Limiting spectral distribution for a class of random matrices
Journal of Multivariate Analysis
Time series: theory and methods
Time series: theory and methods
Time series: data analysis and theory
Time series: data analysis and theory
A well-conditioned estimator for large-dimensional covariance matrices
Journal of Multivariate Analysis
Time Series Analysis and Its Applications (Springer Texts in Statistics)
Time Series Analysis and Its Applications (Springer Texts in Statistics)
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In this paper on developing shrinkage for spectral analysis of multivariate time series of high dimensionality, we propose a new nonparametric estimator of the spectral matrix with two appealing properties. First, compared to the traditional smoothed periodogram our shrinkage estimator has a smaller L"2 risk. Second, the proposed shrinkage estimator is numerically more stable due to a smaller condition number. We use the concept of ''Kolmogorov'' asymptotics where simultaneously the sample size and the dimensionality tend to infinity, to show that the smoothed periodogram is not consistent and to derive the asymptotic properties of our regularized estimator. This estimator is shown to have asymptotically minimal risk among all linear combinations of the identity and the averaged periodogram matrix. Compared to existing work on shrinkage in the time domain, our results show that in the frequency domain it is necessary to take the size of the smoothing span as ''effective sample size'' into account. Furthermore, we perform extensive Monte Carlo studies showing the overwhelming gain in terms of lower L"2 risk of our shrinkage estimator, even in situations of oversmoothing the periodogram by using a large smoothing span.