Multivariate ranks and quantiles

点击翻译按钮获取中文摘要

The notions of ranks, quantiles, comonotonicity and increasing rearrangements of random variables are extraordinarily fruitful concepts in statistics, econometrics and economic theory. However, these concepts seem to rely on the ordering of the real line and their extension to random vectors has proved elusive. Recently, Galichon and Henry (2012) and Ekeland, Galichon and Henry (2012) showed how all these concepts can be reinterpreted in the light of optimal transportation theory, thereby providing canonical multivariate extensions. This research program aims at developing this multivariate extension and its powerful implications for the theory of rank based statistics on the one hand and the measurement of dispersion, inequality and risk and the analysis of optimal risk sharing and insurance on the other. It will be shown that optimal transportation theory can be used to map arbitrary distributions into spherical distributions and allow the definition of generalized rank statistics for distribution free inference in the same way as Hallin and Paindaveine (2006) do for the special case of elliptical distributions. The research will also aim at defining and estimating measures of dispersion of multivariate data as it relates to the exposure and attitude to multiple non substitutable risks and to the measure of inequality in populations, where multiple imperfectly correlated dimensions of well-being are taken into account. This will yield insights into the theory of multiple risk sharing including in the case, where agents have heterogeneous subjective perception of the risks.