at CICSR 208, 2366 Main Mall, UBC
SPEAKER: Professor Peter J. Rousseeuw
TITLE: An Introduction to Regression Depth
ABSTRACT: In this talk we introduce a notion of depth in the regression setting. It provides the rank of any line (plane), rather than ranks of observations or residuals. In simple regression we can compute the depth of any line by an $O(n \log n)$ algorithm. For any bivariate data set $Z_n$ of size $n$ there exists a line with depth at least $n/3$. The largest depth in $Z_n$ can be used as a measure of linearity versus convexity. In both simple and multiple regression we consider the deepest fit, which generalizes the univariate median and is equivariant for monotone transformations of the response. Throughout, the errors may be skewed and non-identically distributed (e.g. heteroskedastic). We also construct depth-based regression quantiles. They estimate the quantiles of y given x, as do the $L^1$-based regression quantiles, but can withstand the effect of leverage points. Using the concept of regression depth, we obtain some new results of discrete geometry.