between the empirical truncated mean and its expectation. A closed form for the test statistic is derived. Assuming known parameters, the mean and the variance of the test are derived under the normality assumption. Moreover, a limiting distributionfor the proposed test as well as an approximation are obtained. Also, based on Monte Carlo simulations, the power of the test is evaluated against stable, symmetric, and skewed classes of distributions. The test proves compatibility with prominent tests and shows higher power for a wide range of alternatives.

A Power Comparison of Various Tests of Univariate Normality on Ex-Gaussian Distributions

“…However, the power of some normality tests is attenuated when dealing with non-normal distributions, particularly when distributions are near-normal and symmetric such as the t and beta distributions, respectively (e.g., Alizadeh Noughabi, & Arghami, 2011;Chaichatschwal & Budsaba, 2007;Farrell & Rogers-Stewart, 2006;Romão, Delgado, & Costa, 2010;Yacizi & Yolacan, 2007;Yap & Sim, 2011;Zghoul & Awad, 2010). That is, some normality tests show low power and this is more likely to occur when the distributions approach normality.…”

“…KS = Kolmogorov-Smirnov test, AD = Anderson-Darling test, and SW = Shapiro-Wilk tests. Y&Y = Yacizi and Yolacan (2007), Z&A = Zghoul and Awad (2010), A&A = Alizadeh and Arghami (2011), R&cols = Romão et al (2010), S = Sürücü (2008), and R&W = Razali and Wah (2010). Only the results of Sürücü (2008) used α = .10.…”