A blockwise empirical likelihood method for time series in frequency domain inference
针对平稳弱相依时间序列的频域推断，提出一种结合分块和周期图经验似然的新方法，可处理线性和非线性过程，无需估计复杂方差，适用于置信区域和假设检验。
Frequency domain analysis of time series is often difficult, as periodogram-based statistics involve non-linear averages with complicated variances. Due to the latter, nonparametric approximations from resampling or empirical likelihood (EL) are useful. However, current versions of periodogram-based EL for time series are highly restricted: these are valid only for linear processes and for special parameters (i.e., ratios). For general frequency domain inference with stationary, weakly dependent time series, we develop a spectral EL (SEL) method by combining two previously separate EL frameworks for time series: block-based EL and periodogram-based EL. This hybridization strategy is new and theoretically non-trivial, particularly as existing block-based EL relies on time domain averages that differ substantially from frequency domain counterparts. We formulate SEL statistics for parameters based on spectral estimating functions and periodogram subsamples. Under mild conditions, SEL log-ratio statistics are shown to be well-defined, admitting chi-square limits. Further, we formally establish an effective bootstrap procedure coupled with SEL. As a result, the SEL method can be used for nonparametric, asymptotically correct confidence regions and tests for frequency domain inference without explicit estimation of intricate variances of periodogram-based statistics. This broadly extends the applicability of EL for time series in three directions: (i) SEL can treat any spectral mean parameters; (ii) SEL is valid for both linear and non-linear processes; and (iii) SEL has a provable bootstrap development, which is rare for time series EL, and provides a novel alternative to other resampling approximations in the frequency domain. Simulation studies suggest the proposed method performs well compared to other non-EL approaches. A real data example demonstrates that SEL has application and extension to complicated scenarios.