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A novel regression method is introduced and studied. The procedure weightssquared residuals based on their magnitude. Unlike the classic least squareswhich treats every squared residual equally important, the new procedureexponentially down-weights squared-residuals that lie far away from the cloudof all residuals and assigns a constant weight (one) to squared-residuals thatlie close to the center of the squared-residual cloud.  The new procedure can keep a good balance between robustness and efficiency,it possesses the highest breakdown point robustness for any regressionequivariant procedure, much more robust than the classic least squares, yetmuch more efficient than the benchmark of robust method, the least trimmedsquares (LTS) of Rousseeuw (1984).  With a smooth weight function, the new procedure could be computed very fastby the first-order (first-derivative) method and the second-order(second-derivative) method.  Assertions and other theoretical findings are verified in simulated and realdata examples.

Weighted least squares regression with the best robustness and high computability