Open Access
December 1996 Robust estimation in structured linear regression
Lamine Mili, Clint W. Coakley
Ann. Statist. 24(6): 2593-2607 (December 1996). DOI: 10.1214/aos/1032181171


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.


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Lamine Mili. Clint W. Coakley. "Robust estimation in structured linear regression." Ann. Statist. 24 (6) 2593 - 2607, December 1996.


Published: December 1996
First available in Project Euclid: 16 September 2002

zbMATH: 0867.62040
MathSciNet: MR1425970
Digital Object Identifier: 10.1214/aos/1032181171

Primary: 62G35
Secondary: 62J05 , 62K99 , 62N99

Keywords: $D$-estimators , Breakdown point , exact fit point , general position , reduced position , robust estimation , structured regression

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 6 • December 1996
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