Data Depth for Regression and Copulas
by Professor Christine Mueller
Abstract: Data depth is one possibility to generalize the univariate Median to multivariate data and other data. The first generalization was given by Tukey 1975 via the notion of half space depth. Rousseeuw and Hubert 1999 extended this approach to regression by using the concept of nonfit. In the first part of this talk, these two steps of generalizations are presented. Afterwards, the extension of regression depth to simplicial regression depth is given. It is shown that the asymptotic distribution of simplicial regression depth is much easier to obtain than that of the regression depth. Hence simplicial regression depth can be used for testing arbitrary hypotheses in regression. Two examples show that these tests have high outlier robustness. In the last part of the talk, data depth is extended to copulas. This extension is only possible via the notion of likelihood depth, the most general concept of data depth which can be applied to many situations. Like in other situations, likelihood depth leads to biased estimators. However, the bias can be corrected. Moreover, tests based on simplicial likelihood depth can be used to test hypotheses about one-dimensional parameters of copulas.
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