A new index to measure positive dependence in trivariate distributions

  • Authors:

  • JesúS E. GarcíA;V. A. GonzáLez-LóPez;R. B. Nelsen

  • Affiliations:

  • Department of Statistics, University of Campinas, Rua Sérgio Buarque de Holanda, 651, Campinas, São Paulo, CEP 13083-859, Brazil;Department of Statistics, University of Campinas, Rua Sérgio Buarque de Holanda, 651, Campinas, São Paulo, CEP 13083-859, Brazil;Department of Mathematical Sciences, Lewis and Clark College, 0615 SW Palatine Hill Road, Portland, OR 97219, USA

  • Venue:
  • Journal of Multivariate Analysis
  • Year:
  • 2013

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Abstract

We introduce a new index to detect dependence in trivariate distributions. The index is based on the maximization of the coefficients of directional dependence over the set of directions. We show how to calculate the index using the three pairwise Spearman's rho coefficients and the three common 3-dimensional versions of Spearman's rho. We obtain the asymptotic distributions of the empirical processes related to the estimators of the coefficients of directional dependence and also we derive the asymptotic distribution of our index. We display examples where the index identifies dependence undetected by the aforementioned 3-dimensional versions of Spearman's rho. The value of the new index and the direction in which the maximal dependence occurs are easily computed and we illustrate with a simulation study and a real data set.