Robust estimation in structured linear regression

Abstract

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, $h_{\mathrm{op}}$, 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 $h_{\mathrm{op}}$. However, when h is smaller than $h_{\mathrm{op}}$, the breakdown point depends in a complicated way on the design as well as on the response.