On the variability of the sample covariance matrix under complex elliptical distributions

2021/01/01 by Elias Raninen, Esa Ollila, David E. Tyler
Computer Science · Mathematics · #Advanced Statistical Methods and Models #Applied mathematics #Bayesian Methods and Mixture Models #Covariance #Covariance function #Covariance intersection #Covariance mapping #Covariance matrix #Elliptical distribution #Equivariant map #Estimation of covariance matrices #Law of total covariance #Mathematics #Matrix (chemical analysis) #Multivariate normal distribution #Pure mathematics #Rational quadratic covariance function #Sample mean and sample covariance #Scatter matrix #Statistical Methods and Bayesian Inference #Statistics #Wishart distribution #math.ST #stat.TH

paper · pdf · doi:10.1109/lsp.2021.3117443

published as IEEE Signal Processing Letters, vol. 28, pp. 2092-2096, 2021

openalex publication_date 2021/08/18 · openalex created_date 2021/11/08 · arxiv created 2021/11/09 · arxiv updated 2021/11/10 · openalex updated_date 2026/08/05

Abstract

We derive the form of the variance-covariance matrix for any affine equivariant matrix-valued statistics when sampling from complex elliptical distributions. We then use this result to derive the variance-covariance matrix of the sample covariance matrix (SCM) as well as its theoretical mean squared error (MSE) when finite fourth-order moments exist. Finally, illustrative examples of the formulas are presented.

Citations

• Rotational Invariant Estimator for General Noisy Matrices
• A well-conditioned estimator for large-dimensional covariance matrices
• The Commutation Matrix: Some Properties and Applications