When shrinking a covariance matrix towards (a multiple) of the identity matrix, the trace of the covariance matrix arises naturally as the optimal scaling factor for the identity target. The trace also appears in other context, for example when measuring the size of a matrix or the amount of uncertainty. Of particular interest is the case when the dimension of the covariance matrix is large. Then the problem arises that the sample covariance matrix is singular if the dimension is larger than the sample size. Another issue is that usually the estimation has to based on correlated time series data. We study the estimation of the trace functional allowing for a high-dimensional time series model, where the dimension is allowed to grow with the sample size—without any constraint. Based on a recent result, we investigate a confidence interval for the trace, which also allows us to propose lower and upper bounds for the shrinkage covariance estimator as well as bounds for the variance of projections. In addition, we provide a novel result dealing with shrinkage towards a diagonal target. We investigate the accuracy of the confidence interval by a simulation study, which indicates good performance, and analyze three stock market data sets to illustrate the proposed bounds, where the dimension (number of stocks) ranges between 32 and 475. Especially, we apply the results to portfolio optimization and determine bounds for the risk associated to the variance-minimizing portfolio.

中文翻译：

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协方差估计的收缩：渐近、置信区间、界限和传感器监控和金融中的应用

当将协方差矩阵向单位矩阵的（倍数）缩小时，协方差矩阵的迹自然会作为单位目标的最佳缩放因子出现。迹线也出现在其他上下文中，例如在测量矩阵大小或不确定性量时。特别有趣的是协方差矩阵的维数很大时的情况。那么问题就出现了，如果维度大于样本大小，样本协方差矩阵是奇异的。另一个问题是，通常估计必须基于相关的时间序列数据。我们研究了允许高维时间序列模型的迹函数的估计，其中允许维度随着样本大小而增长 - 没有任何限制。根据最近的结果，我们研究了轨迹的置信区间，这也允许我们提出收缩协方差估计量的下限和上限以及投影方差的界限。此外，我们提供了一个新的结果来处理对角线目标的收缩。我们通过模拟研究调查了置信区间的准确性，这表明表现良好，并分析了三个股票市场数据集来说明建议的界限，其中维度（股票数量）范围在 32 到 475 之间。特别是，我们应用了结果到投资组合优化并确定与方差最小化投资组合相关的风险界限。我们提供了一个处理对角线目标收缩的新结果。我们通过模拟研究调查了置信区间的准确性，这表明表现良好，并分析了三个股票市场数据集来说明建议的界限，其中维度（股票数量）在 32 到 475 之间。特别是，我们应用了结果到投资组合优化并确定与方差最小化投资组合相关的风险界限。我们提供了一个处理对角线目标收缩的新结果。我们通过模拟研究调查了置信区间的准确性，这表明表现良好，并分析了三个股票市场数据集来说明建议的界限，其中维度（股票数量）范围在 32 到 475 之间。特别是，我们应用了结果到投资组合优化并确定与方差最小化投资组合相关的风险界限。