Simple and Exact Empirical Likelihood Ratio Tests for Normality Based on Moment Relations

Abstract

The empirical likelihood (EL) technique is a powerful nonparametric method with wide theoretical and practical applications. In this article, we use the EL methodology in order to develop simple and efficient goodness-of-fit tests for normality based on the dependence between moments that characterizes normal distributions. The new empirical likelihood ratio (ELR) tests are exact and are shown to be very powerful decision rules based on small to moderate sample sizes. Asymptotic results related to the Type I error rates of the proposed tests are presented. We present a broad Monte Carlo comparison between different tests for normality, confirming the preference of the proposed method from a power perspective. A real data example is provided.

Keywords:

• Characterization theorems
• Empirical likelihood
• Goodness-of-fit
• Omnibus test
• Power study
• Test for normality

Mathematics Subject Classification:

• 62G10
• 62G20
• 62G30

Acknowledgment

The authors are grateful to the Editor and referee for their helpful comments and suggestions that improved this article.

Notes

AD: Anderson-Darling test. KS1: Modified Kolmogorov–Smirnov test \citepL67. LM: Lin–Mudholkar test. SW: Shapiro–Wilk test. SF: Shapiro–Francia test. VG: Vexler–Gurevich test \citepVG10. ELR: ELR test \citepDG07. SEELR: the proposed test.

AD: Anderson–Darling test. KS1: Modified Kolmogorov–Smirnov test \citepL67. LM: Lin–Mudholkar test. SW: Shapiro–Wilk test. SF: Shapiro–Francia test. VG: Vexler–Gurevich test \citepVG10. ELR: ELR test \citepDG07. SEELR: the proposed test.

AD: Anderson–Darling test. KS1: Modified Kolmogorov–Smirnov test \citepL67. LM: Lin–Mudholkar test. SW: Shapiro–Wilk test. SF: Shapiro–Francia test. VG: Vexler–Gurevich test \citepVG10. ELR: ELR test \citepDG07. SEELR: the proposed test.

AD: Anderson–Darling test. KS1: Modified Kolmogorov–Smirnov test (Lilliefors, 1967). LM: Lin–Mudholkar test. SW: Shapiro–Wilk test. SF: Shapiro–Francia test. VG: Vexler–Gurevich test (Vexler and Gurevich, 2010). ELR: ELR test (Dong and Giles, 2007). SEELR: the proposed test.

The actual Type I Error (1) and (2) correspond to size adjusted critical values from Table 1 of Dong and Giles (2007) and asymptotic critical values followed the theoretical result of Dong and Giles (2007).