Small Sample Robust Testing for Normality against Pareto Tails

Abstract

The aim of this article is to introduce the general form (so called RT class) of the robust and classical Jarque–Bera (JB) test based on the location functional. We introduce the two-step procedure which is optimal for testing against the individual or contaminated Pareto alternative. As a reference for such a contamination we consider different Pareto distributions. We also give practical guidelines for robust testing for normality against short- and heavy-tailed alternatives. We concentrate mainly on simulation results for moderate and small samples. However, we also prove consistency and asymptotic distribution for introduced tests. We show that as the suitable measure of nominal level of Pareto tail parameter we may take the t-Hill estimator introduced in the article. To guarantee the consistency of the whole procedure, we also prove the consistency of t-Hill estimator. The introduced general class of robust tests of the normality is illustrated at the selected datasets of financial time series.

Keywords:

• Consistency
• Hill estimator
• t-Hill estimator
• Location functional
• Pareto tail
• Power comparison
• Returns
• Robust tests for normality
• Testing for normality

Mathematics Subject Classification:

• Primary 62F03
• Secondary 62F05, 62F10, 62F12, 62F35, 91G60

Acknowledgment

Research was supported by projects AKTION Austria – Czech Republic Nr. 51p7, Nr. 54p21, Nr. 50p14, and Nr. 54p13.