Empirical Bayes Method for Large Scale Multiple Testing with Heteroscedastic Errors

TL;DR

This paper introduces gg-Mix, an empirical Bayes method for large-scale multiple testing with heteroscedastic errors, which relaxes previous restrictive assumptions and demonstrates superior FDR control and power.

Contribution

The paper proposes a flexible empirical Bayes approach that only assumes independence between means and variances, improving robustness over existing methods.

Findings

gg-Mix controls FDR effectively in simulations

It outperforms existing methods in power

Demonstrated on real data examples

Abstract

In this paper, we address the normal mean inference problem, which involves testing multiple means of normal random variables with heteroscedastic variances. Most existing empirical Bayes methods for this setting are developed under restrictive assumptions, such as the scaled inverse-chi-squared prior for variances and unimodality for the non-null mean distribution. However, when either of these assumptions is violated, these methods often fail to control the false discovery rate (FDR) at the target level or suffer from a substantial loss of power. To overcome these limitations, we propose a new empirical Bayes method, gg-Mix, which assumes only independence between the normal means and variances, without imposing any structural restrictions on their distributions. We thoroughly evaluate the FDR control and power of gg-Mix through extensive numerical studies and demonstrate its superior performance compared to existing methods. Finally, we apply gg-Mix to three real data examples to further illustrate the practical advantages of our approach.