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

Author

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  • Nadja Klein

    (Humboldt Universität zu Berlin)

  • Thomas Kneib

    (Georg-August-Universität Göttingen)

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.

Suggested Citation

  • Nadja Klein & Thomas Kneib, 2020. "Directional bivariate quantiles: a robust approach based on the cumulative distribution function," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 104(2), pages 225-260, June.
  • Handle: RePEc:spr:alstar:v:104:y:2020:i:2:d:10.1007_s10182-019-00355-3
    DOI: 10.1007/s10182-019-00355-3
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    References listed on IDEAS

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    1. Genest, Christian & Segers, Johan, 2010. "On the covariance of the asymptotic empirical copula process," LIDAM Reprints ISBA 2010017, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    2. repec:hal:spmain:info:hdl:2441/4c5431jp6o888pdrcs0fuirl40 is not listed on IDEAS
    3. Marc Hallin & Davy Paindaveine & Miroslav Siman, 2008. "Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth," Working Papers ECARES 2008_042, ULB -- Universite Libre de Bruxelles.
    4. Abdous, B. & Theodorescu, R., 1992. "Note on the spatial quantile of a random vector," Statistics & Probability Letters, Elsevier, vol. 13(4), pages 333-336, March.
    5. Guillaume Carlier & Victor Chernozhukov & Alfred Galichon, 2015. "Vector quantile regression: an optimal transport approach," CeMMAP working papers 58/15, Institute for Fiscal Studies.
    6. Biman Chakraborty, 2001. "On Affine Equivariant Multivariate Quantiles," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 53(2), pages 380-403, June.
    7. Genest, Christian & Segers, Johan, 2010. "On the covariance of the asymptotic empirical copula process," LIDAM Reprints ISBA 2010038, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA).
    8. Oja, Hannu, 1983. "Descriptive statistics for multivariate distributions," Statistics & Probability Letters, Elsevier, vol. 1(6), pages 327-332, October.
    9. Genest, Christian & Segers, Johan, 2010. "On the covariance of the asymptotic empirical copula process," Journal of Multivariate Analysis, Elsevier, vol. 101(8), pages 1837-1845, September.
    10. Einmahl, J. H.J. & Mason, D.M., 1992. "Generalized quantile processes," Other publications TiSEM b2a76bac-045d-457f-869f-d, Tilburg University, School of Economics and Management.
    11. Robert Serfling, 2002. "Quantile functions for multivariate analysis: approaches and applications," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 56(2), pages 214-232, May.
    12. Chen, L. -A. & Welsh, A. H., 2002. "Distribution-Function-Based Bivariate Quantiles," Journal of Multivariate Analysis, Elsevier, vol. 83(1), pages 208-231, October.
    13. Belzunce, F. & Castano, A. & Olvera-Cervantes, A. & Suarez-Llorens, A., 2007. "Quantile curves and dependence structure for bivariate distributions," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 5112-5129, June.
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