Quantile functions for multivariate analysis: approaches and applications

Abstract

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 “medianoriented 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.