Significance testing for canonical correlation analysis in high dimensions

Abstract

We consider the problem of testing for the presence of linear relationships between large sets of random variables based on a post-selection inference approach to canonical correlation analysis. The challenge is to adjust for the selection of subsets of variables having linear combinations with maximal sample correlation. To this end, we construct a stabilized onestep estimator of the euclidean-norm of the canonical correlations maximized over subsets of variables of pre-specified cardinality. This estimator is shown to be consistent for its target parameter and asymptotically normal provided the dimensions of the variables do not grow too quickly with sample size. We also develop a greedy search algorithm to accurately compute the estimator, leading to a computationally tractable omnibus test for the global null hypothesis that there are no linear relationships between any subsets of variables having the pre-specified cardinality. Further, we develop a confidence interval for the target parameter that takes the variable selection into account.

Slicing-free Inverse Regression in High-dimensional Sufficient Dimension Reduction

“…inference(Kim et al 2020), as seen in the post-selection inference of canonical correlation analysis that is done over subsets of variables of pre-specied cardinality(McKeague & Zhang 2020).…”