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 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 convergent fourth-order cumulant function. The significant bias reduction is demonstrated with numerical simulation and an application to real-world data. 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 R and python.

翻译：Welch方法提供了一种统计上一致的功率谱密度估计量。该方法通过对时间序列重叠段计算出的周期图进行平均来实现。对于有限长度时间序列，虽然随着段数的增加，估计量的方差会减小，但其偏差幅度却会增大：在设定段数时，便产生了偏差-方差的权衡。我们通过提出一种新颖的Welch方法去偏化技术来解决这一问题，该技术保持了计算复杂度和渐近一致性，并提升了有限样本性能。我们针对具有有限四阶矩和绝对收敛四阶累积量函数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，我们展示了显著的偏差降低效果。此外，我们的估计量还允许在频率上进行不规则间隔采样，并演示了如何利用这一特性进行信号压缩及进一步降低方差。本文所附代码以R和Python语言提供。