Conference Proceeding

# Maximum Likelihood Covariance Estimation with a Condition Number Constraint

Dept. of Electr. Eng., Stanford Univ., Stanford, CA
Circuits, Systems and Computers, 1977. Conference Record. 1977 11th Asilomar Conference on 12/2006; DOI:10.1109/ACSSC.2006.354997 In proceeding of: Signals, Systems and Computers, 2006. ACSSC '06. Fortieth Asilomar Conference on
Source: IEEE Xplore

ABSTRACT In many signal processing applications, we want to estimate the covariance matrix of a multivariate Gaussian distribution. We often require the estimate to be not only invertible but also well-conditioned. We consider the maximum likelihood estimation of the covariance matrix with a constraint on the condition number. We show that this estimation problem can be reformulated as a convex univariate minimization problem, which admits an analytic solution. This estimation method requires no special assumption on the structure of the true covariance matrix. We demonstrate its good performance in comparison with commonly used estimators, especially when the sample size is small.

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##### Article: A well-conditioned estimator for large-dimensional covariance matrices
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ABSTRACT: Many applied problems require a covariance matrix estimator that is not only invertible, but also well-conditioned (that is, inverting it does not amplify estimation error). For large-dimensional covariance matrices, the usual estimator—the sample covariance matrix—is typically not well-conditioned and may not even be invertible. This paper introduces an estimator that is both well-conditioned and more accurate than the sample covariance matrix asymptotically. This estimator is distribution-free and has a simple explicit formula that is easy to compute and interpret. It is the asymptotically optimal convex linear combination of the sample covariance matrix with the identity matrix. Optimality is meant with respect to a quadratic loss function, asymptotically as the number of observations and the number of variables go to infinity together. Extensive Monte Carlo confirm that the asymptotic results tend to hold well in finite sample.
Journal of Multivariate Analysis. 01/1996;