Summary: A structured linear regression model is one in which there are permanent dependencies among some \(p\) row vectors of the \(n\times p\) 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 \(h\), \(h_{\text{op}}\), 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 \(h_{\text{op}}\). However, when \(h\) is smaller than \(h_{\text{op}}\), the breakdown point depends in a complicated way on the design as well as on the response.