Structured covariances occurring in spectral analysis, filtering and identification need to be estimated from a finite observation record. The corresponding sample covariance usually fails to possess the required structure. This is the case, for instance, in the Byrnes-Georgiou-Lindquist THREE-like tunable, high-resolution spectral estimators. There, the output covariance Sigma of a linear filter is needed to initialize the spectral estimation technique. The sample covariance estimate (Sigma) over cap, however, is usually not compatible with the filter. In this paper, we present a new, systematic way to overcome this difficulty. The new estimate Sigma(o) is obtained by solving an ancillary problem with an entropic-type criterion. Extensive scalar and multivariate simulation shows that this new approach consistently leads to a significant improvement of the spectral estimators performances.