Shrinkage for Covariance Estimation: Asymptotics, Confidence Intervals, Bounds and Applications in Sensor Monitoring and Finance

Abstract: 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.