Directional bivariate quantiles: a robust approach based on the cumulative distribution function

Nadja Klein () and Thomas Kneib
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Nadja Klein: Humboldt Universität zu Berlin
Thomas Kneib: Georg-August-Universität Göttingen

AStA Advances in Statistical Analysis, 2020, vol. 104, issue 2, No 3, 225-260

Abstract: Abstract The definition of multivariate quantiles has gained considerable attention in previous years as a tool for understanding the structure of a multivariate data cloud. Due to the lack of a natural ordering for multivariate data, many approaches have either considered geometric generalisations of univariate quantiles or data depths that measure centrality of data points. Both approaches provide a centre-outward ordering of data points but do no longer possess a relation to the cumulative distribution function of the data generating process and corresponding tail probabilities. We propose a new notion of bivariate quantiles that is based on inverting the bivariate cumulative distribution function and therefore provides a directional measure of extremeness as defined by the contour lines of the cumulative distribution function which define the quantile curves of interest. To determine unique solutions, we transform the bivariate data to the unit square. This allows us to introduce directions along which the quantiles are unique. Choosing a suitable transformation also ensures that the resulting quantiles are equivariant under monotonically increasing transformations. We study the resulting notion of bivariate quantiles in detail, with respect to computation based on linear programming and theoretical properties including asymptotic behaviour and robustness. It turns out that our approach is especially useful for data situations that deviate from the elliptical shape typical for ‘normal-like’ bivariate distributions. Moreover, the bivariate quantiles inherit the robustness of univariate quantiles even in case of extreme outliers.

Keywords: Direction-specific quantiles; Inverse cumulative distribution function; Linear programming; Quantile curves; Robustness (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (2)

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DOI: 10.1007/s10182-019-00355-3

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