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Halfspace depths for scatter, concentration and shape matrices. (English) Zbl 1408.62100

Summary: We propose halfspace depth concepts for scatter, concentration and shape matrices. For scatter matrices, our concept is similar to those from M. Chen et al. [Ann. Stat. 46, No. 5, 1932–1960 (2018; Zbl 1408.62104)] and J. Zhang [J. Multivariate Anal. 82, No. 1, 134–165 (2002; Zbl 1010.62058)]. Rather than focusing, as in these earlier works, on deepest scatter matrices, we thoroughly investigate the properties of the proposed depth and of the corresponding depth regions. We do so under minimal assumptions and, in particular, we do not restrict to elliptical distributions nor to absolutely continuous distributions. Interestingly, fully understanding scatter halfspace depth requires considering different geometries/topologies on the space of scatter matrices. We also discuss, in the spirit of Y. Zuo and R. Serfling [Ann. Stat. 28, No. 2, 461–482 (2000; Zbl 1106.62334)], the structural properties a scatter depth should satisfy, and investigate whether or not these are met by scatter halfspace depth. Companion concepts of depth for concentration matrices and shape matrices are also proposed and studied. We show the practical relevance of the depth concepts considered in a real-data example from finance.

MSC:

62H05 Characterization and structure theory for multivariate probability distributions; copulas
62G35 Nonparametric robustness
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References:

