Quantile functions for multivariate analysis: approaches and applications

Despite the absence of a natural ordering of Euclidean space for dimensions greater than one, the effort to define vector-valued quantile functions for multivariate distributions has generated several approaches. To support greater discrimination in comparing, selecting and using such functions, we introduce relevant criteria, including a notion of "median-oriented quantile function". On this basis we compare recent quantile approaches and several multivariate versions of trimmed mean and interquartile range. We also discuss a univariate "generalized quantile" approach that enables particular features of multivariate distributions, for example scale and kurtosis, to be studied by two-dimensional plots. Methods based on statistical depth functions are found to be especially attractive for quantile-based multivariate inference.