Testing the independence of two random vectors where only one dimension is large

TL;DR

This paper introduces a new independence test for high-dimensional vectors where only one dimension is large, addressing a gap in existing methods that assume both dimensions grow with sample size.

Contribution

A novel independence testing procedure for cases with one large and one small dimension, with proven asymptotic normality and demonstrated effectiveness in real and simulated data.

Findings

The test is consistent and superior to existing methods in simulations.

Applied to RNA-sequencing data, it effectively detects gene independence/dependence.

The procedure is robust to deviations from normality.

Abstract

For testing the independence of two vectors with respective dimensions $p_1$ and $p_2$, the existing literature in high-dimensional statistics all assume that both dimensions $p_1$ and $p_2$ grow to infinity with the sample size. However, as evidenced in the RNA-sequencing data analysis discussed in the paper, it happens frequently that one of the dimension is quite small and the other quite large compared to the sample size. In this paper, we address this new asymptotic framework for the independence test. A new test procedure is introduced and its asymptotic normality is established when the vectors are normal distributed. A Mote-Carlo study demonstrates the consistency of the procedure and exhibits its superiority over some existing high-dimensional procedures. Applied to the RNA-sequencing data mentioned above, we obtain very convincing results on pairwise independence/dependence of gene isoform expressions as attested by prior knowledge established in that field. Lastly, Monte-Carlo experiments show that the procedure is robust against the normality assumption on the population vectors.