Welch's method provides an estimator of the power spectral density that is statistically consistent. This is achieved by averaging over periodograms calculated from overlapping segments of a time series. For a finite length time series, while the variance of the estimator decreases as the number of segments increase, the magnitude of the estimator's bias increases: a bias-variance trade-off ensues when setting the segment number. We address this issue by providing a a novel method for debiasing Welch's method which maintains the computational complexity and asymptotic consistency, and leads to improved finite-sample performance. Theoretical results are given for fourth-order stationary processes with finite fourth-order moments and absolutely continuous fourth-order cumulant spectrum. The significant bias reduction is demonstrated with numerical simulation and an application to real-world data, where several empirical metrics indicate our debiased estimator compares favourably to Welch's. Our estimator also permits irregular spacing over frequency and we demonstrate how this may be employed for signal compression and further variance reduction. Code accompanying this work is available in the R and python languages.

翻译：韦尔奇方法提供了功率谱密度的一种统计一致估计量。该方法通过对时间序列重叠片段计算的周期图进行平均实现。对于有限长度的时间序列，虽然估计量的方差随片段数量增加而减小，但估计量偏差的幅度却随之增大：当设定片段数量时便产生了偏差-方差权衡。我们通过提出一种去偏韦尔奇方法的新方法来解决此问题，该方法保持了计算复杂度和渐近一致性，并改善了有限样本性能。针对具有有限四阶矩和绝对连续四阶累积量谱的四阶平稳过程给出了理论结果。通过数值模拟和实际数据应用展示了显著的偏差降低效果，其中多项经验指标表明我们的去偏估计量优于韦尔奇估计量。我们的估计量还允许频率上的不规则间隔，并展示了如何将其用于信号压缩和进一步方差降低。本文附带的代码以R语言和Python语言提供。