Goodness-of-Fit Tests for High-Dimensional Gaussian Graphical Models via Exchangeable Sampling

TL;DR

This paper develops a flexible, exact goodness-of-fit testing framework for high-dimensional Gaussian graphical models using exchangeable sampling, demonstrating superior power and theoretical optimality in various settings.

Contribution

It introduces a novel exchangeable sampling algorithm for goodness-of-fit tests that works in high dimensions with finite-sample guarantees and improved power.

Findings

Proposed tests outperform existing methods in simulations.

Achieves rate-optimality under different signal patterns.

Demonstrated effectiveness on real-world data.

Abstract

We introduce a general framework for testing goodness-of-fit for Gaussian graphical models in both the low- and high-dimensional settings. This framework is based on a novel algorithm for generating exchangeable copies by conditioning on sufficient statistics. This framework provides exact finite-sample error control regardless of the dimension and allows flexible choices of test statistics to improve power. We explore several candidate test statistics and conduct extensive simulation studies to demonstrate their finite-sample performance compared to existing methods. The proposed tests exhibit superior power, particularly in cases where the true precision matrix deviates from the null hypothesis due to many small nonzero entries. To justify theoretically, we consider a high-dimensional setting where the proposed test achieves rate-optimality under two distinct signal patterns in the precision matrix: (1) dense patterns with many small nonzero entries and (2) strong patterns with at least one large entry. Finally, we illustrate the usefulness of the proposed test through real-world applications.