Multivariate ranks and quantiles

随机变量的秩、分位数、同单调性和递增重排等概念在统计学、计量经济学和经济理论中是非常富有成果的概念。然而，这些概念似乎依赖于真实的线的排序，并且它们对随机向量的扩展已被证明是难以捉摸的。最近，Galichon和亨利（2012年）以及Ekeland，Galichon和亨利（2012年）展示了如何根据最优运输理论重新解释所有这些概念，从而提供规范的多元扩展。该研究计划旨在发展这种多元扩展及其对基于秩的统计理论的强大影响，一方面是分散，不平等和风险的测量，另一方面是最佳风险分担和保险的分析。它将被证明，最优运输理论可以用来映射任意分布到球形分布，并允许广义秩统计的定义，分布自由的推断，以同样的方式作为哈林和Paindaveine（2006）的椭圆分布的特殊情况。这项研究还将旨在界定和估计多变量数据的离散度，因为它涉及对多种不可替代风险的暴露和态度，以及人口不平等的衡量，其中考虑到福祉的多个不完全相关的方面。这将产生深入了解的理论，包括在这种情况下，代理人有异质的主观感知的风险的多重风险分担。

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.