Solving the Empirical Bayes Normal Means Problem with Correlated Noise
This addresses the problem of inaccurate statistical estimates in high-dimensional data analysis for researchers and practitioners, but it is incremental as it extends existing methods to handle correlations.
The paper tackled the Empirical Bayes Normal Means problem with correlated noise, which distorts estimates, by developing new methods that account for unknown correlations, resulting in favorable performance in numerical experiments for large-scale multiple testing and FDR control.
The Normal Means problem plays a fundamental role in many areas of modern high-dimensional statistics, both in theory and practice. And the Empirical Bayes (EB) approach to solving this problem has been shown to be highly effective, again both in theory and practice. However, almost all EB treatments of the Normal Means problem assume that the observations are independent. In practice correlations are ubiquitous in real-world applications, and these correlations can grossly distort EB estimates. Here, exploiting theory from Schwartzman (2010), we develop new EB methods for solving the Normal Means problem that take account of unknown correlations among observations. We provide practical software implementations of these methods, and illustrate them in the context of large-scale multiple testing problems and False Discovery Rate (FDR) control. In realistic numerical experiments our methods compare favorably with other commonly-used multiple testing methods.