Generalized mm-tests for symmetry

Jue Wang, David E. Tyler

Research output: Chapter in Book/Report/Conference proceedingChapter


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.

Original languageEnglish (US)
Title of host publicationModern Nonparametric, Robust and Multivariate Methods
Subtitle of host publicationFestschrift in Honour of Hannu Oja
PublisherSpringer International Publishing
Number of pages16
ISBN (Electronic)9783319224046
ISBN (Print)9783319224039
StatePublished - Jan 1 2015

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)


  • Contiguity
  • Local power
  • M-estimates of location
  • Mixturemodels
  • Skew-symmetry


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