The Bartlett correction is a desirable feature of the likelihood inference, which yields the confidence region for parameters with improved coverage probability. This study examines the Bartlett correction for the frequency domain empirical likelihood (FDEL), based on the Whittle likelihood of linear time series models. Nordman and Lahiri (Ann Stat 34:3019–3050, 2006) showed that the FDEL does not have an ordinary Chi-squared limit when the innovation is non-Gaussian with unknown variance, which restricts the use of the FDEL inference in time series. We show that, by profiling the innovation variance out of the Whittle likelihood function, the FDEL is Chi-squared-distributed and Bartlett correctable. In particular, the order of the coverage error of the confidence region can be reduced from $$O(n^{-1})$$ to $$O(n^{-2})$$ .

中文翻译：

---

具有未知创新方差的时间序列频域经验似然的 Bartlett 校正

Bartlett 校正是似然推断的一个理想特征，它为具有改进覆盖概率的参数产生置信区域。本研究基于线性时间序列模型的惠特尔似然，检查频域经验似然 (FDEL) 的 Bartlett 校正。Nordman 和 Lahiri (Ann Stat 34:3019–3050, 2006) 表明，当创新是非高斯且方差未知时，FDEL 没有普通的卡方极限，这限制了 FDEL 推理在时间序列中的使用。我们表明，通过从 Whittle 似然函数中分析创新方差，FDEL 是卡方分布且 Bartlett 可校正的。特别是，置信区域的覆盖误差的顺序可以从 $$O(n^{-1})$$ 减少到 $$O(n^{-2})$$ 。