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Abstract:
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The univariate concept of quantile function---the inverse of a distribution function---plays a fundamental role in Probability and Statistics. In dimensions two and higher, however, inverting traditional distribution functions does not lead to any satisfactory notion. In their quest for the Grail of an adequate definition, statisticians developed two extremely fruitful theoretical pathways: copula transforms, where marginal quantiles are privileged over global ones, and depth functions, where a center-outward ordering is substituting the more traditional South-West/North-East one. We show how a recent center-outward redefinition, based on measure transportation ideas, of the concept of distribution function, reconciles and fine-tunes these two approaches, and eventually yields a notion of multivariate quantile matching, in arbitrary dimension d, all the properties that make univariate quantiles a successful and vital tool of statistical inference.
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