Asymptotic Independence of the Sum and Maximum of Dependent Random Variables with Applications to High-Dimensional Tests

TL;DR

This paper establishes the asymptotic independence of sums and maxima of dependent variables without strong assumptions and applies it to develop robust high-dimensional tests for means and regression coefficients.

Contribution

It introduces a novel theoretical result on asymptotic independence and leverages it to create new high-dimensional testing procedures that perform well in various data sparsity scenarios.

Findings

Proposed tests perform well in simulations regardless of data sparsity.

Established asymptotic independence without stationarity or strong mixing assumptions.

Demonstrated advantages on real data examples.

Abstract

For a set of dependent random variables, without stationary or the strong mixing assumptions, we derive the asymptotic independence between their sums and maxima. Then we apply this result to high-dimensional testing problems, where we combine the sum-type and max-type tests and propose a novel test procedure for the one-sample mean test, the two-sample mean test and the regression coefficient test in high-dimensional setting. Based on the asymptotic independence between sums and maxima, the asymptotic distributions of test statistics are established. Simulation studies show that our proposed tests have good performance regardless of data being sparse or not. Examples on real data are also presented to demonstrate the advantages of our proposed methods.