Abstract
Comparing the sample mean to the sample median gives rise to a test of univariate symmetry commonly referred to as the MM-test of symmetry. The general idea underlying this test is that the mean and median are equal at symmetric distributions, but not necessarily so at asymmetric distributions. In this paper, we study properties of a more general form of the MM-test of symmetry based upon the comparison of any two location estimators, and in particular upon the comparison of two different M-estimators of location. For these generalized MM-tests of symmetry, the asymptotic null distribution is obtained as well as the asymptotic distribution under local alternatives to symmetry. The local power functions help provide guidelines for choosing good members within the class of MM-tests of symmetry, e.g. in choosing good tuning constants for the M-estimators of location. A study of the local power also shows the advantages of the MM-test relative to other tests of symmetry. The results of the paper are shown to readily extend to testing the symmetry of the error term in a linear model.
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Acknowledgements
David Tyler’s research was supported in part by National Science Foundation Grant DMS-1407751. The authors are extremely grateful to the reviewers for their many helpful comments. In particular, Remark 8.1 was motivated by one of their insightful comments.
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Wang, J., Tyler, D.E. (2015). Generalized MM-Tests for Symmetry. In: Nordhausen, K., Taskinen, S. (eds) Modern Nonparametric, Robust and Multivariate Methods. Springer, Cham. https://doi.org/10.1007/978-3-319-22404-6_8
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DOI: https://doi.org/10.1007/978-3-319-22404-6_8
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