## The Annals of Statistics

### 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.

#### Article information

Source
Ann. Statist., Volume 24, Number 6 (1996), 2593-2607.

Dates
First available in Project Euclid: 16 September 2002

https://projecteuclid.org/euclid.aos/1032181171

Digital Object Identifier
doi:10.1214/aos/1032181171

Mathematical Reviews number (MathSciNet)
MR1425970

Zentralblatt MATH identifier
0867.62040

#### Citation

Mili, Lamine; Coakley, Clint W. Robust estimation in structured linear regression. Ann. Statist. 24 (1996), no. 6, 2593--2607. doi:10.1214/aos/1032181171. https://projecteuclid.org/euclid.aos/1032181171

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• BLACKSBURG, VIRGINIA 24061-0111 BLACKSBURG, VIRGINIA 24061-0439 E-MAIL: lmili@vt.edu E-MAIL: coakley@vt.edu