Nonparametric and Robust Multivariate Analysis via Quantile Functions

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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.