Survey on (Some) Nonparametric and Robust Multivariate Methods

Rather than attempt an encyclopedic survey of nonparametric and robust multivariate methods, we limit to a manageable scope by focusing on just two leading and pervasive themes, descriptive statistics and outlier identification. We set the stage with some perspectives, and we conclude with a look at some open issues and directions. A variety of questions are raised. Is nonparametric inference the goal of nonparametric methods? Are nonparametric methods more important in the multivariate setting than in the univariate case? Should multivariate analyis be carried out componentwise, or with full dimensionality, or pairwise? Do multivariate depth, outlyingness, quantile, and rank functions represent different methodological approaches? Can we have a coherent series of nonparametric multivariate descriptive measures for location, spread, skewness, kurtosis, etc., that are robust and accommodate heavy tailed multivariate data? Can nonparametric outlier identifiers be defined that do not require ellipsoidal contours? What makes an outlier identifier itself robust against outliers? Does outlyingness of a data point with respect to location estimation differ from its outlyingness with respect to dispersion estimation? How do univariate L-functionals extend to the multivariate setting? Does the transformation-retransformation approach pose any issues? How might we conceptualize multivariate descriptive measures and outlier identification methods with respect to arbitrary data spaces, for applications such as functional data analysis, shape fitting, and text analysis? AMS 2000 Subject Classification: Primary 62G10 Secondary 62H99.