Statistica Sinica 14(2004), 431-448

LINEAR FITTING BY SIMPLICIAL INTERCEPT DEPTH

(SID): REFLECTION INVARIANCE AND ROBUSTNESS

Regina Y. Liu, Kesar Singh and Julie H. Teng

Rutgers University

Abstract: This paper introduces the method of simplicial intercept depth (SID) for linear fitting. The SID method is invariant under rotations and reflections. It is also robust against outliers, with its breakdown point bounded away from zero. The paper also presents some comparisons, in terms of robustness, efficiency and invariance, between the SID method and other regression methods such as regression depth, orthogonal regression, $L_1$ regression and least squares. Finally, the paper introduces the simplicial fit plot as a new graphical tool for a visual assessment of the goodness of a given linear fit. The area under the simplicial fit plot of the proposed linear fit corresponds to its SID value. Hence the SID value provides a goodness-of-fit measure for any given linear fit, and can be viewed as a robust analog of the usual coefficient of determination arising from the least squares method.

Key words and phrases: Breakdown, linear fit, reflection invariance, rotation invariance, simplicial intercept depth, simplicial linear fit.