Regression Depth
本文提出回归深度概念，用于给任意回归线（平面）排序，并开发了快速算法；最深的回归线可衡量线性程度，且方法对误差偏态和异方差稳健。
Abstract In this article 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 a fast algorithm. For any bivariate dataset 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 introduce the deepest regression method, which generalizes the univariate median and is equivariant for monotone transformations of the response. Throughout, the errors may be skewed and heteroscedastic. We also consider depth-based regression quantiles. They estimate the quantiles of y given x, as do the Koenker-Bassett regression quantiles, but with the advantage of being robust to leverage outliers. We explore the analogies between depth in regression and in location, where Tukey's halfspace depth is a special case of our general definition. Also, Liu's simplicial depth can be extended to the regression framework.