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The Gauss-Markov theorem considers "best" as "lowest mean square error (MSE)" and a recent version of the theorem shows OLS is not only BLUE but also BUE: https://www.ssc.wisc.edu/~bhansen/papers/gauss.pdf

When the "best" becomes "lowest mean absolute error (MAE)", is it the case that the best estimator is no longer OLS and perhaps quantile (median) regression?

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  • $\begingroup$ The empirical median is a lower-variance (and unbiased) estimator of the mean of a Laplace distribution than the empirical mean (which is the OLS estimator). Estimating the mean corresponds to regression on just an intercept term, so this is a regression. Consequently, the claim about BUE instead of just BLUE is false. $\endgroup$
    – Dave
    Mar 9, 2022 at 11:05

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