[1] Arcones, M. A. and Giné, E. (1993). Limit theorems for \(U\)-processes. Ann. Probab.21 1494–1542. · Zbl 0789.60031 · doi:10.1214/aop/1176989128
[2] Berger, M. (2003). A Panoramic View of Riemannian Geometry. Springer, Berlin. · Zbl 1038.53002
[3] Bhatia, R. (2007). Positive Definite Matrices. Princeton Univ. Press, Princeton, NJ. · Zbl 1133.15017
[4] Bhatia, R. and Holbrook, J. (2006). Riemannian geometry and matrix geometric means. Linear Algebra Appl.413 594–618. · Zbl 1088.15022 · doi:10.1016/j.laa.2005.08.025
[5] Cardot, H., Cénac, P. and Godichon-Baggioni, A. (2017). Online estimation of the geometric median in Hilbert spaces: Nonasymptotic confidence balls. Ann. Statist.45 591–614. · Zbl 1371.62027 · doi:10.1214/16-AOS1460
[6] Cartan, E. (1929). Groupes simples clos et ouverts et géometrie riemannienne. J. Math. Pures Appl.8 1–33.
[7] Chakraborty, A. and Chaudhuri, P. (2014). The spatial distribution in infinite dimensional spaces and related quantiles and depths. Ann. Statist.42 1203–1231. · Zbl 1305.62141 · doi:10.1214/14-AOS1226
[8] Chaudhuri, P. (1996). On a geometric notion of quantiles for multivariate data. J. Amer. Statist. Assoc.91 862–872. · Zbl 0869.62040 · doi:10.1080/01621459.1996.10476954
[9] Chen, M., Gao, C. and Ren, Z. (2018). Robust covariance and scatter matrix estimation under Huber’s contamination model. Ann. Statist. To appear. · Zbl 1408.62104
[10] Claeskens, G., Hubert, M., Slaets, L. and Vakili, K. (2014). Multivariate functional halfspace depth. J. Amer. Statist. Assoc.109 411–423. · Zbl 1367.62162 · doi:10.1080/01621459.2013.856795
[11] Cuevas, A., Febrero, M. and Fraiman, R. (2007). Robust estimation and classification for functional data via projection-based depth notions. Comput. Statist.22 481–496. · Zbl 1195.62032 · doi:10.1007/s00180-007-0053-0
[12] Dang, X. and Serfling, R. J. (2010). Nonparametric depth-based multivariate outlier identifiers, and masking robustness properties. J. Statist. Plann. Inference140 198–213. · Zbl 1191.62084 · doi:10.1016/j.jspi.2009.07.004
[13] Donoho, D. L. and Gasko, M. (1992). Breakdown properties of location estimates based on halfspace depth and projected outlyingness. Ann. Statist.20 1803–1827. · Zbl 0776.62031 · doi:10.1214/aos/1176348890
[14] Dümbgen, L. and Tyler, D. E. (2016). Geodesic convexity and regularized scatter estimators. Available at arXiv:1607.05455v2.
[15] Fan, Y., Jin, J. and Yao, Z. (2013). Optimal classification in sparse Gaussian graphic model. Ann. Statist.41 2537–2571. · Zbl 1294.62061 · doi:10.1214/13-AOS1163
[16] Fan, Y. and Lv, J. (2016). Innovated scalable efficient estimation in ultra-large Gaussian graphical models. Ann. Statist.44 2098–2126. · Zbl 1349.62206 · doi:10.1214/15-AOS1416
[17] Hall, P. and Jin, J. (2010). Innovated higher criticism for detecting sparse signals in correlated noise. Ann. Statist.38 1686–1732. · Zbl 1189.62080 · doi:10.1214/09-AOS764
[18] 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. · Zbl 1183.62088 · doi:10.1214/09-AOS723
[19] He, Y. and Einmahl, J. H. J. (2017). Estimation of extreme depth-based quantile regions. J. R. Stat. Soc. Ser. B. Stat. Methodol.79 449–461. · Zbl 1414.62161
[20] Hubert, M., Rousseeuw, P. J. and Segaert, P. (2015). Multivariate functional outlier detection. Stat. Methods Appl.24 177–202. · Zbl 1441.62124 · doi:10.1007/s10260-015-0297-8
[21] Ilmonen, P. and Paindaveine, D. (2011). Semiparametrically efficient inference based on signed ranks in symmetric independent component models. Ann. Statist.39 2448–2476. · Zbl 1231.62043 · doi:10.1214/11-AOS906
[22] Liu, R. Y. (1990). On a notion of data depth based on random simplices. Ann. Statist.18 405–414. · Zbl 0701.62063 · doi:10.1214/aos/1176347507
[23] 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. · Zbl 0984.62037
[24] López-Pintado, S. and Romo, J. (2009). On the concept of depth for functional data. J. Amer. Statist. Assoc.104 718–734. · Zbl 1388.62139 · doi:10.1198/jasa.2009.0108
[25] Mizera, I. (2002). On depth and deep points: A calculus. Ann. Statist.30 1681–1736. · Zbl 1039.62046 · doi:10.1214/aos/1043351254
[26] Mizera, I. and Müller, C. H. (2004). Location-scale depth. J. Amer. Statist. Assoc.99 949–989. · Zbl 1071.62032 · doi:10.1198/016214504000001312
[27] Nieto-Reyes, A. and Battey, H. (2016). A topologically valid definition of depth for functional data. Statist. Sci.31 61–79. · Zbl 1436.62720
[28] Paindaveine, D. and Van Bever, G. (2014). Inference on the shape of elliptical distributions based on the MCD. J. Multivariate Anal.129 125–144. · Zbl 1360.62280 · doi:10.1016/j.jmva.2014.04.013
[29] Paindaveine, D. and Van Bever, G. (2015). Nonparametrically consistent depth-based classifiers. Bernoulli21 62–82. · Zbl 1359.62258 · doi:10.3150/13-BEJ561
[30] Paindaveine, D. and Van Bever, G. (2018). Supplement to “Halfspace depths for scatter, concentration and shape matrices.” DOI:10.1214/17-AOS1658SUPP. · Zbl 1408.62100
[31] Rousseeuw, P. J. and Hubert, M. (1999). Regression depth. J. Amer. Statist. Assoc.94 388–433. · Zbl 1007.62060 · doi:10.1080/01621459.1999.10474129
[32] Rousseeuw, P. J. and Ruts, I. (1999). The depth function of a population distribution. Metrika49 213–244. · Zbl 1093.62540
[33] Rousseeuw, P. J. and Struyf, A. (2004). Characterizing angular symmetry and regression symmetry. J. Statist. Plann. Inference122 161–173. · Zbl 1040.62041 · doi:10.1016/j.jspi.2003.06.015
[34] Serfling, R. J. (2004). Some perspectives on location and scale depth functions. J. Amer. Statist. Assoc.99 970–973.
[35] Serfling, R. (2010). Equivariance and invariance properties of multivariate quantile and related functions, and the role of standardisation. J. Nonparametr. Stat.22 915–936. · Zbl 1203.62103 · doi:10.1080/10485250903431710
[36] 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. · Zbl 0347.62002
[37] Vardi, Y. and Zhang, C.-H. (2000). The multivariate \(L_{1}\)-median and associated data depth. Proc. Natl. Acad. Sci. USA97 1423–1426. · Zbl 1054.62067 · doi:10.1073/pnas.97.4.1423
[38] Zhang, J. (2002). Some extensions of Tukey’s depth function. J. Multivariate Anal.82 134–165. · Zbl 1010.62058
[39] Zuo, Y. (2003). Projection-based depth functions and associated medians. Ann. Statist.31 1460–1490. · Zbl 1046.62056 · doi:10.1214/aos/1065705115
[40] Zuo, Y. and Serfling, R. (2000). General notions of statistical depth function. Ann. Statist.28 461–482. · Zbl 1106.62334 · doi:10.1214/aos/1016218226
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