Conditional tests for elliptical symmetry using robust estimators

ABSTRACT

This paper presents a procedure for testing the hypothesis that the underlying distribution of the data is elliptical when using robust location and scatter estimators instead of the sample mean and covariance matrix. Under mild assumptions that include elliptical distributions without first moments, we derive the test statistic asymptotic behavior under the null hypothesis and under special alternatives. Numerical experiments allow to compare the behavior of the tests based on the sample mean and covariance matrix with that based on robust estimators, under various elliptical distributions and different alternatives. We also provide a numerical comparison with other competing tests.

KEYWORDS:

• Bootstrap
• conditional test
• elliptical symmetry
• robust estimation

MATHEMATICS SUBJECT CLASSIFICATION:

• 62H15
• 62F35
• 62F40

Acknowledgments

The authors wish to thank two anonymous referees for valuable comments which led to an improved version of the original paper.

Funding

This research was partially supported by Grants W276 and 20020130100279BA from the Universidad of Buenos Aires, pip 112-2011-01-00339 from conicet and pict 2011-0397 and 2014-0351 from anpcyt, Argentina, and also received financial support from the Portuguese National Funds through FCT (Fundação para a Ciência e a Tecnologia) under the scope of project PEst-OE/MAT/UI0822/2011.