The LQD estimator focuses on the order statistic of the set of absolute differences, where has elements and . Following Croux et al. To implement LQD we choose estimates to minimize . Let designate the estimated coefficient vector and let designate the corresponding minimized value of . The given scale value remains unchanged but the weights are updated to match the current coefficient estimates.
Numerical convergence is required for both the values and the weights.
Because redescending -estimators such as the estimator have multiple solutions, starting values affect the results. The particular estimator we use has , and values for , and as given in the indicated row of Table 2 in Hampel et al. The value of fixes the threshold for the magnitude of beyond which an observation is completely rejected by assigning it a weight .