**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.