Improved estimation of the logarithm of the covariance matrix

An improved estimator of certain bilinear forms of the logarithm of the covariance matrix is presented. The new estimator is shown to be consistent, not only for increasing sample size (as traditional estimators), but also when the observation dimension scales up at the same rate as the number of available observations. This characteristic provides very good properties whenever the sample volume and the observation dimension have the same order of magnitude, which is always the case in practical applications. The estimator is derived using the contour-based technique recently introduced in the context of large random matrix theory.