Exact fit points under simple regression with replication

For simple regression with replication and fixed carriers, we determine the highest possible exact fit point of a regression equivariant estimator and show that it is less than 50% (even asymptotically). We determine the optimal value of the quantile index in the Least median of squares and Least trimmed squares estimators and show that these estimators can attain the upper bound on the exact fit point. The main finding of the paper is that the quantile index must be adjusted upward from the usual value in order to achieve this bound.