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On Liu’s simplicial depth and Randles’ interdirections

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  • Serfling, Robert
  • Wang, Yunfei

Abstract

At about the same time (approximately 1989), R. Liu introduced the notion of simplicial depth and R. Randles the notion of interdirections. These completely independent and seemingly unrelated initiatives, serving different purposes in nonparametric multivariate analysis, have spawned significant activity within their quite different respective domains. A surprising and fruitful connection between the two notions is shown. Exploiting the connection, statistical procedures based on interdirections can be modified to use simplicial depth instead, at considerable reduction of computational burden in the case of dimensions 2, 3, and 4. Implications regarding multivariate sign test statistics are discussed in detail, and several other potential applications are noted.

Suggested Citation

  • Serfling, Robert & Wang, Yunfei, 2016. "On Liu’s simplicial depth and Randles’ interdirections," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 235-247.
    • Handle: RePEc:eee:csdana:v:99:y:2016:i:c:p:235-247
    DOI: 10.1016/j.csda.2016.02.002
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    1. Hallin, Marc & Paindaveine, Davy, 2005. "Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 122-163, March.
    2. Aloupis, Greg & Cortes, Carmen & Gomez, Francisco & Soss, Michael & Toussaint, Godfried, 2002. "Lower bounds for computing statistical depth," Computational Statistics & Data Analysis, Elsevier, vol. 40(2), pages 223-229, August.
    3. Robert Serfling, 2010. "Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 22(7), pages 915-936.
    4. Peter J. Rousseeuw & Ida Ruts, 1996. "Bivariate Location Depth," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 45(4), pages 516-526, December.
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