Applications of Mathematics, Vol. 65, No. 3, pp. 287-298, 2020
Scatter halfspace depth: Geometric insights
Stanislav Nagy
Received November 30, 2019. Published online May 25, 2020.
Abstract: Scatter halfspace depth is a statistical tool that allows one to quantify the fitness of a candidate covariance matrix with respect to the scatter structure of a probability distribution. The depth enables simultaneous robust estimation of location and scatter, and nonparametric inference on these. A handful of remarks on the definition and the properties of the scatter halfspace depth are provided. It is argued that the currently used notion of this depth is well suited especially for symmetric random vectors. The scatter halfspace depth closely relates to an appropriate distance of matrix-generated ellipsoids from an upper level set of the (location) halfspace depth function. Several modifications and extensions to the scatter halfspace depth are considered, with their theoretical properties outlined.
References: [1] M. Chen, C. Gao, Z. Ren: Robust covariance and scatter matrix estimation under Huber's contamination model. Ann. Stat. 46 (2018), 1932-1960. DOI 10.1214/17-AOS1607 | MR 3845006 | Zbl 1408.62104
[2] C. Dupin: Applications de géométrie et de méchanique à la marine, aux ponts et chaussées, etc. pour faire suite aux développements de géométrie. Bachelier, Paris (1822). (In French.)
[3] K.-T. Fang, S. Kotz, K.-W. Ng: Symmetric Multivariate and Related Distributions. Monographs on Statistics and Applied Probability 36, Chapman and Hall, London (1990). DOI 10.1201/9781351077040 | MR 1071174 | Zbl 0699.62048
[4] S. Helgason: Integral Geometry and Radon Transforms. Springer, New York (2011). DOI 10.1007/978-1-4419-6055-9 | MR 2743116 | Zbl 1210.53002
[5] M. Meyer, C. Schütt, E. M. Werner: Affine invariant points. Isr. J. Math. 208 (2015), 163-192. DOI 10.1007/s11856-015-1196-2 | MR 3416917 | Zbl 1343.52003
[6] S. Nagy: Scatter halfspace depth for $K$-symmetric distributions. Stat. Probab. Lett. 149 (2019), 171-177. DOI 10.1016/j.spl.2019.02.006 | MR 3921058 | Zbl 1427.62041
[7] S. Nagy: The halfspace depth characterization problem. To appear in Springer Proc. Math. Stat. (2020).
[8] S. Nagy, C. Schütt, E. M. Werner: Halfspace depth and floating body. Stat. Surv. 13 (2019), 52-118. DOI 10.1214/19-ss123 | MR 3973130 | Zbl 1428.62204
[9] D. Paindaveine, G. Van Bever: Halfspace depths for scatter, concentration and shape matrices. Ann. Stat. 46 (2018), 3276-3307. DOI 10.1214/17-AOS1658 | MR 3852652 | Zbl 1408.62100
[10] E. T. Quinto: Singular value decompositions and inversion methods for the exterior Radon transform and a spherical transform. J. Math. Anal. Appl. 95 (1983), 437-448. DOI 10.1016/0022-247X(83)90118-X | MR 716094 | Zbl 0569.44005
[11] R. Schneider: Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications 151, Cambridge University Press, Cambridge (2014). DOI 10.1017/CBO9781139003858 | MR 3155183 | Zbl 1287.52001
[12] J. W. Tukey: Mathematics and the picturing of data. Proceedings of the International Congress of Mathematicians. Vol. 2. Canadian Mathematical Society, Vancouver, 1974, 523-531. MR 0426989 | Zbl 0347.62002
Affiliations: Stanislav Nagy, Charles University, Faculty of Mathematics and Physics, Department of Probability and Mathematical Statistics, Sokolovská 83, 186 75 Praha 8, Czech Republic, e-mail: nagy@karlin.mff.cuni.cz
