The Gauss-Markov theorem tells us that the OLS estimator is the best linear unbiased estimator for the linear regression model.
But suppose I don't care about linearity and unbiasedness. Then is there some other (possible nonlinear/biased) estimator for the linear regression model which is the most efficient under the Gauss-Markov assumptions or some other general set of assumptions?
There is of course one standard result: OLS itself is the best unbiased estimator if in addition to the Gauss-Markov assumptions we also assume that the errors are normally distributed. For some other particular distribution of errors I could compute the corresponding maximum-likelihood estimator.
But I was wondering if there is some estimator which is better-than-OLS in some relatively general set of circumstances?