An adaptive adjustment to the R2 statistic in high-dimensional elliptical models

The R2 statistic and its classic adjusted version, say R*2, tend to overestimate the multiple correlation coefficient when dealing with multivariate data that exhibit heavy tails and tail dependence. This can result in an incorrect significance of correlation in high-dimensional scenarios. A new adaptive adjustment to the R2 statistic is proposed in this paper, which applies to a general population model that covers the family of elliptical distributions and an independent components model. Consistency and asymptotic normality of the new statistic are established under this general model. These findings are then applied to some fundamental inference problems in high dimensions.