# The Catline for Deep Regression

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## Abstract

Motivated by the notion of regression depth (Rousseeuw and Hubert, 1996) we introduce thecatline, a new method for simple linear regression. At any bivariate data setZn={(xi,Â yi);i=1,Â ...,Â n} its regression depth is at leastn/3. This lower bound is attained for data lying on a convex or concave curve, whereas for perfectly linear data the catline attains a depth ofn. We construct anO(nÂ logÂ n) algorithm for the catline, so it can be computed fast in practice. The catline is Fisher-consistent at any linear modely=[beta]x+[alpha]+ein which the error distribution satisfies med(eÂ Â x)=0, which encompasses skewed and/or heteroscedastic errors. The breakdown value of the catline is 1/3, and its influence function is bounded. At the bivariate gaussian distribution its asymptotic relative efficiency compared to theL1line is 79.3% for the slope, and 88.9% for the intercept. The finite-sample relative efficiencies are in close agreement with these values. This combination of properties makes the catline an attractive fitting method.

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