Multivariate ranks and quantiles

秩、分位数、共同性和随机变量的增加重排等概念是统计学、计量经济学和经济理论中非常富有成果的概念。然而，这些概念似乎依赖于实数线的顺序，并且将它们扩展到随机向量已被证明是难以捉摸的。最近，Galichon和Henry（2012）以及Ekeland， Galichon和Henry（2012）展示了如何根据最优运输理论重新解释所有这些概念，从而提供了规范的多元扩展。本研究计划旨在发展这一多元推广及其对基于秩的统计理论，以及对离散度、不平等和风险的测量以及最优风险分担和保险的分析的强大意义。将证明最优运输理论可用于将任意分布映射为球形分布，并允许定义广义秩统计量用于分布自由推理，就像Hallin和Paindaveine（2006）对椭圆分布的特殊情况所做的那样。该研究还将致力于定义和估计多元数据分散的措施，因为它涉及到对多种不可替代风险的暴露和态度，以及对人口不平等的衡量，其中考虑到福祉的多个不完全相关的维度。这将产生对多重风险分担理论的见解，包括在代理人对风险有异质主观感知的情况下。

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.