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Estimating the Spectral Density at Frequencies Near Zero
arXiv - STAT - Methodology Pub Date : 2022-08-03 , DOI: arxiv-2208.02300
Tucker McElroy, Dimitris Politis

Estimating the spectral density function $f(w)$ for some $w\in [-\pi, \pi]$ has been traditionally performed by kernel smoothing the periodogram and related techniques. Kernel smoothing is tantamount to local averaging, i.e., approximating $f(w)$ by a constant over a window of small width. Although $f(w)$ is uniformly continuous and periodic with period $2\pi$, in this paper we recognize the fact that $w=0$ effectively acts as a boundary point in the underlying kernel smoothing problem, and the same is true for $w=\pm \pi$. It is well-known that local averaging may be suboptimal in kernel regression at (or near) a boundary point. As an alternative, we propose a local polynomial regression of the periodogram or log-periodogram when $w$ is at (or near) the points 0 or $\pm \pi$. The case $w=0$ is of particular importance since $f(0)$ is the large-sample variance of the sample mean; hence, estimating $f(0)$ is crucial in order to conduct any sort of inference on the mean.

中文翻译:

估计频率接近零的光谱密度

估计一些 $w\in [-\pi, \pi]$ 的谱密度函数 $f(w)$ 传统上是通过核平滑周期图和相关技术来执行的。核平滑等同于局部平均,即在一个小宽度的窗口上通过一个常数来近似$f(w)$。尽管 $f(w)$ 是一致连续的和周期性的,周期为 $2\pi$,但在本文中,我们认识到 $w=0$ 有效地充当了底层核平滑问题的边界点,同样如此对于 $w=\pm \pi$。众所周知,在边界点处(或附近)的核回归中,局部平均可能不是最优的。作为替代方案,我们建议当 $w$ 位于(或接近)点 0 或 $\pm \pi$ 时,对周期图或对数周期图进行局部多项式回归。$w=0$ 的情况特别重要,因为 $f(0)$ 是样本均值的大样本方差;因此,估计 $f(0)$ 对于对均值进行任何类型的推断至关重要。
更新日期:2022-08-05
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