Testing independence in high dimensions with sums of rank correlations

Dennis Leung, Mathias Drton

Research output: Contribution to journalArticlepeer-review

37 Scopus citations

Abstract

We treat the problem of testing independence between m continuous variables when m can be larger than the available sample size n. We consider three types of test statistics that are constructed as sums or sums of squares of pairwise rank correlations. In the asymptotic regime where both m and n tend to infinity, a martingale central limit theorem is applied to show that the null distributions of these statistics converge to Gaussian limits, which are valid with no specific distributional or moment assumptions on the data. Using the framework of U-statistics, our result covers a variety of rank correlations including Kendall's tau and a dominating term of Spearman's rank correlation coefficient (rho), but also degenerate U-statistics such as Hoeffding's D, or the Tau; of Bergsma and Dassios [Bernoulli 20 (2014) 1006-1028]. As in the classical theory for U-statistics, the test statistics need to be scaled differently when the rank correlations used to construct them are degenerate U-statistics. The power of the considered tests is explored in rate-optimality theory under a Gaussian equicorrelation alternative as well as in numerical experiments for specific cases of more general alternatives.

Original languageEnglish
Pages (from-to)280-307
Number of pages28
JournalAnnals of Statistics
Volume46
Issue number1
DOIs
StatePublished - Feb 2018
Externally publishedYes

Keywords

  • High-dimensional statistics
  • Independence
  • Minimax optimality
  • Rank correlations
  • U-statistics

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