In this article, we propose a simple estimator of cross‐covariance matrices for a multi‐variate time series with an unknown mean based on a linear combination of the circular sample cross‐covariance estimator. Our estimator is exactly unbiased when the data generating process follows a vector moving average (VMA) model with an order less than one half of the sampling period, and is nearly unbiased if such VMA model can approximate the data generating process well. In addition, our estimator is shown to be asymptotically equivalent to the conventional sample cross‐covariance estimator. Via simulation, we show that the proposed estimator can to a large extent eliminate the finite sample bias of cross‐covariance estimates, while not necessarily increase the mean squared error.

中文翻译：

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一个简单的几乎没有偏协方差的估计量

在本文中，我们基于圆形样本交叉协方差估计量的线性组合，为带有未知均值的多元时间序列提出了一个简单的交叉协方差矩阵估计。当数据生成过程遵循矢量移动平均（VMA）模型且其阶次小于采样周期的一半时，我们的估计器将完全无偏，如果这种VMA模型可以很好地近似数据生成过程，则我们的估计器将几乎无偏。另外，我们的估计量被证明与传统样本互协方差估计量渐近等效。通过仿真，我们表明所提出的估计器可以在很大程度上消除交叉协方差估计的有限样本偏差，而不必增加均方误差。