Sankhya: The Indian Journal of Statistics

1995, Volume 57, Series A, Pt. 2, pp. 221--226

A NOTE ON THE SMOOTHNESS OF L1-ESTIMATORS FOR THE LINEAR MODEL

By

 STEVEN P. ELLIS, University of Rochester and Ohio State University

SUMMARY.  Let X be a fixed n x p matrix of full rank and let y be an n x1 column vector. If we fit the model $y= X \beta +$ error, by choosing the p x 1 column vector $\beta$ to minimize the sum of the absolute values of the residuals (L1-estimation), then in general the set of minimizing $\beta$'s is a polyhedron $\hat{B}(y) \subset \Re^p$. Let  $\hat{\beta}_{c}(y)$ and $\hat{\beta}_{s}(y)$ denote the centroid and Steiner points, respectively, of $\hat{B}(y)$. It is to be shown that both $\hat{\beta}_{c}(y)$ and $\hat{\beta}_{s}(y)$ are (uniformly order 1) Lipschitz as functions of y. (The Lipschitz constants depend on X.)

AMS (1991) subject classification.  Primary 62J05, secondary 62F35.

Key words and phrases. Centroid, least absolute value, minimum absolute value, Steiner point.

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