Robust Maximum Association Estimators

The maximum association between two multivariate variables X and Y is defined as the maximal value that a bivariate association measure between one-dimensional projections αtX and βtY 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 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.

两个多元变量X与Y之间的最大关联度，定义为一维投影αᵀX与βᵀY之间的二元关联测度所能达到的最大值。若将皮尔逊相关（Pearson correlation）用作投影指标，则可得到第一典型相关系数。本文提出采用更具稳健性的关联测度，例如斯皮尔曼秩相关（Spearman’s rank correlation）、肯德尔秩相关（Kendall’s rank correlation），或源自二元散点矩阵（bivariate scatter matrices）的关联测度。我们对所提出的最大关联测度，以及求取最大关联度的系数所对应的估计量的稳健性展开了研究。在Y为单变量这一重要特殊情形下，最大秩相关估计量所得的回归估计量对响应变量的单调变换（monotonic transformations of the response）保持不变。我们推导得到该特殊情形下的渐近方差（asymptotic variances）。研究表明，最大秩相关估计量兼具良好的有效性与稳健性。模拟实验与真实数据案例验证了这类估计量的稳健性以及处理非线性关系的检验效能。本文配套补充材料可在线获取。