Omnibus test for normality based on the Edgeworth expansion

Area of Science:

• Statistics
• Statistical Inference
• Probability Theory

Background:

• Normality assumption is crucial for statistical inference and signal processing.
• Existing normality tests have limitations.
• Detecting departures from normality is essential for robust analysis.

Purpose of the Study:

• Develop a novel statistical test for normality.
• Improve detection of non-normal distributions.
• Evaluate the proposed test against existing methods.

Main Methods:

• Utilized the Edgeworth expansion for distribution approximation.
• Constructed a new test statistic based on cumulants and the first four moments.
• Compared the proposed test with established normality tests (D'Agostino-Pearson, Jarque-Bera, Shapiro-Wilk).
• Analyzed various platykurtic and leptokurtic distributions (generalized Gaussian, mixed Gaussian, α-stable, Student's t).

Main Results:

• The proposed test shows superior power for platykurtic distributions.
• For leptokurtic distributions, the test performs comparably to leading methods.
• Demonstrated efficacy on real-world data examples.

Conclusions:

• The novel Edgeworth expansion-based test offers a valuable addition to normality testing.
• The test effectively identifies deviations from normality, particularly for platykurtic cases.
• The method is robust and applicable to real data analysis.