We introduce innovative inference procedures for analyzing time series data. Our methodology enables density approximation and composite hypothesis testing based on Whittle's estimator, a widely applied M-estimator in the frequency domain. Its core feature involves the (general Legendre transform of the) cumulant generating function of the Whittle likelihood score, as obtained using an approximated distribution of the periodogram ordinates. We present a testing algorithm that significantly expands the applicability of the state-of-the-art saddlepoint test, while maintaining the numerical accuracy of the saddlepoint approximation. Additionally, we demonstrate connections between our findings and three other prevalent frequency domain approaches: the bootstrap, empirical likelihood, and exponential tilting. Numerical examples using both simulated and real data illustrate the advantages and accuracy of our methodology.

翻译：我们引入了创新的推断程序用于分析时间序列数据。该方法基于Whittle估计量（一种在频域中广泛应用的M估计量），实现了密度近似与复合假设检验。其核心特征在于利用了Whittle似然得分的累积生成函数（通过广义勒让德变换），该函数通过周期图坐标的近似分布获得。我们提出了一种检验算法，在保持鞍点近似数值精度的同时，显著拓展了最先进的鞍点检验法的适用范围。此外，我们还论证了本研究结果与另外三种主流频域方法（自助法、经验似然法和指数倾斜法）之间的内在联系。通过模拟数据与真实数据的数值示例，验证了该方法在精度与适用性方面的优势。