Geometric aspects of robust testing for normality and sphericity

ABSTRACT

Stochastic robustness of control systems under random excitation motivates challenging developments in geometric approach to robustness. The assumption of normality is rarely met when analyzing real data and thus the use of classic parametric methods with violated assumptions can result in the inaccurate computation of p-values, effect sizes, and confidence intervals. Therefore, quite naturally, research on robust testing for normality has become a new trend. Robust testing for normality can have counterintuitive behavior, some of the problems have been introduced in Stehlík et al. [Chemometrics and Intelligent Laboratory Systems 130 (2014): 98–108]. Here we concentrate on explanation of small-sample effects of normality testing and its robust properties, and embedding these questions into the more general question of testing for sphericity. We give geometric explanations for the critical tests. It turns out that the tests are robust against changes of the density generating function within the class of all continuous spherical sample distributions.

KEYWORDS:

• Huberization, trimming
• Lehmann–Bickel functional
• Monte Carlo simulations
• power comparison
• robust tests for normality
• normality
• sphericity

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H15
• 62H10

Funding

M. Stehlík acknowledges the support of Fondecyt Proyecto Regular No. 1151441. M. Stehlík also acknowledges LIT-2016-1-SEE-023 project “Modeling complex dependencies: how to make strategic multicriterial decisions?/Modec.” L. Střelec acknowledges the support of the Czech Science Foundation (GA CR, project No. GA16-07089S).