Inference for High-Dimensional Sparse Spectral Precision Matrices

2026-06-06

AI Summary

This study addresses the challenges in inferring conditional dependence structures of high-dimensional stationary time series in the frequency domain, which are hindered by truncation and smoothing biases arising from finite-sample discrete Fourier transforms (DFTs) and the difficulty of estimating complex-valued spectral precision matrices in high dimensions. To overcome these issues, the authors propose a debiased complex-valued graphical Lasso estimator based on the full likelihood across neighboring frequency points. The method enables high-dimensional inference of sparse spectral precision matrices at fixed frequencies and achieves entrywise consistent covariance estimation through cross-frequency information aggregation. It constitutes the first approach to perform full-likelihood inference directly on the DFT, effectively controlling regularization, truncation, and smoothing biases. Simulations demonstrate accurate confidence interval coverage across all non-zero frequencies, higher statistical power than existing methods, and false discovery rates close to nominal levels.

Abstract

Gaussian graphical models in the spectral domain offer a principled approach for recovering conditional dependence structures in stationary high-dimensional time series. Inference on the spectral precision matrix at a fixed frequency enables tests of frequency-specific conditional associations among time series components. The problem is challenging because finite-sample discrete Fourier transforms induce truncation and smoothing biases, while the complex-valued nature of the spectral precision matrix complicates high-dimensional variance estimation, rendering methods for i.i.d. samples not directly applicable. Existing approaches do not provide full likelihood-based inference for the discrete Fourier transforms. We propose a high-dimensional inference framework for sparse spectral precision matrices using the full likelihood of neighboring discrete Fourier transforms. We construct a debiased complex graphical lasso estimator at any fixed frequency. Using asymptotic theory for quadratic forms of multivariate time series, we establish its asymptotic normality and construct entry-wise consistent covariance estimators by aggregating information across neighboring frequencies. The key theoretical contribution is the simultaneous control of regularization, finite-sample truncation, and smoothing biases, enabling valid inference. Simulation studies show reliable coverage away from zero frequency and improved detection power over the benchmark, with false discovery rates near the desired level.

Problem

• spectral precision matrix
• high-dimensional inference
• Gaussian graphical models
• conditional dependence
• time series

Innovation

• spectral precision matrix
• debiased graphical lasso
• high-dimensional time series
• frequency-specific inference
• asymptotic normality