Annals of Statistics

Monge–Kantorovich depth, quantiles, ranks and signs

Victor Chernozhukov, Alfred Galichon, Marc Hallin, and Marc Henry

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Abstract

We propose new concepts of statistical depth, multivariate quantiles, vector quantiles and ranks, ranks and signs, based on canonical transportation maps between a distribution of interest on $\mathbb{R}^{d}$ and a reference distribution on the $d$-dimensional unit ball. The new depth concept, called Monge–Kantorovich depth, specializes to halfspace depth for $d=1$ and in the case of spherical distributions, but for more general distributions, differs from the latter in the ability for its contours to account for non-convex features of the distribution of interest. We propose empirical counterparts to the population versions of those Monge–Kantorovich depth contours, quantiles, ranks, signs and vector quantiles and ranks, and show their consistency by establishing a uniform convergence property for empirical (forward and reverse) transport maps, which is the main theoretical result of this paper.

Article information

Source
Ann. Statist., Volume 45, Number 1 (2017), 223-256.

Dates
Received: January 2015
Revised: February 2016
First available in Project Euclid: 21 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.aos/1487667622

Digital Object Identifier
doi:10.1214/16-AOS1450

Mathematical Reviews number (MathSciNet)
MR3611491

Zentralblatt MATH identifier
06710510

Subjects
Primary: 62M15: Spectral analysis 62G35: Robustness

Keywords
Statistical depth vector quantiles vector ranks multivariate signs empirical transport maps uniform convergence of empirical transport

Citation

Chernozhukov, Victor; Galichon, Alfred; Hallin, Marc; Henry, Marc. Monge–Kantorovich depth, quantiles, ranks and signs. Ann. Statist. 45 (2017), no. 1, 223--256. doi:10.1214/16-AOS1450. https://projecteuclid.org/euclid.aos/1487667622


