Robust Maximum Association Estimators

ABSTRACT

The maximum association between two multivariate variables $\mathbf{X}$ and $\mathbf{Y}$ is defined as the maximal value that a bivariate association measure between one-dimensional projections ${\alpha }^t\mathbf{X}$ and ${\beta }^t\mathbf{Y}$ can attain. Taking the Pearson correlation as projection index results in the first canonical correlation coefficient. We propose to use more robust association measures, such as Spearman’s or Kendall’s rank correlation, or association measures derived from bivariate scatter matrices. We study the robustness of the proposed maximum association measures and the corresponding estimators of the coefficients yielding the maximum association. In the important special case of $\mathbf{Y}$ being univariate, maximum rank correlation estimators yield regression estimators that are invariant against monotonic transformations of the response. We obtain asymptotic variances for this special case. It turns out that maximum rank correlation estimators combine good efficiency and robustness properties. Simulations and a real data example illustrate the robustness and the power for handling nonlinear relationships of these estimators. Supplementary materials for this article are available online.

KEYWORDS:

• Influence function
• Projection pursuit
• Rank correlation
• Regression
• Robustness

Supplementary Materials

A supplementary report contains further technical details, a proof of Theorem 1, as well as an extensive collection of numerical results.

Acknowledgment

Parts of this research were done while Andreas Alfons was a Postdoctoral Research Fellow, Faculty of Economics and Business, KU Leuven, Leuven, Belgium. The authors thank the associate editor and three anonymous referees for their constructive remarks that substantially improved their article.