Nonparametric and Robust Multivariate Analysis via Quantile Functions

这个项目开发的概念，观点和工具，导致一个概念上有充分依据的理论和方法的非参数和强大的数据分析在任意维度。正在开发的一般框架包括一维分析的传统工具的扩展，以及独特的高维背景下的工具。 一个理论的“中位数为导向的分位数函数”的概率解释类似于单变量分位数正在追求。 一个中心的方法是基于统计深度函数。 一个主要的次要目标是使基于深度的方法达到确定的完成程度。 特别关注的主题包括多元分位数函数，深度函数，向量值L-统计，矩阵值尺度统计，广义分位数过程和广义L-统计。通过函数表示的线性化技术被使用。总体而言，该项目推进了使用分位数方法的非参数和稳健的多元分析，并解决了一系列特定的应用。随着多元统计建模的应用范围不断扩大，多元概率分布和数据的处理变得越来越重要和中心。 大量的单变量统计分析都是用统计量来进行的，这些统计量很容易解释。 使用铅笔已经成为基本的。 本计画将此方法延伸至更高的维度，以支持一个连贯且有意义的多元数据分析与解释的“平行”方法。 因此，在不同的应用领域的多变量统计的用户获得了一种具有根本重要性和概念上很好理解的工具。

This project develops concepts, perspectives, and tools leading toward a conceptually well-founded theory and methodology of nonparametric and robust data analysis in arbitrary dimension. The general framework being developed includes extensions of the traditional tools of one-dimensional analysis as well as tools unique to the higher-dimensional context. A theory of "median oriented quantile functions" having probabilistic interpretations similar to univariate quantiles is being pursued. A central approach is based on statistical depth functions. A major secondary objective is to bring the depth-based approach to a definitive degree of completion. Topics receiving special focus include multivariate quantile functions, depth functions, vector-valued L-statistics, matrix-valued scale statistics, generalized quantile processes, and generalized L-statistics. Linearization techniques via functional representations are used. Overall, the project advances nonparametric and robust multivariate analysis using quantile methods and addresses a range of specific applications.As the scope of application of multivariate statistical modeling has widened, the treatment of multivariate probability distributions and data has become increasingly important and central. A great deal of univariate statistical analysis is carried out in terms of percentiles, which lend themselves easily to interpretation. The use of percentiles has become fundamental. This project extends such methodology to higher dimension, to support a coherent and meaningful "percentiles" approach to the analysis and interpretation of multivariate data. Users of multivariate statistics in diverse fields of application thus acquire a tool that is of fundamental importance and conceptually well understood.