LAUSR

Spectral estimation provides key insights into the frequency domain characteristics of a time series. Naive nonparametric estimates of the spectral density, such as the periodogram, are inconsistent, and the more advanced lag window or multitaper estimators are often still too noisy. We propose an $L_1$ penalized quasi-likelihood Whittle framework based on multitaper spectral estimates which performs semiparametric spectral estimation for regularly sampled univariate stationary time series. Our new approach circumvents the problematic Gaussianity assumption required by least square approaches and achieves sparsity for a wide variety of basis functions. We present an alternating direction method of multipliers (ADMM) algorithm to efficiently solve the optimization problem, and develop universal threshold and generalized information criterion (GIC) strategies for efficient tuning parameter selection that outperform cross-validation methods. Theoretically, a fast convergence rate for the proposed spectral estimator is established. We demonstrate the utility of our methodology on simulated series and to the spectral analysis of electroencephalogram (EEG) data.