Detection of non-Gaussianity

Abstract

We develop two tests sensitive to various departures from composite goodness-of-fit hypothesis of normality. The tests are based on the sums of squares of some components naturally arising in decomposition of the Shapiro–Wilk-type statistic. Each component itself has diagnostic properties. The numbers of squared components in sums are determined via some novel selection rules based on the data. The new solutions prove to be effective tools in detecting a broad spectrum of sources of non-Gaussianity. We also discuss two variants of the new tests adjusted to verification of simple goodness-of-fit hypothesis of normality. These variants also compare well to popular competitors.

Keywords:

• Akaike selection rule
• correlation test
• goodness-of-fit
• higher criticism
• L-statistic
• model selection
• sample quantile function
• Shapiro–Wilk test
• test of normality
• Wasserstein distance

2010 Mathematics Subject Classification:

• 62F03
• 62G10

Supplemental data

Supplemental data for this article can be accessed http://dx.doi.org/10.1080/00949655.2014.983110.

Acknowledgments

This work was supported by Grant N N201 608440 from the National Science Centre, Poland. Calculations have been carried out in Wrocław Centre for Networking and Supercomputing (http://www.wcss.wroc.pl), grant No. 199. We are deeply grateful for kind cooperation of the Centre. We also thank the referee for useful remarks.