关于核谱估计中带宽选择的局部到零交叉验证对数似然准则的期望
On the Expectation of the Local-to-Zero Cross-Validated Log Likelihood Criterion for Bandwidth Selection in Kernel Spectral Estimation
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中文总结 AI 辅助
研究核谱估计零频率下基于局部到零交叉验证对数似然准则的带宽选择，通过关注关键项期望，证实在$4/5\lt c\lt1$时其收敛到渐近均方误差，为HAC标准误差估计中使用该准则提供依据，结果有别于现有CVLL文献。
中文摘要 AI 辅助
我们考虑基于局部到零版本的交叉验证对数似然（CVLL）准则进行零频率下核谱估计器的数据驱动带宽选择。修改后的版本是$\mbox{CVLL}_c$，基于从1到$n^c$（$0\lt c\lt1$，$n$为样本量）的傅里叶频率求和。我们关注$\mbox{CVLL}_c$泰勒级数展开中关键项的期望，并表明在$4/5\lt c\lt1$的情况下，它收敛到零频率下谱估计器的相应渐近均方误差。这为在异方差和自相关一致（HAC）标准误差估计中使用局部CVLL准则提供了一些依据。我们的理论结果不同于现有关于CVLL的文献，因为那些结果利用了CVLL是全局的这一事实，对$(0,\pi)$内所有频率求和，而不是像$\mbox{CVLL}_c$那样局部到零频率。
英文摘要
We consider data-driven bandwidth selection for a kernel spectral estimator at zero frequency based on a local-to-zero version of the cross validated log-likelihood (CVLL) criterion. The modified version is $\mbox{CVLL}_c$, based on a sum over Fourier frequencies from $1$ to $n^c$ with $0<c<1$, where $n$ is the sample size. We focus on the expectation of a key term in a Taylor series expansion for $\mbox{CVLL}_c$ and show that in the case $4/5 < c < 1$ it converges to the corresponding asymptotic mean squared error of the spectral estimator at zero frequency. This provides some justification for the use of the local CVLL criterion for Heteroskedasticity and Autocorrelation Consistent (HAC) standard error estimation. Our theoretical results do not follow from existing literature on CVLL because those results exploit the fact that CVLL is global, summing over all frequencies in $(0,π)$ rather than local-to-zero frequency, as is the case for $\mbox{CVLL}_c$.
