Robust estimation in structured linear regression

A structured linear regression model is one in which there are permanent dependencies among some p row vectors of the n × p design matrix. To study structured linear regression, we introduce a new class of robust estimators, called D-estimators, which can be regarded as a generalization of the least median of squares and least trimmed squares estimators. They minimize a dispersion function of the ordered absolute residuals up to the rank h. We investigate their breakdown point and exact fit point as a function of h in structured linear regression. It is found that the D- and S-estimators can achieve the highest possible breakdown point for h appropriately chosen. It is shown that both the maximum breakdown point and the corresponding optimal value of h, hₒₚ, are sample dependent. They hinge on the design but not on the response. The relationship between the breakdown point and the design vanishes when h is strictly larger than hₒₚ. However, when h is smaller than hₒₚ, the breakdown point depends in a complicated way on the design as well as on the response.