Regression Depth
提出回归深度概念，用于衡量任意直线或平面的秩，并基于此开发最深回归方法，能处理偏态和异方差误差，且对杠杆异常值稳健。
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.