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Abstract: The use of quantiles to obtain insights about multivariate data is addressed.It is argued that incisive insights can be obtained by considering directionalquantiles, the quantiles of projections. Directional quantile envelopes areproposed as a way to condense this kind of information; it is demonstrated thatthey are essentially halfspace Tukey depth levels sets, coinciding forelliptic distributions in particular multivariate normal with densitycontours. Relevant questions concerning their indexing, the possibility of thereverse retrieval of directional quantile information, invariance with respectto affine transformations, and approximation-asymptotic properties are studied.It is argued that the analysis in terms of directional quantiles and theirenvelopes offers a straightforward probabilistic interpretation and thusconveys a concrete quantitative meaning; the directional definition can beadapted to elaborate frameworks, like estimation of extreme quantiles anddirectional quantile regression, the regression of depth contours oncovariates. The latter facilitates the construction of multivariate growthcharts-the question that motivated all the development.

Autor: Linglong Kong, Ivan Mizera

Fuente: https://arxiv.org/

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