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References

  • [1] Agostinelli, C. and Romanazzi, M. (2011). Local depth. J. Statist. Plann. Inference 141 817–830.
  • [2] Aurenhammer, F., Hoffmann, F. and Aronov, B. (1998). Minkowski-type theorems and mean-square clustering. Algorithmica 20 61–76.
  • [3] Benamou, J.-D. and Brenier, Y. (2000). A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem. Numer. Math. 84 375–393.
  • [4] Brenier, Y. (1991). Polar factorization and monotone rearrangement of vector-valued functions. Comm. Pure Appl. Math. 44 375–417.
  • [5] Carlier, G., Chernozhukov, V. and Galichon, A. (2016). Vector quantile regression: An optimal transport approach. Ann. Statist. 44 1165–1192.
  • [6] Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc. 91 862–872.
  • [7] Chen, Y., Dang, X., Peng, H. and Bart, H. L. J. (2009). Outlier detection with the kernelized spatial depth function. IEEE Trans. Pattern Anal. Mach. Intell. 31 288–305.
  • [8] Chernozhukov, V., Galichon, A., Hallin, M. and Henry, M. (2016). Supplement to “Monge–Kantorovich depth, quantiles, ranks and signs.” DOI:10.1214/16-AOS1450SUPP.
  • [9] Cuesta-Albertos, J. A. and Nieto-Reyes, A. (2008). The random Tukey depth. Comput. Statist. Data Anal. 52 4979–4988.
  • [10] Decurninge, A. (2014). Multivariate quantiles and multivariate $L$-moments. Preprint. Available at arXiv:1409.6013.
    arXiv: 1409.6013
  • [11] Deneen, L. and Shute, G. (1988). Polygonizations of point sets in the plane. Discrete Comput. Geom. 3 77–87.
  • [12] Doksum, K. (1974). Empirical probability plots and statistical inference for nonlinear models in the two-sample case. Ann. Statist. 2 267–277.
  • [13] Doksum, K. A. and Sievers, G. L. (1976). Plotting with confidence: Graphical comparisons of two populations. Biometrika 63 421–434.
  • [14] Donoho, D. L. (1982). Breakdown properties of multivariate location estimators. Qualifying paper, Harvard Univ.
  • [15] Donoho, D. L. and Gasko, M. (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Statist. 20 1803–1827.
  • [16] Dutta, S., Ghosh, A. K. and Chaudhuri, P. (2011). Some intriguing properties of Tukey’s halfspace depth. Bernoulli 17 1420–1434.
  • [17] Edelsbrunner, H., Kirkpatrick, D. G. and Seidel, R. (1983). On the shape of a set of points in the plane. IEEE Trans. Inform. Theory 29 551–559.
  • [18] Ekeland, I., Galichon, A. and Henry, M. (2012). Comonotonic measures of multivariate risks. Math. Finance 22 109–132.
  • [19] Galichon, A. and Henry, M. (2012). Dual theory of choice with multivariate risks. J. Econom. Theory 147 1501–1516.
  • [20] Ghosh, A. K. and Chaudhuri, P. (2005). On maximum depth and related classifiers. Scand. J. Statist. 32 327–350.
  • [21] Grünbaum, B. (1994). Hamiltonian polygons and polyhedra. Geombinatorics 3 83–89.
  • [22] Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests. Academic Press, New York.
  • [23] Hallin, M. and Paindaveine, D. (2002). Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks. Ann. Statist. 30 1103–1133.
  • [24] Hallin, M. and Paindaveine, D. (2004). Rank-based optimal tests of the adequacy of an elliptic VARMA model. Ann. Statist. 32 2642–2678.
  • [25] Hallin, M. and Paindaveine, D. (2005). Affine-invariant aligned rank tests for the multivariate general linear model with VARMA errors. J. Multivariate Anal. 93 122–163.
  • [26] Hallin, M. and Paindaveine, D. (2006). Semiparametrically efficient rank-based inference for shape. I. Optimal rank-based tests for sphericity. Ann. Statist. 34 2707–2756.
  • [27] Hallin, M. and Paindaveine, D. (2008). Optimal rank-based tests for homogeneity of scatter. Ann. Statist. 36 1261–1298.
  • [28] Hallin, M., Paindaveine, D. and Šiman, M. (2010). Multivariate quantiles and multiple-output regression quantiles: From $L_{1}$ optimization to halfspace depth. Ann. Statist. 38 635–669.
  • [29] Hallin, M. and Werker, B. J. M. (2003). Semi-parametric efficiency, distribution-freeness and invariance. Bernoulli 9 137–165.
  • [30] Hassairi, A. and Regaieg, O. (2008). On the Tukey depth of a continuous probability distribution. Statist. Probab. Lett. 78 2308–2313.
  • [31] Hlubinka, D., Kotík, L. and Vencálek, O. (2010). Weighted halfspace depth. Kybernetika (Prague) 46 125–148.
  • [32] Judd, K. L. (1998). Numerical Methods in Economics. MIT Press, Cambridge, MA.
  • [33] Koenker, R. and Bassett, G. Jr. (1978). Regression quantiles. Econometrica 46 33–50.
  • [34] Koltchinskii, V. and Dudley, R. (1992). On spatial quantiles. Unpublished manuscript.
  • [35] Koshevoy, G. and Mosler, K. (1997). Zonoid trimming for multivariate distributions. Ann. Statist. 25 1998–2017.
  • [36] Koshevoy, G. A. (2002). The Tukey depth characterizes the atomic measure. J. Multivariate Anal. 83 360–364.
  • [37] Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist. 18 405–414.
  • [38] Liu, R. Y. (1992). Data depth and multivariate rank tests. In $L_{1}$-Statistical Analysis and Related Methods (Neuchâtel, 1992) (Y. Dodge, ed.) 279–294. North-Holland, Amsterdam.
  • [39] Liu, R. Y., Parelius, J. M. and Singh, K. (1999). Multivariate analysis by data depth: Descriptive statistics, graphics and inference. Ann. Statist. 27 783–858.
  • [40] Liu, R. Y. and Singh, K. (1993). A quality index based on data depth and multivariate rank tests. J. Amer. Statist. Assoc. 88 252–260.
  • [41] Mahalanobis, P. C. (1936). On the generalized distance in statistics. Proc. Natl. Acad. Sci., India 12 49–55.
  • [42] McCann, R. J. (1995). Existence and uniqueness of monotone measure-preserving maps. Duke Math. J. 80 309–323.
  • [43] Mosler, K. (2002). Multivariate Dispersion, Central Regions and Depth: The Lift Zonoid Approach. Lecture Notes in Statistics 165. Springer, Berlin.
  • [44] Möttönen, J. and Oja, H. (1995). Multivariate sign and rank methods. J. Nonparametr. Stat. 5 201–213.
  • [45] Oja, H. (1983). Descriptive statistics for multivariate distributions. Statist. Probab. Lett. 1 327–332.
  • [46] Paindaveine, D. and Šiman, M. (2012). Computing multiple-output regression quantile regions. Comput. Statist. Data Anal. 56 840–853.
  • [47] Paindaveine, D. and van Bever, G. (2013). From depth to local depth. J. Amer. Statist. Assoc. 108 1105–1119.
  • [48] Politis, D., Romano, J. P. and Wolf, M. (1999). Weak convergence of dependent empirical measures with application to subsampling in function spaces. J. Statist. Plann. Inference 79 179–190.
  • [49] Rockafellar, R. T. (1997). Convex Analysis. Princeton Landmarks in Mathematics. Princeton Univ. Press, Princeton, NJ.
    Mathematical Reviews (MathSciNet): MR1451876
    Zentralblatt MATH: 0932.90001
  • [50] Rockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 317. Springer, Berlin.
  • [51] Serfling, R. (2002). Quantile functions for multivariate analysis: Approaches and applications. Stat. Neerl. 56 214–232.
  • [52] Singh, K. (1991). Majority depth. Unpublished manuscript.
  • [53] Stahel, W. (1981). Robuste Schätzungen: Infinitesimale Optimalität und Schätzungen von Kovarianzmatrizen. Ph.D. thesis, Univ. Zürich.
  • [54] Tukey, J. W. (1975). Mathematics and the picturing of data. In Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2 523–531. Canad. Math. Congress, Montreal, Que.
  • [55] van Zwet, W. R. (1964). Convex Transformations of Random Variables. Mathematisch Centrum, Amsterdam.
  • [56] van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence. Springer, New York.
  • [57] Vardi, Y. and Zhang, C.-H. (2000). The multivariate $L_{1}$-median and associated data depth. Proc. Natl. Acad. Sci. USA 97 1423–1426 (electronic).
  • [58] Villani, C. (2003). Topics in Optimal Transportation. Graduate Studies in Mathematics 58. Amer. Math. Soc., Providence, RI.
  • [59] Villani, C. (2008). Stability of a 4th-order curvature condition arising in optimal transport theory. J. Funct. Anal. 255 2683–2708.
  • [60] Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin.
  • [61] Zuo, Y. (2003). Projection-based depth functions and associated medians. Ann. Statist. 31 1460–1490.
  • [62] Zuo, Y. and Serfling, R. (2000). General notions of statistical depth function. Ann. Statist. 28 461–482.

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