Two classes of MUSIC-like estimators are considered. One class, called weighted norm MUSIC, possesses an optimizing functional, or null spectrum, which is the product of the MUSIC null spectrum and an angle-dependent weight. The second class, which is denoted the Dr estimator, has an optimizing functional that is dependent on a parameter r and is a generalized distance between two particular vectors in the signal subspace. It is shown that the asymptotic mean-square errors of these estimators are the same as MUSIC. By determining an appropriate weight, based on a derived large-sample expression for the estimator bias, a weighted norm MUSIC estimator is found that gives zero bias of order N~], where N is the sample size. Using an approximate relation between the two types of estimators under consideration, a data-dependent parameter r(9) is derived for the Dr estimator, which results in small bias over a wide range of signal-to-noise ratios (SNR's) for two closely spaced sources.