Regression Depth
作者:
PeterJ. Rousseeuw,
Mia Hubert,
期刊:
Journal of the American Statistical Association
(Taylor Available online 1999)
卷期:
Volume 94,
issue 446
页码: 388-402
ISSN:0162-1459
年代: 1999
DOI:10.1080/01621459.1999.10474129
出版商: Taylor & Francis Group
关键词: Asymmetric error distribution;Deepest regression;Depth envelopes;Depth quantiles;Geometry;Halfspace depth;Heteroscedasticity;Robust regression;Simplicial depth
数据来源: Taylor
摘要:
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 datasetZnof sizenthere exists a line with depth at leastn/3. The largest depth inZncan 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 ofygivenx, 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.
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